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\begin{document}

\title{Weil-\'{e}tale cohomology and zeta-values of arithmetic schemes at negative integers}
\titlemark{Weil-\'{e}tale cohomology and zeta-values of arithmetic schemes at negative integers}

\emsauthor{1}{
    \givenname{Alexey}
    \surname{Beshenov}
    \mrid{1113833}}{A.~Beshenov}

\Emsaffil{1}{
    \address{Cosmos 2674}
    \zip{44520}
    \city{Guadalajara}
    \country{Mexico}
    \affemail{cadadr@gmail.com}}

\classification[14F42,14F20]{19F27}
\keywords{Weil-\'{e}tale cohomology, motivic cohomology, special values of zeta-functions}

\begin{abstract}
    Following the ideas of Flach and Morin (2018), we state a
    conjecture in terms of Weil-\'{e}tale cohomology for the vanishing order
    and special value of the zeta function $\zeta (X,s)$ at $s = n < 0$,
    where $X$ is a separated scheme of finite type over $\Spec \ZZ$.
    We prove that the conjecture is compatible with closed-open decompositions
    of schemes and with affine bundles, and consequently,
    that it holds for cellular schemes over certain one-dimensional bases.
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

Let $X$ be an \textbf{arithmetic scheme}, by which we mean in this paper that
it is a separated scheme of finite type $X \to \Spec \ZZ$. Then the
corresponding \textbf{zeta function} is defined by
\begin{equation}
    \label{eqn:Euler-product-for-zeta}
    \zeta (X,s) = \prod_{\substack{x \in X \\ \text{closed pt.}}}
    \frac{1}{1 - N (x)^{-s}}.
\end{equation}
Here, for a closed point $x \in X$, the norm
\begin{equation*}
    N (x) = |\kappa (x)| = |\mathcal{O}_{X,x}/\mathfrak{m}_{X,x}|
\end{equation*}
is the size of the corresponding residue field. The product converges for
$\Re s > \dim X$, and conjecturally admits a meromorphic continuation to the
whole complex plane. Basic facts and conjectures about zeta functions of schemes
can be found in \cite{Serre-1965}.

Of particular interest are the so-called special values of $\zeta (X,s)$ at
integers $s = n \in \ZZ$, also known as the \textbf{zeta-values} of $X$. To
define these, we assume that $\zeta (X,s)$ admits a meromorphic continuation
around $s = n$. We denote by
\begin{equation*}
    d_n = \ord_{s=n} \zeta (X,s)
\end{equation*}
the \textbf{vanishing order} of $\zeta (X,s)$ at $s = n$. That is, $d_n > 0$
(resp. $d_n < 0$) if $\zeta (X,s)$ has a zero (resp. pole) of order $d_n$
at $s = n$.

The \textbf{special value} of $\zeta (X,s)$ at $s = n$ is defined as the
leading nonzero coefficient of the Taylor expansion:
\begin{equation*}
    \zeta^* (X,n) = \lim_{s
        \to n} (s - n)^{-d_n}\,\zeta (X,s).
\end{equation*}

Early on, Lichtenbaum conjectured that both numbers $\ord_{s = n} \zeta (X,s)$
and $\zeta^* (X,n)$ should have a cohomological interpretation related to the
\'{e}tale motivic cohomology of $X$ (see e.g. \cite{Lichtenbaum-1984} for
varieties over finite fields).

This is made precise in Lichtenbaum's Weil-\'{e}tale program. It suggests the
existence of \textbf{Weil-\'{e}tale cohomology}, which is a suitable
modification of motivic cohomology that encodes the information about
the vanishing order and the special value of $\zeta (X,s)$ at $s = n$.
For Lichtenbaum's recent work on this topic, we refer the reader to
\cite{Lichtenbaum-2005,Lichtenbaum-2009-Euler-char,Lichtenbaum-2009-number-rings,Lichtenbaum-2021}.

The case of varieties over finite fields $X/\FF_q$ is now well understood
thanks to the work of Geisser
\cite{Geisser-2004,Geisser-2006,Geisser-2010-arithmetic-homology}.

Flach and Morin considered the case of proper, regular arithmetic schemes $X$.
In \cite{Flach-Morin-2012} they have studied the corresponding Weil-\'{e}tale
topos. Later, in \cite{Morin-2014} Morin gave an explicit construction of
Weil-\'{e}tale cohomology groups $H^i_\Wc (X, \ZZ)$ for a proper and regular
arithmetic scheme $X$. This construction was further generalized by Flach and
Morin in \cite{Flach-Morin-2018} to groups $H^i_\Wc (X, \ZZ(n))$ with weights
$n \in \ZZ$, again for a proper and regular $X$.

Motivated by the work of Flach and Morin, the author constructed in
\cite{Beshenov-Weil-etale-1} Weil-\'{e}tale cohomology groups
$H^i_\Wc (X, \ZZ (n))$ for any arithmetic scheme $X$
(removing the assumption that $X$ is proper or regular) and
strictly negative weights $n < 0$. The construction is based
on the following assumption.

\begin{conjecture}
    $\mathbf{L}^c (X_\et,n)$: given an arithmetic scheme $X$ and $n < 0$,
    the cohomology groups $H^i (X_\et, \ZZ^c (n))$ are finitely generated
    for all $i \in \ZZ$.
\end{conjecture}

For the known cases, see \cite[\S 8]{Beshenov-Weil-etale-1}. Under this
conjecture, we constructed in \cite[\S 7]{Beshenov-Weil-etale-1} perfect
complexes of abelian groups $R\Gamma_\Wc (X, \ZZ(n))$ and the corresponding
cohomology groups
\begin{equation*}
    H^i_\Wc (X, \ZZ(n)) \dfn H^i (R\Gamma_\Wc (X, \ZZ(n))).
\end{equation*}

This text investigates the conjectural relation of our Weil-\'{e}tale
cohomology to the special value of $\zeta (X,s)$ at $s = n < 0$. Specifically,
we make the following conjectures.

\begin{enumerate}
    \item[1)] $\mathbf{VO} (X,n)$ (Conjecture~\ref{conjecture:VO(X,n)}):
          \emph{The vanishing order is given by the weighted alternating sum of ranks}
          \begin{equation*}
              \ord_{s=n} \zeta (X,s) =
              \sum_{i\in \ZZ} (-1)^i \cdot i \cdot \rk_\ZZ H_\Wc^i (X, \ZZ(n)).
          \end{equation*}

    \item[2)] A consequence of $\mathbf{B} (X,n)$ (Conjecture~\ref{conjecture:B(X,n)})
          and Lemma~\ref{lemma:smile-theta}:
          \emph{After tensoring the cohomology groups $H_\Wc^i (X, \ZZ (n))$
            with $\RR$, we obtain a long exact sequence of
            finite dimensional real vector spaces}
          \begin{equation*}
              \cdots \to H_\Wc^{i-1} (X, \RR (n)) \xrightarrow{\smile\theta}
              H_\Wc^i (X, \RR (n)) \xrightarrow{\smile\theta}
              H_\Wc^{i+1} (X, \RR (n)) \to \cdots
          \end{equation*}

          It follows that there is a canonical isomorphism
          \begin{equation*}
              \lambda\colon \RR \xrightarrow{\cong}
              (\det_\ZZ R\Gamma_\Wc (X, \ZZ (n))) \otimes \RR.
          \end{equation*}
          Here $\det_\ZZ R\Gamma_\Wc (X, \ZZ (n))$ is the determinant of the
          perfect complex of abelian groups $R\Gamma_\Wc (X, \ZZ (n))$,
          in the sense of Knudsen and Mumford \cite{Knudsen-Mumford-1976}.
          In particular, $\det_\ZZ R\Gamma_\Wc (X, \ZZ (n))$ is a free
          $\ZZ$-module of rank $1$. For the convenience of the reader,
          we give a brief overview of determinants in \S\ref{app:determinants}.

    \item[3)] $\mathbf{C} (X,n)$ (Conjecture~\ref{conjecture:C(X,n)}):
          \emph{The special value is determined up to sign by}
          \begin{equation*}
              \lambda (\zeta^* (X, n)^{-1}) \cdot \ZZ =
                \det_\ZZ R\Gamma_\Wc (X, \ZZ (n)).
          \end{equation*}
\end{enumerate}

If $X$ is proper and regular, then our construction of
$R\Gamma_\Wc (X, \ZZ (n))$ and the above conjectures agree with those of
Flach and Morin from \cite{Flach-Morin-2018}. Apart from removing the assumption
that $X$ is proper and regular, a novelty of this work is that we prove
the compatibility of the conjectures with operations on schemes, in particular
with closed-open decompositions $Z \not\hookrightarrow X \hookleftarrow U$,
where $Z \subset X$ is a closed subscheme and $U = X\setminus Z$
is the open complement, and with affine bundles $\AA_X^r = \AA_\ZZ^r \times X$
(see Proposition~\ref{prop:compatibility-of-VO(X,n)} and
Theorem~\ref{thm:compatibility-of-C(X,n)}). This gives a machinery for starting
from the cases of schemes for which the conjectures are known and constructing
new schemes for which the conjectures also hold. As an application, we prove in
\S\ref{sec:unconditional-results} the following result
(Theorem~\ref{thm:main-result}).

\begin{maintheorem*}
    Let $B$ be a one-dimensional arithmetic scheme, such that each of
    the generic points $\eta \in B$ satisfies one of the following properties:
    \begin{enumerate}
        \item[a)] $\fchar \kappa (\eta) = p > 0$;

        \item[b)] $\fchar \kappa (\eta) = 0$, and $\kappa (\eta)/\QQ$ is an abelian number
              field.
    \end{enumerate}
    If $X$ is a $B$-cellular arithmetic scheme with smooth quasi-projective
    fiber $X_{\red,\CC}$, then Conjectures~$\mathbf{VO} (X,n)$ and
    $\mathbf{C} (X,n)$ hold unconditionally for any $n < 0$.
\end{maintheorem*}

In fact, this result is established for a larger class of arithmetic schemes
$\mathcal{C} (\ZZ)$; we refer to \S\ref{sec:unconditional-results} for more
details.

\subsection*{Outline of the paper}

In \S\ref{sec:regulator} we define the regulator morphism, based on the
construction of Kerr, Lewis, and M\"{u}ller-Stach
\cite{Kerr-Lewis-Muller-Stach-2006}, and state the associated
Conjecture~$\mathbf{B} (X,n)$.

Then \S\ref{sec:vanishing-order-conjecture} is devoted to
Conjecture~$\mathbf{VO} (X,n)$ about the vanishing order. We also explain why
it is consistent with a conjecture of Soul\'{e}, and with the vanishing order
arising from the expected functional equation.

In \S\ref{sec:special-value-conjecture} we state Conjecture~$\mathbf{C} (X,n)$
about the special value.

We explain in \S\ref{sec:finite-fields} that if $X$ is a variety over a finite
field, then Conjecture~$\mathbf{C} (X,n)$ is consistent with the conjectures
considered by Geisser in
\cite{Geisser-2004,Geisser-2006,Geisser-2010-arithmetic-homology}, and it
follows from Conjecture~$\mathbf{L}^c (X_\et,n)$.

Then we prove in \S\ref{sec:compatibility-with-operations} that Conjectures
$\mathbf{VO} (X,n)$ and $\mathbf{C} (X,n)$ are compatible with basic operations
on schemes: disjoint unions, closed-open decompositions, and affine bundles.
Using these results, we conclude in \S\ref{sec:unconditional-results} with a
class of schemes for which the conjectures hold unconditionally.

For the convenience of the reader, Appendix~\ref{app:determinants} gives a
brief overview of basic definitions and facts related to the determinants of
complexes.

\subsection*{Notation}

In this paper, $X$ always denotes an \textbf{arithmetic scheme} (separated, of
finite type over $\Spec \ZZ$), and $n$ is always a strictly negative integer.

We denote by
\begin{equation*}
    R\Gamma_\fg (X, \ZZ (n)) \quad\text{and}\quad R\Gamma_\Wc (X, \ZZ (n))
\end{equation*}
the complexes of abelian groups constructed in \cite{Beshenov-Weil-etale-1}
under Conjecture~$\mathbf{L}^c (X_\et,n)$. We set
\begin{align*}
    H^i_\fg (X, \ZZ (n)) & \dfn H^i (R\Gamma_\fg (X, \ZZ (n))), \\
    H^i_\Wc (X, \ZZ (n)) & \dfn H^i (R\Gamma_\Wc (X, \ZZ (n))).
\end{align*}

By \cite[Proposition 5.5 and 7.12]{Beshenov-Weil-etale-1}, these cohomology
groups are finitely generated, assuming Conjecture~$\mathbf{L}^c (X_\et,n)$;
moreover, the groups $H^i_\Wc (X, \ZZ(n))$ are bounded, and $H^i_\fg (X, \ZZ
        (n))$ are bounded from below and finite $2$-torsion for $i \gg 0$.

Briefly, the construction fits in the following diagram of distinguished
triangles in the derived category $\mathbf{D} (\ZZ)$:
\begin{equation*}
    \begin{tikzcd}[column sep=1.5em]
        &[-3em] \RHom (R\Gamma (X_\et, \ZZ^c (n)), \QQ [-2]) \ar{d}{\alpha_{X,n}} \ar{r} &[-2.5em] 0 \ar{d} \\
        & R\Gamma_c (X_\et, \ZZ(n)) \ar{d}\ar{r}{u_\infty^*} & R\Gamma_c (G_\RR, X (\CC), \ZZ(n))\ar{d}{id} \\
        R\Gamma_\Wc (X, \ZZ (n)) \ar{r} & R\Gamma_\fg (X, \ZZ(n)) \ar[dashed]{r}{i_\infty^*}\ar{d} & R\Gamma_c (G_\RR, X (\CC), \ZZ(n)) \ar{r} \ar{d} & {[1]} \\
        & \RHom (R\Gamma (X_\et, \ZZ^c (n)), \QQ [-1]) \ar{r} & 0
    \end{tikzcd}
\end{equation*}
For more details, see \cite{Beshenov-Weil-etale-1}. For real coefficients,
we set
\begin{align*}
    R\Gamma_\fg (X, \RR (n)) & \dfn R\Gamma_\fg (X, \ZZ (n)) \otimes \RR, \\
    R\Gamma_\Wc (X, \RR (n)) & \dfn R\Gamma_\Wc (X, \ZZ (n)) \otimes \RR.
\end{align*}
Accordingly,
\begin{align*}
    H^i_\fg (X, \RR (n)) & \dfn H^i (R\Gamma_\fg (X, \RR (n))) = H^i_\fg (X, \ZZ (n)) \otimes \RR, \\
    H^i_\Wc (X, \RR (n)) & \dfn H^i (R\Gamma_\Wc (X, \RR (n))) = H^i_\Wc (X, \ZZ (n)) \otimes \RR.
\end{align*}

By $X (\CC)$ we denote the space of complex points of $X$ with the usual
analytic topology. It carries a natural action of $G_\RR = \Gal (\CC/\RR)$ via
the complex conjugation. For a subring $A \subseteq \RR$ we denote by $A (n)$
the $G_\RR$-module $(2\pi i)^n\,A$, and also the corresponding constant
$G_\RR$-equivariant sheaf on $X (\CC)$.

We denote by $R\Gamma_c (X (\CC), A (n))$ the cohomology with compact support
with $A (n)$-coefficients, and its $G_\RR$-equivariant version is defined by
\begin{equation*}
    R\Gamma_c (G_\RR, X (\CC), A (n)) \dfn
        R\Gamma (G_\RR, R\Gamma_c (X (\CC), A (n))).
\end{equation*}
For real coefficients, we have
\begin{equation*}
    H_c^i (G_\RR, X (\CC), \RR (n)) =
    H^i_c (X (\CC), \RR (n))^{G_\RR},
\end{equation*}
where the $G_\RR$-action on $H^i_c (X (\CC), \RR (n))$ naturally comes from
the corresponding action on $X (\CC)$ and $\RR (n)$.

\textbf{Borel--Moore homology} is defined as the dual to cohomology with compact
support. We are interested in the real coefficients:
\begin{align*}
    R\Gamma_\BM (X (\CC), \RR (n))        & \dfn \RHom (R\Gamma_c (X (\CC), \RR (n)), \RR),        \\
    R\Gamma_\BM (G_\RR, X (\CC), \RR (n)) & \dfn \RHom (R\Gamma_c (G_\RR, X (\CC), \RR (n)), \RR).
\end{align*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{ack}
    Parts of this work are based on my doctoral thesis, which I wrote under the
    supervision of Baptiste Morin (Universit\'{e} de Bordeaux) and Bas Edixhoven
    (Universiteit Leiden). I am very grateful to them for their support
    in working on this project. I am also indebted to Matthias Flach,
    as the ideas for this work came from \cite{Flach-Morin-2018}.
    I thank Stephen Lichtenbaum and Niranjan Ramachandran who kindly agreed
    to act as reviewers for my thesis and provided me with useful comments
    and suggestions. Finally, I thank Diosel L\'{o}pez Cruz,
    Cristhian Garay L\'{o}pez, Maxim Mornev, and Carlos Vargas for many
    fruitful mathematical discussions.

    This paper was edited while I visited the Center for Research in Mathematics
    (CIMAT), Guanajuato. I personally thank Pedro Luis del \'{A}ngel and
    Xavier G\'{o}mez Mont for their hospitality.
\end{ack}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Regulator morphism and Conjecture~$\mathbf{B} (X,n)$}
\label{sec:regulator}

In order to formulate the special value conjecture, we need a regulator
morphism from motivic cohomology to Deligne(--Beilinson) (co)homology. Such
regulators were originally introduced by Bloch in \cite{Bloch-1986-Lefschetz},
and here we use the construction of Kerr, Lewis, and M\"{u}ller-Stach
\cite{Kerr-Lewis-Muller-Stach-2006}, which works at the level of complexes.
We will simply call it ``the KLM regulator''. It works under the assumption that
$X_{\red,\CC}$ is a smooth quasi-projective variety.

For simplicity, in this section we assume that $X$ is reduced (motivic
cohomology does not distinguish between $X$ and $X_\red$), and that $X_\CC$ is
connected of dimension $d_\CC$ (otherwise, the arguments below can be applied
to each connected component). We fix a compactification by a normal crossing
divisor
\begin{equation*}
    \begin{tikzcd}
        X_\CC \ar[right hook->]{r}{j} & \overline{X}_\CC & D \ar[left hook->]{l}
    \end{tikzcd}
\end{equation*}
The KLM regulator has the form of a morphism in the derived category
\begin{equation}
    \label{eqn:KLM-morphism-1}
    z^p (X_\CC, -\bullet) \otimes \QQ \to
    {}' C_\mathcal{D}^{2p - 2d_\CC + \bullet} (\overline{X}_\CC, D, \QQ (p-d_\CC)).
\end{equation}

Here $z^p (X_\CC, -\bullet)$ denotes the Bloch's cycle complex
\cite{Bloch-1986}. To define it, consider the algebraic simplex
$\Delta_\CC^i = \Spec \CC [t_0,\ldots,t_i]/(1 - \sum_j t_j)$.
Then, $z^p (X_\CC, i)$ is freely generated by algebraic cycles
$Z \subset X_\CC \times_{\Spec \CC} \Delta_\CC^i$ of codimension $p$
which intersect the faces properly.
It is more convenient for us to work with
\begin{equation*}
    z_{d_\CC - p} (X_\CC, i) = z^p (X_\CC, i),
\end{equation*}
generated by
cycles $Z \subset X_\CC \times_{\Spec \CC} \Delta_\CC^i$ of dimension $p+i$.

The complex ${}' C_\mathcal{D}^{\bullet} (\overline{X}_\CC, D, \QQ (k))$ on the
right-hand side of \eqref{eqn:KLM-morphism-1} computes Deligne(--Beilinson)
homology, as defined by Jannsen \cite{Jannsen-1988}. If we take
$p = d_\CC + 1 - n$, tensor it with $\RR$ and shift it by $2n$, we obtain
\begin{equation}
    \label{eqn:KLM-morphism-2}
    z_{n-1} (X_\CC, -\bullet) \otimes \RR [2n] \to
    {}' C_\mathcal{D}^{2 + \bullet} (\overline{X}_\CC, D, \RR (1-n)).
\end{equation}

\begin{remark}
    Some comments are in order.

    \begin{enumerate}
        \item Originally, the KLM regulator is defined using a cubical version
            of cycle complexes, but these are quasi-isomorphic to the usual
            simplicial cycle complexes by \cite{Levine-1994}, so we make
            no distinction here. For an explicit simplicial version of the
            KLM regulator, see \cite{Kerr-Lewis-Lopatto-2018}.

        \item The KLM regulator is defined as a true morphism of complexes
            (not just a morphism in the derived category) on a subcomplex
            $z^r_\RR (X_\CC, \bullet) \subset z^r (X_\CC, \bullet)$.
            This inclusion becomes a quasi-isomorphism if we pass to rational
            coefficients. In the original paper
            \cite{Kerr-Lewis-Muller-Stach-2006} this is stated without
            tensoring with $\QQ$, but the omission is acknowledged later
            in \cite{Kerr-Lewis-2007}. For our purposes, it suffices to have
            a regulator with coefficients in $\RR$.

        \item The case of a smooth quasi-projective $X_\CC$, where one must
            consider a compactification by a normal crossing divisor as above,
            is treated in \cite[\S 5.9]{Kerr-Lewis-Muller-Stach-2006}.
    \end{enumerate}
\end{remark}

Now we make a small digression to identify the right-hand side of
\eqref{eqn:KLM-morphism-2}. Under our assumption that $n < 0$, Deligne homology
is equivalent to Borel--Moore homology.

\begin{lemma}
    For any $n < 0$ there is a quasi-isomorphism
    \begin{multline*}
        {}' C^\bullet_\mathcal{D} (\overline{X_\CC}, D, \RR (1-n)) \cong
        R\Gamma_\BM (X (\CC), \RR (n)) [-1] \\
        \dfn \RHom (R\Gamma_c (X (\CC), \RR (n)), \RR) [-1].
    \end{multline*}
    Moreover, it respects the natural actions of $G_\RR$ on both complexes.

    \begin{proof}
        From the proof of \cite[Theorem~1.15]{Jannsen-1988}, for any $k \in \ZZ$
        we have a quasi-isomorphism
        \begin{equation}
            \label{eqn:Jannsen-Theorem-1.15}
            {}' C^\bullet_\mathcal{D} (\overline{X_\CC}, D, \RR (k)) \cong
            R\Gamma (\overline{X} (\CC), \RR (k + d_\CC)_{\DB, (\overline{X}_\CC,X_\CC)}) [2d_\CC],
        \end{equation}
        where
        \begin{equation*}
            \RR (k + d_\CC)_{\DB, (\overline{X}_\CC,X_\CC)} =
            \Cone \left(\begin{array}{c}
                R j_* \RR (k + d_\CC) \\
                \oplus                \\
                \Omega^{\geqslant k + d_\CC}_{\overline{X} (\CC)} (\log D)
            \end{array}
            \xrightarrow{\epsilon - \iota}
            R j_* \Omega_{X (\CC)}^\bullet \right) [-1]
        \end{equation*}
        is the sheaf whose hypercohomology on $\overline{X} (\CC)$ gives
        Deligne--Beilinson cohomology (see \cite{Esnault-Viehweg-1988} for more
        details).

        Here $\Omega^\bullet_{\overline{X} (\CC)}$ denotes the usual de Rham
        complex of holomorphic differential forms, and
        $\Omega^\bullet_{\overline{X} (\CC)} (\log D)$ is the complex of forms
        with at most logarithmic poles along $D (\CC)$. The
        latter complex is filtered by subcomplexes $\Omega^{\geqslant
                \bullet}_{\overline{X} (\CC)} (\log D)$. The morphism $\epsilon\colon R j_* \RR
            (k) \to R j_* \Omega^\bullet_{X (\CC)}$ is induced by the canonical morphism of
        sheaves $\RR (k) \to \mathcal{O}_{X (\CC)}$, and $\iota$ is induced by the
        natural inclusion $\Omega^\bullet_{\overline{X} (\CC)} (\log D)
            \xrightarrow{\cong} j_* \Omega_{X (\CC)}^\bullet = R j_* \Omega_{X
                (\CC)}^\bullet$, which is a quasi-isomorphism of filtered complexes.

        We are interested in the case of $k > 0$ when the part $\Omega^{\geqslant k +
                d_\CC}_{\overline{X} (\CC)} (\log D)$ vanishes, and we obtain
        \begin{align}
            \notag \RR (k + d_\CC)_{\DB, (\overline{X}_\CC,X_\CC)} & \cong
            R j_* \Cone \Bigl(\RR (k + d_\CC)
            \xrightarrow{\epsilon}
            \Omega_{X (\CC)}^\bullet \Bigr) [-1]                                                                              \\
            \notag                                                 & \cong R j_* \Cone \Bigl(\RR (k + d_\CC) \xrightarrow{\epsilon}
            \Omega_{X (\CC)}^\bullet [-1] \Bigr)                                                                              \\
            \label{eqn:deligne-homology-1}                         & \cong R j_* \Cone \Bigl(\RR (k + d_\CC) \to \CC [-1] \Bigr)    \\
            \label{eqn:deligne-homology-2}                         & \cong R j_* \RR (k + d_\CC - 1) [-1]
        \end{align}
        Here \eqref{eqn:deligne-homology-1} comes from the Poincar\'{e} lemma
        $\CC \cong \Omega_{X (\CC)}^\bullet$ and \eqref{eqn:deligne-homology-2}
        from the short exact sequence of $G_\RR$-modules
        $\RR (k + d_\CC) \rightarrowtail \CC \twoheadrightarrow \RR (k + d_\CC - 1)$.

        Returning to \eqref{eqn:Jannsen-Theorem-1.15} for $k = 1-n$, we find that
        \begin{align*}
            {}' C^\bullet_\mathcal{D} (\overline{X_\CC}, D, \RR (1-n)) & \cong
            R\Gamma (X (\CC), \RR (d_\CC - n)) [2d_\CC-1]                                                                      \\
                                                                       & \cong \RHom (R\Gamma_c (X (\CC), \RR (n)), \RR) [-1].
        \end{align*}
        Here the final isomorphism is Poincar\'{e} duality.
        All the above is $G_\RR$-equivariant.
    \end{proof}
\end{lemma}

Returning now to \eqref{eqn:KLM-morphism-2}, the previous lemma allows us to
reinterpret the KLM regulator as
\begin{equation}
    \label{eqn:KLM-morphism-3}
    z_{n-1} (X_\CC, -\bullet) \otimes \RR [2n] \to
    R\Gamma_\BM (X (\CC), \RR (n)), \RR) [1].
\end{equation}

We have
\begin{multline}
    \label{eqn:KLM-morphism-4}
    z_{n-1} (X_\CC, -\bullet) \otimes \RR [2n] =
    z_{n-1} (X_\CC, -\bullet) \otimes \RR [2n-2] [2] \\
    = \Gamma (X_{\CC,\et}, \RR^c (n-1)) [2],
\end{multline}
where the complex of sheaves $\RR^c (p)$ is defined by
$U \rightsquigarrow z_p (U, -\bullet) \otimes \RR [2p]$.
By \'{e}tale cohomological descent \cite[Theorem~3.1]{Geisser-2010},
\begin{equation}
    \label{eqn:KLM-morphism-5}
    \Gamma (X_{\CC,\et}, \RR^c (n-1)) \cong R\Gamma (X_{\CC,\et}, \RR^c (n-1)).
\end{equation}
(We note that \cite[Theorem~3.1]{Geisser-2010} holds unconditionally, since the
Beilinson--Lichtenbaum conjecture follows from the Bloch--Kato conjecture, which
is now a theorem; see also \cite{Geisser-2004-Dedekind} where the consequences
of Bloch--Kato for motivic cohomology are deduced.)

Finally, the base change from $X$ to $X_\CC$ naturally maps cycles $Z \subset X
    \times \Delta_\ZZ^i$ of dimension $n$ to cycles in $X_\CC \times_{\Spec \CC}
    \Delta_\CC^i$ of dimension $n-1$, so that there is a morphism
\begin{equation}
    \label{eqn:KLM-morphism-6}
    R\Gamma (X_\et, \RR^c (n)) \to R\Gamma (X_{\CC,\et}, \RR^c (n-1)) [2].
\end{equation}

\begin{remark}
    Assuming that $X$ is flat and has pure Krull dimension $d$, we have
    $\RR^c (n)^X = \RR (d-n)^X [2d]$, where $\RR (\bullet)$ is the usual cycle
    complex defined by $z^n (\text{\textvisiblespace}, -\bullet) [-2n]$.
    Similarly, $\RR^c (n)^{X_\CC} = \RR (d_\CC-n)^{X_\CC} [2d_\CC]$, with
    $d_\CC = d - 1$. With this renumbering, the morphism
    \eqref{eqn:KLM-morphism-6} becomes
    \begin{equation*}
        R\Gamma (X_\et, \RR (d-n)) [2d] \to R\Gamma (X_{\CC,\et}, \RR (d-n)) [2d].
    \end{equation*}
    This probably looks more natural, but we make no additional assumptions about
    $X$ and work exclusively with complexes $A^c (\bullet)$ defined in terms of
    dimension of algebraic cycles, rather than $A (\bullet)$ defined in terms of
    codimension.
\end{remark}

\begin{definition}
    Given an arithmetic scheme $X$ with smooth quasi-projective $X_\CC$ and
    $n < 0$, consider the composition of morphisms
    \begin{multline*}
        R\Gamma (X_\et, \RR^c (n)) \xrightarrow{\text{\eqref{eqn:KLM-morphism-6}}}
        R\Gamma (X_{\CC,\et}, \RR^c (n-1)) [2] \stackrel{\text{\eqref{eqn:KLM-morphism-5}}}{\cong}
        \Gamma (X_{\CC,\et}, \RR^c (n-1)) [2] \\
        \stackrel{\text{\eqref{eqn:KLM-morphism-4}}}{=}
        z_{n-1} (X_\CC, -\bullet)_\RR [2n] \xrightarrow{\text{\eqref{eqn:KLM-morphism-3}}}
        R\Gamma_\BM (X (\CC), \RR (n)), \RR) [1].
    \end{multline*}
    Moreover, we take the $G_\RR$-invariants, which gives us the
    \textbf{(\'{e}tale) regulator}
    \begin{equation*}
        \Reg_{X,n}\colon R\Gamma (X_\et, \RR^c (n)) \to
        R\Gamma_\BM (G_\RR, X(\CC), \RR (n)) [1].
    \end{equation*}
\end{definition}

Now we state our conjecture about the regulator, which will play an important
role in everything that follows.

\begin{conjecture}
    \label{conjecture:B(X,n)}
    $\mathbf{B} (X,n)$: given an arithmetic scheme $X$ with smooth
    quasi-projective $X_\CC$ and $n < 0$, the regulator morphism $\Reg_{X,n}$
    induces a quasi-isomorphism of complexes of real vector spaces
    \begin{equation*}
        \Reg_{X,n}^\vee\colon R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1] \to
        \RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR).
    \end{equation*}
\end{conjecture}

\begin{remark}
    If $X/\FF_q$ is a variety over a finite field, then $X (\CC) = \emptyset$,
    so the regulator map is not interesting. Indeed, in our setting, its purpose
    is to take care of the archimedian places of $X$. In this case
    $\mathbf{B} (X,n)$ implies that $H^i (X_\et, \ZZ^c (n))$ are torsion groups.
    However, by \cite[Proposition~4.2]{Beshenov-Weil-etale-1},
    Conjecture~$\mathbf{L}^c (X_\et, n)$ already implies that
    $H^i (X_\et, \ZZ^c (n))$ are finite groups.
\end{remark}

\begin{remark}
    \label{rmk:regulator-is-defined-for-XC-smooth-quasi-proj}
    We reiterate that our construction of $\Reg_{X,n}$ works for $X_{\red,\CC}$
    smooth quasi-projective. In everything that follows, whenever the regulator
    morphism or Conjecture~$\mathbf{B} (X,n)$ is brought, we tacitly assume this
    restriction. This is rather unfortunate, since Weil-\'{e}tale cohomology was
    constructed in \cite{Beshenov-Weil-etale-1} for any arithmetic scheme,
    assuming only Conjecture~$\mathbf{L}^c (X_\et,n)$. Defining the regulator for
    singular $X_{\red,\CC}$ is an interesting project for future work.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Vanishing order Conjecture $\mathbf{VO} (X,n)$}
\label{sec:vanishing-order-conjecture}

Assuming that $\zeta (X,s)$ admits a meromorphic continuation around $s = n <
    0$, we make the following conjecture for the vanishing order at $s = n$.

\begin{conjecture}
    \label{conjecture:VO(X,n)}
    $\mathbf{VO} (X,n)$: one has
    \begin{equation*}
        \ord_{s=n} \zeta (X,s) =
        \sum_{i \in \ZZ} (-1)^i \cdot i \cdot \rk_\ZZ H^i_\Wc (X, \ZZ (n)).
    \end{equation*}
\end{conjecture}

We note that the right-hand side makes sense under Conjecture~$\mathbf{L}^c
    (X_\et,n)$, which implies that $H^i_\Wc (X, \ZZ (n))$ are finitely generated
groups, trivial for $|i| \gg 0$; see
\cite[Proposition~7.12]{Beshenov-Weil-etale-1}.

\begin{remark}
    Conjecture~$\mathbf{VO} (X,n)$ is similar to
    \cite[Conjecture~5.11]{Flach-Morin-2018}. If $X$ is proper and regular, then
    $\mathbf{VO} (X,n)$ is the same as Flach and Morin's vanishing order
    conjecture. Indeed, the latter is
    \begin{equation}
        \label{eqn:FM-vanishing-order}
        \ord_{s = n} \zeta (X,s) =
        \sum_{i\in \ZZ} (-1)^i \cdot i \cdot \dim_\RR H^i_{\ar,c} (X, \widetilde{\RR}(n)),
    \end{equation}
    where
    \begin{equation*}
        R\Gamma_{\ar,c} (X, \widetilde{\RR}(n)) \dfn
        R\Gamma_c (X, \RR(n)) \oplus R\Gamma_c (X, \RR(n)) [-1].
    \end{equation*}
    Moreover, \cite[Proposition 4.14]{Flach-Morin-2018}, gives a distinguished
    triangle
    \begin{multline*}
        R\Gamma_\dR (X_\RR/\RR) / \Fil^n [-2] \to
        R\Gamma_{\ar,c} (X, \widetilde{\RR}(n)) \to
        R\Gamma_\Wc (X, \ZZ(n)) \otimes \RR \\
        \to R\Gamma_\dR (X_\RR/\RR) / \Fil^n [-1]
    \end{multline*}
    So, in case of $n < 0$ we have
    $R\Gamma_{\ar,c} (X, \widetilde{\RR}(n)) \cong
        R\Gamma_\Wc (X, \ZZ(n)) \otimes \RR$ and
    \eqref{eqn:FM-vanishing-order} is exactly Conjecture~$\mathbf{VO} (X,n)$.
\end{remark}

\begin{remark}
    The alternating sum in Conjecture~$\mathbf{VO} (X,n)$ is the so-called
    \textbf{secondary Euler characteristic}
    \begin{equation*}
        \chi' (R\Gamma_\Wc (X, \ZZ (n))) \dfn
        \sum_{i \in \ZZ} (-1)^i \cdot i \cdot \rk_\ZZ H^i_\Wc (X, \ZZ (n)).
    \end{equation*}
    The calculations below show that the usual Euler characteristic of $R\Gamma_\Wc
        (X, \ZZ (n))$ vanishes, assuming Conjectures~$\mathbf{L}^c (X_\et,n)$ and
    $\mathbf{B} (X,n)$. See \cite{Ramachandran-2016} for more details on the
    secondary Euler characteristic and its occurrences in nature.
\end{remark}

Under the regulator conjecture, our vanishing order formula takes the form of
the usual Euler characteristic of equivariant cohomology $R\Gamma_c (G_\RR,
    X(\CC), \RR (n))$ or motivic cohomology $R\Gamma (X_\et, \ZZ^c (n)) [1]$.

\begin{proposition}
    \label{prop:VO(X,n)-assuming-B(X,n)}
    Assuming $\mathbf{L}^c (X_\et, n)$ and $\mathbf{B} (X,n)$,
    Conjecture~$\mathbf{VO} (X,n)$ is equivalent to
    \begin{align*}
        \ord_{s=n} \zeta (X,s) & = \chi (R\Gamma_c (G_\RR, X(\CC), \RR (n))
        = \sum_{i \in \ZZ} (-1)^i \dim_\RR H^i_c (X(\CC), \RR (n))^{G_\RR}  \\
                               & = -\chi (R\Gamma (X_\et, \ZZ^c (n)))
        = \sum_{i \in \ZZ} (-1)^{i+1} \rk_\ZZ H^i (X_\et, \ZZ^c (n)).
    \end{align*}
    Moreover, we have
    \begin{equation*}
        \chi (R\Gamma_\Wc (X, \ZZ(n))) = 0.
    \end{equation*}

    \begin{proof}
        Thanks to \cite[Proposition~7.13]{Beshenov-Weil-etale-1}, the Weil-\'{e}tale
        complex tensored with $\RR$ splits as
        \begin{equation*}
            R\Gamma_\Wc (X,\RR (n)) \cong
            \RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR) [-1] \oplus
            R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1].
        \end{equation*}
        Assuming Conjecture~$\mathbf{B} (X,n)$, we also have a quasi-isomorphism
        \begin{equation*}
            R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1] \cong
            \RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR),
        \end{equation*}
        so that
        \begin{equation*}
            \dim_\RR H^i_\Wc (X,\RR(n)) =
            \dim_\RR H^{i-1}_c (X (\CC), \RR (n))^{G_\RR} +
            \dim_\RR H^{i-2}_c (X (\CC), \RR (n))^{G_\RR}.
        \end{equation*}
        Thus, we can rewrite the sum
        \begin{align*}
            \sum_{i \in \ZZ} (-1)^i \cdot i \cdot \rk_\ZZ H^i_\Wc (X, \ZZ (n)) & = \sum_{i \in \ZZ} (-1)^i \cdot i \cdot \dim_\RR H^i_\Wc (X, \RR (n))                            \\
                                                                               & = \sum_{i \in \ZZ} (-1)^i \cdot i \cdot
            \dim_\RR H^{i-1}_c (X (\CC), \RR (n))^{G_\RR}                                                                                                                         \\
                                                                               & \quad\quad + \sum_{i \in \ZZ} (-1)^i \cdot i \cdot \dim_\RR H^{i-2}_c (X (\CC), \RR (n))^{G_\RR} \\
                                                                               & = -\sum_{i \in \ZZ} (-1)^i \, \dim_\RR H^{i-1}_c (X (\CC), \RR (n))^{G_\RR}                      \\
                                                                               & = \chi (R\Gamma_c (G_\RR, X (\CC), \RR (n)).
        \end{align*}
        Similarly,
        \begin{align*}
            \sum_{i \in \ZZ} (-1)^i \cdot i \cdot \rk_\ZZ H^i_\Wc (X, \ZZ (n)) & = \chi (\RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR) [1]) \\
                                                                               & = -\chi (R\Gamma (X_\et, \ZZ^c (n))).
        \end{align*}
        These considerations also show that the usual Euler characteristic of
        $R\Gamma_\Wc (X, \ZZ(n))$ vanishes.
    \end{proof}
\end{proposition}

\begin{remark}
    Conjecture~$\mathbf{VO} (X,n)$ is related to a conjecture of Soul\'{e}
    \cite[Conjecture~2.2]{Soule-1984-ICM}, which originally reads in terms of
    $K'$-theory
    \begin{equation*}
        \ord_{s=n} \zeta (X,s) =
        \sum_{i \in \ZZ} (-1)^{i+1} \, \dim_\QQ K'_i (X)_{(n)}.
    \end{equation*}
    As explained in \cite[Remark~43]{Kahn-2005}, this can be rewritten in terms of
    Borel--Moore motivic homology as $$\sum_{i \in \ZZ} (-1)^{i+1} \, \dim_\QQ
        H_i^{BM} (X, \QQ (n)).$$ In our setting, $H^i (X_\et, \ZZ^c (n))$ plays the
    role of Borel--Moore homology, which explains the formula
    \begin{equation*}
        \ord_{s=n} \zeta (X,s) =
        \sum_{i \in \ZZ} (-1)^{i+1} \rk_\ZZ H^i (X_\et, \ZZ^c (n)).
    \end{equation*}
\end{remark}

\begin{remark}[{\cite[Proposition~5.13]{Flach-Morin-2018}}]
    \label{rmk:archimedian-euler-factor}
    As for the formula
    \begin{equation*}
        \ord_{s=n} \zeta (X,s) =
        \sum_{i \in \ZZ} (-1)^i \dim_\RR H^i_c (X(\CC), \RR (n))^{G_\RR},
    \end{equation*}
    it essentially means that the vanishing order at $s = n < 0$ comes from the
    archimedian $\Gamma$-factor appearing in the (hypothetical) functional
    equation, as explained in \cite[\S\S 3,4]{Serre-1970} (see also \cite[\S
        4]{Flach-Morin-2020}).

    Indeed, under the assumption that $X_\CC$ is a smooth projective variety, we
    consider the Hodge decomposition
    \begin{equation*}
        H^i (X (\CC), \CC) = \bigoplus_{p+q = i} H^{p,q},
    \end{equation*}
    which carries an action of $G_\RR = \{ id, \sigma \}$ such that $\sigma
        (H^{p,q}) = H^{q,p}$. We set $h^{p,q} = \dim_\CC H^{p,q}$. For $p = i/2$ we
    consider the eigenspace decomposition $H^{p,p} = H^{p,+} \oplus H^{p,-}$, where
    \begin{align*}
        H^{p,+} & = \{ x \in H^{p,p} \mid \sigma (x) = (-1)^p\,x \},     \\
        H^{p,-} & = \{ x \in H^{p,p} \mid \sigma (x) = (-1)^{p+1}\,x \},
    \end{align*}
    and set $h^{p,\pm} = \dim_\CC H^{p,\pm}$ accordingly.
    The completed zeta function
    \begin{equation*}
        \zeta (\overline{X}, s) = \zeta (X, s)\,\zeta (X_\infty, s)
    \end{equation*}
    is expected to satisfy a functional equation of the form
    \begin{equation*}
        A^{\frac{d-s}{2}}\,\zeta (\overline{X},d-s) =
        A^{\frac{s}{2}}\,\zeta (\overline{X},s).
    \end{equation*}
    Here
    \begin{gather*}
        \zeta (X_\infty, s) = \prod_{i\in \ZZ} L_\infty (H^i (X),s)^{(-1)^i}, \\
        L_\infty (H^i (X), s) =
        \prod_{p = i/2} \Gamma_\RR (s - p)^{h^{p,+}}\,\Gamma_\RR (s-p+1)^{h^{p,-}} \,
        \prod_{\substack{p + q = i \\ p < q}} \Gamma_\CC (s - p)^{h^{p,q}}, \\
        \Gamma_\RR (s) = \pi^{-s/2} \, \Gamma (s/2), \quad
        \Gamma_\CC (s) = (2\pi)^{-s} \, \Gamma (s).
    \end{gather*}

    Therefore, the expected vanishing order at $s = n < 0$ is
    \begin{align*}
        \ord_{s=n} \zeta (X,s) & = -\ord_{s=n} \zeta (X_\infty,s)                                 \\
                               & = -\sum_{i\in \ZZ} (-1)^i \ord_{s=n} L_\infty (H^i (X), s)       \\
                               & = \sum_{i\in \ZZ} (-1)^i \Bigl(\sum_{p = i/2} h^{p,(-1)^{n-p}} +
        \sum_{\substack{p + q = i                                                                 \\ p < q}} h^{p,q}\Bigr).
    \end{align*}
    The last equality follows from the fact that $\Gamma (s)$ has simple poles at
    all $s = n \le 0$. We have
    \begin{align*}
        \dim_\RR H^i (X (\CC), \RR (n))^{G_\RR} & = \dim_\RR H^i (X (\CC), \RR)^{\sigma = (-1)^n}               \\
                                                & = \dim_\CC H^i (X (\CC), \CC)^{\sigma = (-1)^n}               \\
                                                & = \sum_{p = i/2} h^{p,(-1)^{n-p}} + \sum_{\substack{p + q = i \\ p < q}} h^{p,q}.
    \end{align*}
    Here the terms $h^{p,q}$ with $p < q$ come from $\sigma (H^{p,q}) = H^{q,p}$,
    while $h^{p,(-1)^{n-p}}$ come from the action on $H^{p,p}$.
    We see that our conjectural formula recovers the expected vanishing order.
\end{remark}

Let us look at some particular examples when the meromorphic continuation for
$\zeta (X,s)$ is known.

\begin{example}
    \label{example:VO(X,n)-for-number-rings}
    Suppose that $X = \Spec \mathcal{O}_F$ is the spectrum of the ring of integers
    of a number field $F/\QQ$. Let $r_1$ be the number of real embeddings
    $F \hookrightarrow \RR$ and $r_2$ be the number of conjugate pairs of complex
    embeddings $F \hookrightarrow \CC$. The space $X (\CC)$ with the action of
    complex conjugation can be visualized as follows:
    \begin{equation*}
        \begin{tikzpicture}
            \matrix(m)[matrix of math nodes, row sep=1em, column sep=1em,
                text height=1ex, text depth=0.2ex]{
                ~       & ~       & ~      & ~       & ~ & \bullet & \bullet & \cdots & \bullet                          \\
                \bullet & \bullet & \cdots & \bullet                                                                     \\
                ~       & ~       & ~      & ~       & ~ & \bullet & \bullet & \cdots & \bullet \\};

            \draw[->] (m-2-1) edge[loop above,min distance=10mm] (m-2-1);
            \draw[->] (m-2-2) edge[loop above,min distance=10mm] (m-2-2);
            \draw[->] (m-2-4) edge[loop above,min distance=10mm] (m-2-4);

            \draw[->] (m-1-6) edge[bend left] (m-3-6);
            \draw[->] (m-1-7) edge[bend left] (m-3-7);
            \draw[->] (m-1-9) edge[bend left] (m-3-9);

            \draw[->] (m-3-6) edge[bend left] (m-1-6);
            \draw[->] (m-3-7) edge[bend left] (m-1-7);
            \draw[->] (m-3-9) edge[bend left] (m-1-9);

            \draw [decorate,decoration={brace,amplitude=5pt,mirror}] ($(m-3-1)+(-0.5em,-0.5em)$) -- ($(m-3-4)+(0.5em,-0.5em)$);
            \draw [decorate,decoration={brace,amplitude=5pt,mirror}] ($(m-3-6)+(-0.5em,-0.5em)$) -- ($(m-3-9)+(0.5em,-0.5em)$);

            \draw ($(m-3-1)!.5!(m-3-4)$) node[yshift=-2em,anchor=base] {$r_1$ points};
            \draw ($(m-3-6)!.5!(m-3-9)$) node[yshift=-2em,anchor=base] {$2 r_2$ points};
        \end{tikzpicture}
    \end{equation*}

    The complex $R\Gamma_c (X (\CC), \RR (n))$ consists of a single $G_\RR$-module
    in degree $0$ given by $$\RR (n)^{\oplus r_1} \oplus (\RR (n) \oplus \RR
        (n))^{\oplus r_2},$$ with the action of $G_\RR$ on the first summand $\RR
        (n)^{\oplus r_1}$ via the complex conjugation and the action on the second
    summand $(\RR (n) \oplus \RR (n))^{\oplus r_2}$ via $(x,y) \mapsto
        (\overline{y}, \overline{x})$. The corresponding real space of fixed points has
    dimension
    \begin{equation*}
        \dim_\RR H^0_c (G_\RR, X (\CC), \RR (n)) = \begin{cases}
            r_2,       & n \text{ odd},  \\
            r_1 + r_2, & n \text{ even}, \\
        \end{cases}
    \end{equation*}
    which indeed coincides with the vanishing order of the Dedekind zeta function
    $\zeta (X,s) = \zeta_F (s)$ at $s = n < 0$.

    On the motivic cohomology side, for $n < 0$ the groups $H^i (X_\et, \ZZ^c (n))$
    are finite, except for $i = -1$, where by \cite[Proposition~4.14]{Geisser-2017}
    \begin{equation*}
        \rk_\ZZ H^{-1} (X_\et, \ZZ^c (n)) = \begin{cases}
            r_2,       & n \text{ odd},  \\
            r_1 + r_2, & n \text{ even}.
        \end{cases}
    \end{equation*}
\end{example}

\begin{example}
    Suppose that $X$ is a variety over a finite field $\FF_q$. Then the vanishing
    order conjecture is not very interesting, because the formula yields
    \begin{align*}
        \ord_{s=n} \zeta (X,s) & = \sum_{i \in \ZZ} (-1)^i \dim_\RR H^i_c (X(\CC), \RR (n))^{G_\RR} \\
                               & = \sum_{i \in \ZZ} (-1)^{i+1} \rk_\ZZ H^i (X_\et, \ZZ^c (n)) = 0,
    \end{align*}
    since $X (\CC) = \emptyset$, and also because $\mathbf{L}^c (X_\et, n)$
    implies $\rk_\ZZ H^i (X_\et, \ZZ^c (n)) = 0$ for all $i$ in the case of
    varieties over finite fields, as observed in
    \cite[Proposition~4.2]{Beshenov-Weil-etale-1}. Therefore, the conjecture
    simply asserts that $\zeta (X,s)$ has no zeros or poles at $s = n < 0$.  This
    is indeed the case. We have $\zeta (X,s) = Z (X,q^{-s})$, where
    \begin{equation*}
        Z (X,t) = \exp \Bigl(\sum_{k\ge 1} \frac{\# X (\FF_{q^k})}{k}\,t^k\Bigr)
    \end{equation*}
    is the Hasse--Weil zeta function. According to Deligne's work on Weil's
    conjectures \cite{Deligne-Weil-II}, the zeros and poles of $Z (X,s)$ satisfy
    $|s| = q^{-w/2}$, where $0 \le w \le 2 \dim X$
    (see e.g. \cite[pp.\,26--27]{Katz-1994}). In particular, $q^{-s}$ for
    $s = n < 0$ is neither a zero nor a pole of $Z (X,s)$.

    We also note that our definition of $H^i_\Wc (X, \ZZ(n))$, and pretty much
    everything said above, only makes sense for $n < 0$. Already for $n = 0$, for
    example, the zeta function of a smooth projective curve $X/\FF_q$ has a simple
    pole at $s = 0$.
\end{example}

\begin{example}
    Let $X = E$ be an integral model of an elliptic curve over $\QQ$. Then, as a
    consequence of the modularity theorem
    (Wiles--Breuil--Conrad--Diamond--Taylor), it is known that $\zeta (E,s)$
    admits a meromorphic continuation satisfying the functional equation with the
    $\Gamma$-factors discussed in Remark~\ref{rmk:archimedian-euler-factor}.
    In this particular case $\ord_{s=n} \zeta (E,s) = 0$ for all $n < 0$. This is
    consistent with the fact that
    $\chi (R\Gamma_c (G_\RR, E (\CC), \RR (n))) = 0$.

    Indeed, the equivariant cohomology groups $H^i_c (E (\CC), \RR (n))^{G_\RR}$
    are the following:
    \begin{center}
        \renewcommand{\arraystretch}{1.5}
        \begin{tabular}{rccc}
            \hline
                      & $i = 0$ & $i = 1$ & $i = 2$ \\
            \hline
            $n$ even: & $\RR$   & $\RR$   & $0$     \\
            $n$ odd:  & $0$     & $\RR$   & $\RR$   \\
            \hline
        \end{tabular}
    \end{center}
    ---see, for example, the calculation in \cite[Lemma~A.6]{Siebel-2019}.
\end{example}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 4
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Special value Conjecture~$\mathbf{C} (X,n)$}
\label{sec:special-value-conjecture}

\begin{definition}
    We define a morphism of complexes
    \begin{equation*}
        \smile\theta\colon R\Gamma_\Wc (X,\ZZ(n)) \otimes \RR \to
        R\Gamma_\Wc (X,\ZZ(n)) [1] \otimes \RR
    \end{equation*}
    using the splitting \cite[Proposition~7.13]{Beshenov-Weil-etale-1}
    \begin{equation*}
        R\Gamma_\Wc (X, \RR (n)) \cong
        \RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR) [-1] \oplus
        R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1]
    \end{equation*}
    as follows:
    \begin{equation*}
        \begin{tikzcd}
            R\Gamma_\Wc (X, \RR(n)) \ar{d}{\cong}\ar[dashed]{r}{\smile\theta} & R\Gamma_\Wc (X, \RR(n)) [1]\ar{d}{\cong} \\
            \RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR) [-1] & \RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR) \\[-2em]
            \oplus & \oplus \\[-2em]
            R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1]\ar{uur}[description]{\Reg_{X,n}^\vee} & R\Gamma_c (G_\RR, X (\CC), \RR (n))
        \end{tikzcd}
    \end{equation*}
\end{definition}

\begin{lemma}
    \label{lemma:smile-theta}
    Assuming Conjectures $\mathbf{L}^c (X_\et,n)$ and $\mathbf{B} (X,n)$, the
    morphism $\smile\theta$ induces a long exact sequence of finite dimensional
    real vector spaces
    \begin{equation*}
        \cdots \to H^{i-1}_\Wc (X, \RR (n))
        \xrightarrow{\smile\theta}
        H^i_\Wc (X, \RR (n))
        \xrightarrow{\smile\theta}
        H^{i+1}_\Wc (X, \RR (n)) \to \cdots
    \end{equation*}

    \begin{proof}
        We obtain a sequence
        \begin{equation*}
            \begin{tikzcd}[column sep=1em]
                \cdots\ar{r} & H^i_\Wc (X, \RR(n))\ar{d}{\cong}\ar[dashed]{r}{\smile\theta} & H^{i+1}_\Wc (X, \RR(n))\ar{d}{\cong} \ar{r} & \cdots \\
                & \Hom (H^{-i-1} (X_\et, \ZZ^c (n)), \RR) & \Hom (H^{-i-2} (X_\et, \ZZ^c (n)), \RR) \\[-2em]
                \cdots & \oplus & \oplus & \cdots \\[-2em]
                & H_c^{i-1} (G_\RR, X (\CC), \RR (n))\ar{uur}[description]{\cong} & H_c^i (G_\RR, X (\CC), \RR (n))
            \end{tikzcd}
        \end{equation*}
        The diagonal arrows are isomorphisms according to $\mathbf{B} (X,n)$, so the
        sequence is exact.
    \end{proof}
\end{lemma}

The Weil-\'{e}tale complex $R\Gamma_\Wc (X, \ZZ(n))$ is defined in \cite[\S
    7]{Beshenov-Weil-etale-1} up to a \emph{non-unique} isomorphism in the derived
category $\mathbf{D} (\ZZ)$ via a distinguished triangle
\begin{equation}
    \label{eqn:triangle-defining-RGamma-Wc}
    R\Gamma_\Wc (X, \ZZ(n)) \to R\Gamma_\fg (X,\ZZ(n)) \xrightarrow{i_\infty}
    R\Gamma_c (G_\RR, X (\CC), \ZZ (n)) \to [1]
\end{equation}
This is rather awkward, and there should be a better, more canonical
construction of $R\Gamma_\Wc (X, \ZZ(n))$. For our purposes, however, this is
not much of a problem, since the special value conjecture is not formulated in
terms of $R\Gamma_\Wc (X, \ZZ (n))$, but in terms of its determinant
$\det_\ZZ R\Gamma_\Wc (X, \ZZ(n))$ (see Appendix~\ref{app:determinants}), which
is well-defined.

\begin{lemma}
    \label{lemma:determinant-of-RGamma-Wc-well-defined}
    The determinant $\det_\ZZ R\Gamma_\Wc (X, \ZZ(n))$ is defined up to a
    canonical isomorphism.

    \begin{proof}
        Two different choices for the mapping fiber in
        \eqref{eqn:triangle-defining-RGamma-Wc} yield an isomorphism of
        distinguished triangles
        \begin{equation*}
            \begin{tikzcd}
                R\Gamma_\Wc (X, \ZZ(n)) \ar{r}\ar{d}{f}[swap]{\cong} & R\Gamma_\fg (X,\ZZ(n)) \ar{r}{i_\infty}\ar{d}{id} & R\Gamma_c (G_\RR, X (\CC), \ZZ (n)) \ar{r}\ar{d}{id} & {[1]}\ar{d}{f}[swap]{\cong} \\
                R\Gamma_\Wc (X, \ZZ(n))' \ar{r} & R\Gamma_\fg (X,\ZZ(n)) \ar{r}{i_\infty} & R\Gamma_c (G_\RR, X (\CC), \ZZ (n)) \ar{r} & {[1]}
            \end{tikzcd}
        \end{equation*}
        The idea is to use functoriality of determinants with respect to isomorphisms
        of distinguished triangles (see Appendix~\ref{app:determinants}). The only
        technical problem is that whenever $X (\RR) \ne \emptyset$, the complexes
        $R\Gamma_\fg (X,\ZZ(n))$ and $R\Gamma_c (G_\RR, X (\CC), \ZZ (n))$ are not
        perfect, but may have finite $2$-torsion in $H^i (-)$ for arbitrarily big $i$
        (in \cite{Beshenov-Weil-etale-1} we called such complexes \textbf{almost
            perfect}). On the other hand, the determinants are defined only for perfect
        complexes. Fortunately, $H^i (i_\infty^*)$ is an isomorphism for $i \gg 0$ by
        the boundedness of $H^i_\Wc (X, \ZZ(n))$
        \cite[Proposition~7.12]{Beshenov-Weil-etale-1}, so that for $m$ big enough we
        can take the corresponding canonical truncations $\tau_{\le m}$:
        \begin{equation*}
            \begin{tikzcd}[row sep=1.5em, column sep=0.75em, font=\small]
                \tau_{\le m} R\Gamma_\Wc (X, \ZZ(n)) \ar{r}\ar{d}{\cong} & \tau_{\le m} R\Gamma_\fg (X,\ZZ(n)) \ar{r}\ar{d} & \tau_{\le m} R\Gamma_c (G_\RR, X (\CC), \ZZ (n)) \ar{r}\ar{d} & {[1]}\ar{d}{\cong} \\
                R\Gamma_\Wc (X, \ZZ(n)) \ar{r}\ar{d} & R\Gamma_\fg (X,\ZZ(n)) \ar{r}{i_\infty}\ar{d} & R\Gamma_c (G_\RR, X (\CC), \ZZ (n)) \ar{r}\ar{d} & {[1]}\ar{d} \\
                0 \ar{r}\ar{d} & \tau_{\ge m+1} R\Gamma_\fg (X,\ZZ(n)) \ar{r}{\cong}\ar{d} & \tau_{\ge m+1} R\Gamma_c (G_\RR, X (\CC), \ZZ (n)) \ar{r}\ar{d} & 0\ar{d} \\
                {[1]} \ar{r} & {[1]} \ar{r} & {[1]} \ar{r} & {[2]}
            \end{tikzcd}
        \end{equation*}

        The truncations give us (rotating the triangles)
        \begin{equation*}
            \begin{tikzcd}[column sep=1em,font=\small]
                \tau_{\le m} R\Gamma_c (G_\RR, X (\CC), \ZZ (n))[-1] \ar{r}\ar{d}{id} & R\Gamma_\Wc (X, \ZZ (n))\ar{d}{f}[swap]{\cong}\ar{r} & \tau_{\le m} R\Gamma_\fg (X,\ZZ(n)) \ar{d}{id}\ar{r} & {[0]} \ar{d}{id} \\
                \tau_{\le m} R\Gamma_c (G_\RR, X (\CC), \ZZ (n))[-1] \ar{r} & R\Gamma_\Wc (X, \ZZ (n))' \ar{r} & \tau_{\le m} R\Gamma_\fg (X,\ZZ(n)) \ar{r} & {[0]}
            \end{tikzcd}
        \end{equation*}
        By the functoriality of determinants with respect to isomorphisms of
        distinguished triangles (see Appendix~\ref{app:determinants}), we have a
        commutative diagram
        \begin{equation*}
            \begin{tikzcd}
                \begin{array}{c} \det_\ZZ \tau_{\le m} R\Gamma_c (G_\RR, X (\CC), \ZZ (n))[-1] \\ \otimes \\ \det_\ZZ \tau_{\le m} R\Gamma_\fg (X,\ZZ(n)) \end{array} \ar{r}{i}[swap]{\cong} \ar{d}{id} & \det_\ZZ R\Gamma_\Wc (X,\ZZ(n)) \ar{d}{\det_\ZZ (f)}[swap]{\cong} \\
                \begin{array}{c} \det_\ZZ \tau_{\le m} R\Gamma_c (G_\RR, X (\CC), \ZZ (n))[-1] \\ \otimes \\ \det_\ZZ \tau_{\le m} R\Gamma_\fg (X, \ZZ (n)) \end{array} \ar{r}{i'}[swap]{\cong} & \det_\ZZ R\Gamma_\Wc (X,\ZZ(n))
            \end{tikzcd}
        \end{equation*}
        so that $\det_\ZZ (f) = i'\circ i^{-1}$.
    \end{proof}
\end{lemma}

\begin{lemma}
    The non-canonical splitting
    \begin{equation*}
        R\Gamma_\Wc (X, \RR(n)) \cong
        \RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR) [-1] \oplus
        R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1]
    \end{equation*}
    from \cite[Proposition~7.13]{Beshenov-Weil-etale-1} yields a canonical
    isomorphism of determinants
    \begin{equation*}
        \det_\RR R\Gamma_\Wc (X, \RR (n)) \cong
        \begin{array}{c}
            \det_\RR \RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR) [-1] \\
            \otimes_\RR                                           \\
            \det_\RR R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1]
        \end{array}
    \end{equation*}

    \begin{proof}
        This is similar to the previous lemma; in fact, after tensoring with $\RR$,
        we obtain perfect complexes of real vector spaces, so the truncations are no
        longer needed. By \cite[Proposition~7.4]{Beshenov-Weil-etale-1} we have
        $i_\infty^* \otimes \RR = 0$, so there is an isomorphism of triangles
        \begin{equation}
            \label{eqn:splitting-of-RGamma-Wc-triangles}
            \begin{tikzcd}[row sep=1em]
                R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1] \ar{r}{id}\ar{d} & R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1]\ar{d} \\
                R\Gamma_\Wc (X, \RR (n)) \ar[dashed]{r}{f}[swap]{\cong}\ar{d} & \begin{array}{c} \RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR) [-1] \\ \oplus \\ R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1] \end{array}\ar{d} \\
                R\Gamma_\fg (X, \RR (n)) \ar{r}{g\otimes \RR}[swap]{\cong}\ar{d} & \RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR) [-1] \ar{d} \\
                R\Gamma_c (G_\RR, X (\CC), \RR (n)) \ar{r}{id} & R\Gamma_c (G_\RR, X (\CC), \RR (n))
            \end{tikzcd}
        \end{equation}
        Here the third horizontal arrow comes from the triangle defining
        $R\Gamma_\fg (X, \ZZ(n))$:
        \begin{multline*}
            \RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR) [-2] \xrightarrow{\alpha_{X,n}}
            R\Gamma_c (X_\et, \ZZ (n)) \to
            R\Gamma_\fg (X, \ZZ(n)) \\
            \xrightarrow{g} \RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR) [-1]
        \end{multline*}
        tensored with $\RR$ (see \cite[Proposition~5.7]{Beshenov-Weil-etale-1}).
        The distinguished column on the right-hand side of
        \eqref{eqn:splitting-of-RGamma-Wc-triangles} is the direct sum
        \begin{equation*}
            \begin{tikzcd}[row sep=1.5em]
                R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1]\ar{d}{id} &[-2em] &[-2em] 0\ar{d} \\
                R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1]\ar{d} & \oplus & \RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR) [-1]\ar{d}{id} \\
                0 \ar{d} & & \RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR) [-1] \ar{d} \\
                R\Gamma_c (G_\RR, X (\CC), \RR (n)) & & 0
            \end{tikzcd}
        \end{equation*}
        The splitting isomorphism $f$ in \eqref{eqn:splitting-of-RGamma-Wc-triangles}
        is not canonical at all. However, after taking the determinants, we obtain a
        commutative diagram (see Appendix~\ref{app:determinants})
        \begin{equation*}
            \begin{tikzcd}[column sep=1em, font=\small]
                \begin{array}{c} \det_\RR R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1] \\ \otimes_\RR \\ \det_\RR R\Gamma_\fg (X, \RR(n)) \end{array} \ar{r}{i}[swap]{\cong} \ar{d}{id \otimes \det_\RR (g\otimes \RR)}[swap]{\cong} & \det_\RR R\Gamma_\Wc (X, \RR (n)) \ar{d}{\det_\RR (f)}[swap]{\cong}\ar[dashed]{dl}{\cong} \\
                \begin{array}{c} \det_\RR R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1] \\ \otimes_\RR \\ \det_\RR \RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR) [-1] \end{array} \ar{r}{i'}[swap]{\cong} & \det_\RR \left(\!\!\!\begin{array}{c} \RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR) [-1] \\ \oplus \\ R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1] \end{array}\!\!\!\right)
            \end{tikzcd}
        \end{equation*}
        The dashed diagonal arrow is the desired canonical isomorphism.
    \end{proof}
\end{lemma}

\begin{definition}
    Given an arithmetic scheme $X$ and $n < 0$, assume
    Conjectures~$\mathbf{L}^c (X_\et, n)$ and $\mathbf{B} (X,n)$. Consider the
    quasi-isomorphism
    \begin{multline}\small
        \label{eqn:definition-of-lambda}
        \left(\!\!\!\begin{array}{c} R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-2] \\ \oplus \\ R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1] \end{array}\!\!\!\right)
        \xrightarrow[\cong]{\Reg_{X,n}^\vee [-1] \oplus id}
        \left(\!\!\!\begin{array}{c} \RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR) [-1] \\ \oplus \\ R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1] \end{array}\!\!\!\right) \\
        \\
        \xrightarrow[\cong]{\text{split}} R\Gamma_\Wc (X, \RR (n)).
    \end{multline}
    Note that the first complex has determinant
    \begin{equation*}
        \det_\RR \left(\!\!\begin{array}{c} R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-2] \\ \oplus \\ R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1] \end{array}\!\!\right) \cong
        \begin{array}{c} \det_\RR R\Gamma_c (G_\RR, X (\CC), \RR (n)) \\ \otimes_\RR \\ (\det_\RR R\Gamma_c (G_\RR, X (\CC), \RR (n)))^{-1} \end{array} \cong \RR,
    \end{equation*}
    and for the last complex in \eqref{eqn:definition-of-lambda}, by the
    compatibility with base change, we have a canonical isomorphism
    \begin{equation*}
        \det_\RR R\Gamma_\Wc (X, \RR (n)) \cong
        (\det_\ZZ R\Gamma_\Wc (X, \ZZ (n))) \otimes \RR.
    \end{equation*}
    Therefore, after taking the determinants, the quasi-isomorphism
    \eqref{eqn:definition-of-lambda} induces a canonical isomorphism
    \begin{equation}
        \label{eqn:morphism-lambda}
        \lambda = \lambda_{X,n}\colon \RR \xrightarrow{\cong}
        (\det_\ZZ R\Gamma_\Wc (X, \ZZ (n))) \otimes \RR.
    \end{equation}
\end{definition}

\begin{remark}
    An equivalent way to define $\lambda$ is
    \begin{multline*}
        \lambda\colon \RR \xrightarrow{\cong}
        \bigotimes_{i\in \ZZ} (\det_\RR H^i_\Wc (X, \RR (n)))^{(-1)^i} \\
        \xrightarrow{\cong} \Bigl(\bigotimes_{i\in \ZZ} (\det_\ZZ H^i_\Wc (X, \ZZ (n)))^{(-1)^i}\Bigr) \otimes \RR \\
        \xrightarrow{\cong} (\det_\ZZ R\Gamma_\Wc (X, \ZZ (n))) \otimes \RR,
    \end{multline*}
    where the first isomorphism comes from Lemma~\ref{lemma:smile-theta}.
\end{remark}

We are ready to state the main conjecture of this paper. The determinant
$\det_\ZZ R\Gamma_\Wc (X, \ZZ (n)))$ is a free $\ZZ$-module of rank $1$, and
the isomorphism \eqref{eqn:morphism-lambda} canonically embeds it in $\RR$. We
conjecture that this embedding gives the special value of $\zeta (X,s)$ at $s =
    n$ in the following sense.

\begin{conjecture}
    \label{conjecture:C(X,n)}
    $\mathbf{C} (X,n)$: let $X$ be an arithmetic scheme and $n < 0$ a strictly
    negative integer. Assuming Conjectures~$\mathbf{L}^c (X_\et, n)$,
    $\mathbf{B} (X,n)$ and the meromorphic continuation of $\zeta (X,s)$ around
    $s = n < 0$, the corresponding special value is determined up to sign by
    \begin{equation*}
        \lambda (\zeta^* (X,n)^{-1}) \cdot \ZZ =
        \det_\ZZ R\Gamma_\Wc (X, \ZZ (n)),
    \end{equation*}
    where $\lambda$ is the canonical isomorphism \eqref{eqn:morphism-lambda}.
\end{conjecture}

\begin{remark}
    This conjecture is similar to \cite[Conjecture~5.12]{Flach-Morin-2018}.
    When $X$ is proper and regular, the above conjecture is the same as the
    special value conjecture of Flach and Morin, which for $n \in \ZZ$ reads
    \begin{equation}
        \label{eqn:FM-special-value}
        \lambda_\infty \Bigl(\zeta^* (X,n)^{-1} \cdot C (X,n) \cdot \ZZ\Bigr) =
        \Delta (X/\ZZ, n).
    \end{equation}
    Here the fundamental line $\Delta (X/\ZZ,n)$ is defined via
    \begin{equation*}
        \Delta (X/\ZZ,n) \dfn
        \det_\ZZ R\Gamma_\Wc (X, \ZZ(n)) \otimes
        \det_\ZZ R\Gamma_\dR (X/\ZZ)/\Fil^n.
    \end{equation*}
    If $n < 0$, then $\Delta (X/\ZZ,n) = \det_\ZZ R\Gamma_\Wc (X, \ZZ(n))$.
    Moreover, $C (X,n)$ in \eqref{eqn:FM-special-value} is a rational number,
    defined via $\prod_p |c_p (X,n)|_p$. Here $c_p (X,n) \in
        \QQ_p^\times/\ZZ_p^\times$ are the local factors described in \cite[\S
        5.4]{Flach-Morin-2018}, but \cite[Proposition~5.8]{Flach-Morin-2018} states
    that if $n \le 0$, then $c_p (X,n) \equiv 1 \pmod{\ZZ_p^\times}$ for all $p$.
    Therefore, $C (X,n) = 1$ in our situation. Finally, the trivialization
    isomorphism $\lambda_\infty$ is defined exactly as our $\lambda$. Therefore,
    \eqref{eqn:FM-special-value} for $n < 0$ agrees with Conjecture~$\mathbf{C}
        (X,n)$.

    Flach and Morin prove that their conjecture is consistent with the Tamagawa
    number conjecture of Bloch--Kato--Fontaine--Perrin-Riou
    \cite{Fontaine-Perrin-Riou-1994}; see \cite[\S 5.6]{Flach-Morin-2018} for the
    details.
\end{remark}

\begin{remark}
    Some canonical isomorphisms of determinants involve multiplication by $\pm 1$,
    so it is no surprise that the resulting conjecture is stated up to sign
    $\pm 1$. This is not a major problem, however, since the sign can be recovered
    from the (conjectural) functional equation.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Case of varieties over finite fields}
\label{sec:finite-fields}

For varieties over finite fields, our special value conjecture corresponds to
the conjectures studied by Geisser in
\cite{Geisser-2004,Geisser-2006,Geisser-2010-arithmetic-homology}.

\begin{proposition}
    \label{prop:C(X,n)-over-finite-fields}
    If $X/\FF_q$ is a variety over a finite field, then under the assumption
    $\mathbf{L}^c (X_\et,n)$, the special value conjecture $\mathbf{C} (X,n)$ is
    equivalent to
    \begin{align}
        \label{eqn:special-value-for-X/Fq}
        \notag \zeta^* (X,n) & = \pm \prod_{i \in \ZZ} |H_\Wc^i (X, \ZZ(n))|^{(-1)^i}         \\
                             & = \pm \prod_{i \in \ZZ} |H^i (X_\et, \ZZ^c (n))|^{(-1)^i}      \\
        \notag               & = \pm \prod_{i \in \ZZ} |H_i^c (X_\ar, \ZZ (n))|^{(-1)^{i+1}},
    \end{align}
    where $H_i^c (X_\ar, \ZZ (n))$ are Geisser's arithmetic homology groups
    defined in \cite{Geisser-2010-arithmetic-homology}.

    \begin{proof}
        Assuming $\mathbf{L}^c (X_\et,n)$, we have, thanks to
        \cite[Proposition~7.7]{Beshenov-Weil-etale-1}
        \begin{equation*}
            H^i_\Wc (X, \ZZ (n)) \cong
            \Hom (H^{2-i} (X_\et, \ZZ^c (n)), \QQ/\ZZ) \cong
            \Hom (H_{i-1}^c (X_\ar, \ZZ (n)), \QQ/\ZZ).
        \end{equation*}
        The cohomology groups involved are finite and vanish for $|i| \gg 0$ by
        \cite[Proposition~4.2]{Beshenov-Weil-etale-1}, and by
        Lemma~\ref{lemma:determinant-for-torsion-cohomology} the determinant is given
        by
        \begin{equation*}
            \begin{tikzcd}[row sep=0.75em, column sep=0pt]
                \det_\ZZ R\Gamma_\Wc (X, \ZZ (n)) \ar[equals]{d} & \subset & \det_\ZZ R\Gamma_\Wc (X, \ZZ (n)) \otimes \QQ \ar[equals]{d} \\
                \frac{1}{m}\ZZ & \subset & \QQ
            \end{tikzcd}
        \end{equation*}
        where
        \begin{equation*}
            m = \prod_{i \in \ZZ} |H_\Wc^i (X, \ZZ(n))|^{(-1)^i}. \qedhere
        \end{equation*}
    \end{proof}
\end{proposition}

\begin{remark}
    Formulas like \eqref{eqn:special-value-for-X/Fq} were proposed by Lichtenbaum
    early on in \cite{Lichtenbaum-1984}.
\end{remark}

\begin{theorem}
    \label{thm:C(X,n)-over-finite-fields}
    Let $X/\FF_q$ be a variety over a finite field satisfying
    Conjecture~$\mathbf{L}^c (X_\et, n)$ for $n < 0$. Then
    Conjecture~$\mathbf{C} (X,n)$ holds.
\end{theorem}

We note that \eqref{prop:C(X,n)-over-finite-fields} is equivalent to the
special value formula that appears in
\cite[Theorem~4.5]{Geisser-2010-arithmetic-homology}. Conjecture~$\mathbf{P}_0
    (X)$ in the statement of \cite[Theorem~4.5]{Geisser-2010-arithmetic-homology}
is implied by our Conjecture~$\mathbf{L}^c (X_\et,n)$ thanks to
\cite[Proposition~4.1]{Geisser-2010-arithmetic-homology}. Geisser's proof
eventually reduces to Milne's work \cite{Milne-1986}, but for our case of $s =
    n < 0$, the situation is simpler, and we can give a direct explanation, using
earlier results of B\'{a}yer and Neukirch \cite{Bayer-Neukirch-1978} concerning
Grothendieck's trace formula.

\begin{proof}
    By the previous proposition, the conjecture reduces to
    \begin{equation*}
        \zeta (X,n) = \prod_{i \in \ZZ} |H^i (X_\et, \ZZ^c (n))|^{(-1)^i}.
    \end{equation*}

    By duality \cite[Theorem~I]{Beshenov-Weil-etale-1} $$|H^{2-i} (X_\et, \ZZ^c
        (n))| = |H^i_c (X_\et, \ZZ (n))|,$$ where
    \begin{equation*}
        \ZZ (n) \dfn
        \bigoplus_{\ell \ne p} \QQ_\ell/\ZZ_\ell (n) [-1] \dfn
        \bigoplus_{\ell \ne p} \mu_{\ell^\infty}^{\otimes n} [-1] \dfn
        \bigoplus_{\ell \ne p} \varinjlim_r \mu_{\ell^r}^{\otimes n} [-1],
    \end{equation*}
    and $p$ is the characteristic of the base field. Now $H^i_c (X_\et, \QQ_\ell
        (n)) = 0$ for $n < 0$, and therefore $H^i_c (X_\et, \ZZ_\ell (n)) \cong
        H^{i-1}_c (X_\et, \QQ_\ell/\ZZ_\ell (n))$. This means that our formula can be
    written as
    \begin{equation}
        \label{eqn:zeta-X/Fq-product-formula}
        \zeta (X,n) =
        \prod_{\ell \ne p} \prod_{i \in \ZZ} |H^i_c (X_\et, \ZZ_\ell (n))|^{(-1)^i}.
    \end{equation}

    Grothendieck's trace formula (see \cite{Grothendieck-FL} or
    \cite[Rapport]{SGA4-1-2}) reads
    \begin{equation*}
        Z (X,t) =
        \prod_{i \in \ZZ} \det \bigl(1 - tF \bigm| H^i_c (\overline{X}, \QQ_\ell)\bigr)^{(-1)^{i+1}},
    \end{equation*}
    where $\overline{X} \dfn X \times_{\Spec \FF_q} \overline{\FF}_q$ and $F$ is
    the Frobenius acting on $H^i_c (\overline{X}, \QQ_\ell)$. Substituting $t =
        q^{-n}$,
    \begin{equation*}
        \zeta (X,n) =
        \prod_{i \in \ZZ} \det \bigl(1 - q^{-n} F \bigm| H^i_c (\overline{X}, \QQ_\ell)\bigr)^{(-1)^{i+1}}.
    \end{equation*}
    Then, by the proof of \cite[Theorem~(3.1)]{Bayer-Neukirch-1978}, for each $\ell
        \ne p$, we obtain
    \begin{equation}
        \label{eqn:bayer-neukirch}
        |\zeta (X,n)|_\ell =
        \prod_{i \in \ZZ} |H^i_c (X, \ZZ_\ell (n))|^{(-1)^{i+1}}.
    \end{equation}

    On the other hand, for $n < 0$ we have
    \begin{equation}
        \label{eqn:zeta-X/Fq-p-part}
        |\zeta (X,n)|_p = 1.
    \end{equation}
    This fact can be justified, without assuming that $X$ is smooth or projective,
    e.g., using Kedlaya's trace formula for rigid cohomology
    \cite[p.\,1446]{Kedlaya-2006}, which gives
    \begin{equation*}
        Z (X,t) = \prod_i P_i (t)^{(-1)^{i+1}},
        \quad\text{where }
        P_i (t) \in \ZZ[t] \text{ and } P_i (0) = 1.
    \end{equation*}
    In particular, $P_i (q^{-n}) \equiv 1 \pmod{p}$.

    The product formula recovers from \eqref{eqn:bayer-neukirch} and
    \eqref{eqn:zeta-X/Fq-p-part} our special value formula
    \eqref{eqn:zeta-X/Fq-product-formula}.
\end{proof}

\begin{remark}
    The fact that $|\zeta (X,n)|_p = 1$, as observed in the argument above,
    explains why our Weil-\'{e}tale cohomology ignores the $p$-primary part in
    some sense.
\end{remark}

Let us consider a few examples to see how the special value conjecture works
over finite fields.

\begin{example}
    \label{example:C(X,n)-for-Spec-Fq}
    If $X = \Spec \FF_q$, then $\zeta (X,s) = \frac{1}{1 - q^{-s}}$. In this case
    for $n < 0$ we obtain
    \begin{equation}
        \label{eqn:motivic-cohomology-of-Fq}
        H^i (\Spec \FF_{q,\et}, \ZZ^c (n)) \cong
        \begin{cases}
            \ZZ/(q^{-n} - 1), & i = 1,  \\
            0,                & i \ne 1
        \end{cases}
    \end{equation}
    (see, for example, \cite[Example~4.2]{Geisser-2017}). Therefore, formula
    \eqref{eqn:special-value-for-X/Fq} indeed recovers $\zeta (X,n)$ up to sign.

    Similarly, if we replace $\Spec \FF_q$ with $\Spec \FF_{q^m}$, considered as a
    variety over $\FF_q$, then $\zeta (\Spec \FF_{q^m},s) = \zeta (\Spec \FF_q,
        ms)$, and \eqref{eqn:motivic-cohomology-of-Fq} also changes accordingly.
\end{example}

\begin{example}
    Consider $X = \PP^1_{\FF_q}/(0\sim 1)$, or equivalently, a nodal cubic.
    The zeta function is $\zeta (X,s) = \frac{1}{1 - q^{1-s}}$.  We can calculate
    the groups $H^i (X_\et, \ZZ^c (n))$ using the blowup square
    \begin{equation*}
        \begin{tikzcd}
            \Spec \FF_q \sqcup \Spec \FF_q \ar{r}\ar{d}\tikzpb & \PP^1_{\FF_q} \ar{d} \\
            \Spec \FF_q \ar{r} & X
        \end{tikzcd}
    \end{equation*}
    This is similar to \cite[\S 8, Example~2]{Geisser-2006}. Geisser uses the
    eh-topology and long exact sequences associated to abstract blowup squares
    \cite[Proposition~3.2]{Geisser-2006}. In our case the same reasoning works,
    because by \cite[Theorem~I]{Beshenov-Weil-etale-1}, one has $H^i (X_\et, \ZZ^c
        (n)) \cong \Hom (H^{2-i}_c (X_\et, \ZZ (n)),\QQ/\ZZ)$, where $\ZZ (n) =
        \varinjlim_{p\nmid m} \mu_m^{\otimes n} [-1]$, and \'{e}tale cohomology and
    eh-cohomology coincide for such sheaves by \cite[Theorem~3.6]{Geisser-2006}.

    Using the projective bundle formula, we calculate from
    \eqref{eqn:motivic-cohomology-of-Fq}
    \begin{equation*}
        H^i (\PP^1_{\FF_q,\et}, \ZZ^c (n)) \cong \begin{cases}
            \ZZ/(q^{1-n} - 1), & i = -1,      \\
            \ZZ/(q^{-n} - 1),  & i = +1,      \\
            0,                 & i \ne \pm 1.
        \end{cases}
    \end{equation*}
    By the argument from \cite[\S 8, Example~2]{Geisser-2006}, the short exact
    sequences
    \begin{equation*}
        0 \to H^i (\PP^1_{\FF_q,\et}, \ZZ^c (n)) \to
        H^i (X_\et, \ZZ^c (n)) \to
        H^{i+1} ((\Spec \FF_q)_\et, \ZZ^c (n)) \to 0
    \end{equation*}
    give
    \begin{equation*}
        H^i (X_\et, \ZZ^c (n)) \cong \begin{cases}
            \ZZ/(q^{1-n} - 1), & i = -1,           \\
            \ZZ/(q^{-n} - 1),  & i = 0,1,          \\
            0,                 & \text{otherwise}.
        \end{cases}
    \end{equation*}
    The formula \eqref{eqn:special-value-for-X/Fq} gives the correct value $\zeta
        (X,n)$.
\end{example}

\begin{example}
    In general, if $X/\FF_q$ is a curve, then Conjecture~$\mathbf{L}^c (X_\et,n)$
    holds; see for example \cite[Proposition~4.3]{Geisser-2017}. The cohomology
    $H^i (X_\et, \ZZ^c(n))$ is concentrated in degrees $-1, 0, +1$ by duality
    \cite[Theorem~I]{Beshenov-Weil-etale-1} and the reasons of cohomological
    dimension, and the special value formula is
    \begin{equation*}
        \zeta^* (X,n) =
        \pm \frac{|H^0 (X_\et, \ZZ^c (n))|}{|H^{-1} (X_\et, \ZZ^c (n))|\cdot |H^1 (X_\et, \ZZ^c (n))|}.
    \end{equation*}
\end{example}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 6
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Compatibility with operations on schemes}
\label{sec:compatibility-with-operations}

The following basic properties follow from the definition of $\zeta (X,s)$
(formula \eqref{eqn:Euler-product-for-zeta}).

\begin{enumerate}
    \item[1)] \textbf{Disjoint unions}: if $X = \coprod_{1 \le i \le r} X_i$ is a
          finite disjoint union of arithmetic schemes, then
          \begin{equation}
              \label{eqn:zeta-function-for-disjoint-unions}
              \zeta (X,s) = \prod_{1 \le i \le r} \zeta (X_i,s).
          \end{equation}
          In particular,
          \begin{align*}
              \ord_{s=n} \zeta (X,s) & = \sum_{1 \le i \le r} \ord_{s=n} \zeta (X_i,s), \\
              \zeta^* (X,n)          & = \prod_{1 \le i \le r} \zeta^* (X_i,n).
          \end{align*}

    \item[2)] \textbf{Closed-open decompositions}: if $Z \subset X$ is a closed
          subscheme and $U = X\setminus Z$ is its open complement, then we say that we
          have a \textbf{closed-open decomposition} and write
          $Z \not\hookrightarrow X \hookleftarrow U$. In this case
          \begin{equation}
              \label{eqn:zeta-function-for-closed-open-decompositions}
              \zeta (X,s) = \zeta (Z,s) \cdot \zeta (U,s).
          \end{equation}
          In particular,
          \begin{align*}
              \ord_{s=n} \zeta (X,s) & = \ord_{s=n} \zeta (Z,s) + \ord_{s=n} \zeta (U,s), \\
              \zeta^* (X,n)          & = \zeta^* (Z,n) \cdot \zeta^* (U,n).
          \end{align*}

    \item[3)] \textbf{Affine bundles}: for any $r \ge 0$ the zeta function of the
          relative affine space $\AA^r_X = \AA^r_\ZZ \times X$ satisfies
          \begin{equation}
              \label{eqn:zeta-function-for-affine-space}
              \zeta (\AA^r_X, s) = \zeta (X, s-r).
          \end{equation}
          In particular,
          \begin{align*}
              \ord_{s=n} \zeta (\AA^r_X, s) & = \ord_{s=n-r} \zeta (X, s), \\
              \zeta^* (\AA^r_X, n)          & = \zeta^* (X, n-r).
          \end{align*}
\end{enumerate}

This suggests that Conjectures~$\mathbf{VO} (X,n)$ and $\mathbf{C} (X,n)$
should also satisfy the corresponding compatibilities. We verify in this
section that this is indeed the case.

\begin{lemma}
    \label{lemma:compatibility-of-Lc(X,n)}
    Let $n < 0$.

    \begin{enumerate}
        \item[1)] If $X = \coprod_{1 \le i \le r} X_i$ is a finite disjoint union of
              arithmetic schemes, then
              \begin{equation*}
                  \mathbf{L}^c (X_\et,n) \iff \mathbf{L}^c (X_{i,\et},n)\text{ for all }i.
              \end{equation*}

        \item[2)] For a closed-open decomposition
              $Z \not\hookrightarrow X \hookleftarrow U$, if two of the three conjectures
              \begin{equation*}
                  \mathbf{L}^c (X_\et,n), \quad
                  \mathbf{L}^c (Z_\et,n), \quad
                  \mathbf{L}^c (U_\et, n)
              \end{equation*}
              are true, then the third is also true.

        \item[3)] For an arithmetic scheme $X$ and any $r \ge 0$, one has
              \begin{equation*}
                  \mathbf{L}^c (\AA^r_{X,\et}, n) \iff \mathbf{L}^c (X_\et, n-r).
              \end{equation*}
    \end{enumerate}

    \begin{proof}
        See the proof of \cite[Proposition~5.10]{Morin-2014}.
    \end{proof}
\end{lemma}

\begin{lemma}
    \label{lemma:compatibility-of-B(X,n)}
    Let $n < 0$.

    \begin{enumerate}
        \item[1)] If $X = \coprod_{1 \le i \le r} X_i$ is a finite disjoint union of
              arithmetic schemes, then
              \begin{multline*}
                  \Reg_{X,n} = \bigoplus_{1 \le i \le r} \Reg_{X_i,n}\colon\\
                  \bigoplus_{1 \le i \le r} R\Gamma (X_{i,\et}, \RR^c (n)) \to
                  \bigoplus_{i \le i \le r} R\Gamma_\BM (G_\RR, X_i (\CC), \RR (n)) [1].
              \end{multline*}
              In particular,
              \begin{equation*}
                  \mathbf{B} (X,n) \iff \mathbf{B} (X_i,n)\text{ for all }i.
              \end{equation*}

        \item[2)] For a closed-open decomposition of arithmetic schemes
              $Z \not\hookrightarrow X \hookleftarrow U$, the corresponding regulators
              give a morphism of distinguished triangles

              \begin{equation*}
                  \begin{tikzcd}[column sep=4em]
                      R\Gamma (Z_\et, \RR^c (n)) \ar{d}\ar{r}{\Reg_{Z,n}} & R\Gamma_\BM (G_\RR, Z (\CC), \RR (n)) [1] \ar{d} \\
                      R\Gamma (X_\et, \RR^c (n)) \ar{d}\ar{r}{\Reg_{X,n}} & R\Gamma_\BM (G_\RR, X (\CC), \RR (n)) [1] \ar{d} \\
                      R\Gamma (U_\et, \RR^c (n)) \ar{d}\ar{r}{\Reg_{U,n}} & R\Gamma_\BM (G_\RR, U (\CC), \RR (n)) [1] \ar{d} \\
                      R\Gamma (Z_\et, \RR^c (n)) [1]\ar{r}{\Reg_{Z,n} [1]} & R\Gamma_\BM (G_\RR, Z (\CC), \RR (n)) [2]
                  \end{tikzcd}
              \end{equation*}
              In particular, if two of the three conjectures
              \begin{equation*}
                  \mathbf{B} (X,n), \quad
                  \mathbf{B} (Z,n), \quad
                  \mathbf{B} (U,n)
              \end{equation*}
              are true, then the third is also true.

        \item[3)] For any $r \ge 0$, the diagram
              \begin{equation*}
                  \begin{tikzcd}
                      R\Gamma (X_\et, \RR^c (n-r)) [2r] \ar{d}{\Reg_{X,n-r}}\ar{r}{\cong} & R\Gamma (\AA^r_{X,\et}, \RR^c (n))\ar{d}{\Reg_{\AA^r_X,n}} \\
                      R\Gamma_\BM (G_\RR, X (\CC), \RR (n-r)) [2r] \ar{r}{\cong} & R\Gamma_\BM (G_\RR, \AA^r_X (\CC), \RR (n))
                  \end{tikzcd}
              \end{equation*}
              commutes. In particular, one has $$\mathbf{B} (\AA^r_X, n) \iff \mathbf{B} (X,
                  n-r).$$
    \end{enumerate}

    \begin{proof}
        Part 1) is clear because all cohomologies that enter the definition of
        $\Reg_{X,n}$ decompose into direct sums over $i = 1,\ldots r$.  Parts 2) and
        3) boil down to the corresponding functoriality properties for the KLM
        morphism \eqref{eqn:KLM-morphism-1}, namely that it commutes with proper
        pushforwards and flat pullbacks by
        \cite[Lemma~3 and~4]{Weisschuh-2017}. For closed-open decompositions, the
        distinguished triangle
        \begin{equation*}
            R\Gamma (Z_\et, \RR^c (n)) \to R\Gamma (X_\et, \RR^c (n)) \to
            R\Gamma (U_\et, \RR^c (n)) \to R\Gamma (Z_\et, \RR^c (n)) [1]
        \end{equation*}
        comes exactly from the proper pushforward along $Z \hookrightarrow X$ and flat
        pullback along $U \hookrightarrow X$ (see \cite[Corollary~7.2]{Geisser-2010}
        and \cite[\S 3]{Bloch-1986}). Similarly, the quasi-isomorphism $R\Gamma (X_\et,
            \RR^c (n-r)) [2r] \cong R\Gamma (\AA^r_{X,\et}, \RR^c (n))$ results from the
        flat pullback along $p\colon \AA^r_X \to X$.
    \end{proof}
\end{lemma}

\begin{proposition}
    \label{prop:compatibility-of-VO(X,n)}
    For each arithmetic scheme $X$ below and $n < 0$, assume
    $\mathbf{L}^c (X_\et,n)$, $\mathbf{B} (X,n)$, and the meromorphic continuation
    of $\zeta (X,s)$ around $s = n$.

    \begin{enumerate}
        \item[1)] If $X = \coprod_{1 \le i \le r} X_i$ is a finite disjoint union of
              arithmetic schemes, then
              \begin{equation*}
                  \mathbf{VO} (X,n) \iff \mathbf{VO} (X_i,n)\text{ for all }i.
              \end{equation*}

        \item[2)] For a closed-open decomposition
              $Z \not\hookrightarrow X \hookleftarrow U$,
              if two of the three conjectures
              \begin{equation*}
                  \mathbf{VO} (X,n), \quad
                  \mathbf{VO} (Z,n), \quad
                  \mathbf{VO} (U,n)
              \end{equation*}
              are true, then the third is also true.

        \item[3)] For any $r \ge 0$, one has
              \begin{equation*}
                  \mathbf{VO} (\AA^r_X, n) \iff \mathbf{VO} (X, n-r).
              \end{equation*}
    \end{enumerate}

    \begin{proof}
        We have already observed in Proposition~\ref{prop:VO(X,n)-assuming-B(X,n)}
        that under Conjecture~$\mathbf{B} (X,n)$ we can rewrite $\mathbf{VO} (X,n)$
        as
        \begin{equation*}
            \ord_{s=n} \zeta (X,s) = \chi (R\Gamma_c (G_\RR, X(\CC), \RR (n))).
        \end{equation*}

        In part 1), we have $$\ord_{s=n} \zeta (X,s) = \sum_{1 \le i \le r} \ord_{s=n}
            \zeta (X_i,s),$$ and for the corresponding $G_\RR$-equivariant cohomology,
        \begin{equation*}
            R\Gamma_c (G_\RR, X(\CC), \RR (n)) =
            \bigoplus_{1 \le i \le r} R\Gamma_c (G_\RR, X(\CC), \RR (n)).
        \end{equation*}
        The statement follows from the additivity of the Euler characteristic:
        \begin{equation*}
            \begin{tikzcd}[column sep=5em]
                \ord_{s=n} \zeta (X,s) \ar[equals]{r}{\mathbf{VO} (X,n)}\ar[equals]{d} & \chi (R\Gamma_c (G_\RR, X(\CC), \RR (n))) \ar[equals]{d} \\
                \sum\limits_{1 \le i \le r} \ord_{s=n} \zeta (X_i,s) \ar[equals]{r}{\forall i \mathbf{VO} (X_i,n)} & \sum\limits_{1 \le i \le r} \chi (R\Gamma_c (G_\RR, X_i (\CC), \RR (n)))
            \end{tikzcd}
        \end{equation*}

        Similarly in part 2), we can consider the distinguished triangle
        \begin{multline*}
            R\Gamma_c (G_\RR, U (\CC), \RR (n)) \to
            R\Gamma_c (G_\RR, X (\CC), \RR (n)) \to
            R\Gamma_c (G_\RR, Z (\CC), \RR (n)) \\
            \to R\Gamma_c (G_\RR, U (\CC), \RR (n)) [1]
        \end{multline*}
        and the additivity of the Euler characteristic gives
        \begin{equation*}
            \begin{tikzcd}[column sep=4em]
                \ord_{s=n} \zeta (X,s) \ar[equals]{r}{\mathbf{VO} (X,n)}\ar[equals]{d} & \chi (R\Gamma_c (G_\RR, X(\CC), \RR (n))) \ar[equals]{d} \\
                \ord_{s=n} \zeta (Z,s) \ar[equals]{r}{\mathbf{VO} (Z,n)} & \chi (R\Gamma_c (G_\RR, Z (\CC), \RR (n))) \\[-2em]
                + & + \\[-2em]
                \ord_{s=n} \zeta (U,s) \ar[equals]{r}{\mathbf{VO} (U,n)} & \chi (R\Gamma_c (G_\RR, U (\CC), \RR (n)))
            \end{tikzcd}
        \end{equation*}

        Finally, in part 3), we assume for simplicity that $X_\CC$ is connected of
        dimension $d_\CC$. Then the Poincar\'{e} duality and homotopy invariance of the
        usual cohomology without compact support give us
        \begin{multline*}
            R\Gamma_c (G_\RR, \AA^r (\CC) \times X (\CC), \RR (n)) \\
            \stackrel{\text{P.D.}}{\cong}
            \RHom (R\Gamma (G_\RR, \AA^r (\CC) \times X (\CC), \RR (d_\CC + r - n)), \RR) [-2d_\CC - 2r] \\
            \stackrel{\text{H.I.}}{\cong}
            \RHom (R\Gamma (G_\RR, X (\CC), \RR (d_\CC + r - n)), \RR) [-2d_\CC - 2r] \\
            \stackrel{\text{P.D.}}{\cong}
            R\Gamma_c (G_\RR, X (\CC), \RR (n - r)) [-2r].
        \end{multline*}
        The twist $[-2r]$ is even and therefore has no effect on the Euler
        characteristic, so that we obtain
        \begin{equation*}
            \begin{tikzcd}[column sep=4em]
                \ord_{s=n} \zeta (\AA^r_X,s) \ar[equals]{r}{\mathbf{VO} (\AA^r_X,n)}\ar[equals]{d} & \chi (R\Gamma_c (G_\RR, \AA^r (\CC) \times X(\CC), \RR (n))) \ar[equals]{d} \\
                \ord_{s=n-r} \zeta (X,s) \ar[equals]{r}{\mathbf{VO} (X,n-r)} & \chi (R\Gamma_c (G_\RR, X (\CC), \RR (n-r)))
            \end{tikzcd}
        \end{equation*}
    \end{proof}
\end{proposition}

Our next goal is to prove similar compatibilities for Conjecture~$\mathbf{C}
    (X,n)$, as was just done for $\mathbf{VO} (X,n)$. We split the proof into three
technical lemmas \ref{lemma:lambda-and-disjoint-unions},
\ref{lemma:lambda-and-closed-open-decompositions},
\ref{lemma:lambda-and-affine-bundles}, each for the corresponding
compatibility.

\begin{lemma}
    \label{lemma:lambda-and-disjoint-unions}
    Let $n < 0$ and let $X = \coprod_{1 \le i \le r} X_i$ be a finite disjoint
    union of arithmetic schemes. Assume $\mathbf{L}^c (X_\et,n)$ and
    $\mathbf{B} (X,n)$. Then there is a quasi-isomorphism of complexes
    \begin{equation}
        \label{eqn:RGamma-Wc-and-disjoint-unions}
        \bigoplus_{1 \le i \le r} R\Gamma_\Wc (X_i, \ZZ(n)) \cong
        R\Gamma_\Wc (X, \ZZ(n)),
    \end{equation}
    which after taking the determinants gives a commutative diagram

    \begin{equation}
        \label{eqn:lambda-and-disjoint-unions}
        \begin{tikzcd}
            \RR \otimes_\RR \cdots \otimes_\RR \RR\ar{d}{\lambda_{X_1,n}\otimes\cdots\otimes\lambda_{X_r,n}}[swap]{\cong} \ar{r}{x_1\otimes\cdots\otimes x_r \mapsto x_1\cdots x_r}[swap]{\cong} & \RR \ar{d}{\lambda_{X,n}}[swap]{\cong} \\
            \bigotimes\limits_{1 \le i \le r} (\det_\ZZ R\Gamma_\Wc (X_i, \ZZ(n))) \otimes \RR \ar{r}{\cong} & (\det_\ZZ R\Gamma_\Wc (X_i, \ZZ(n))) \otimes \RR
        \end{tikzcd}
    \end{equation}

    \begin{proof}
        For $X = \coprod_{1 \le i \le r} X_i$, all cohomologies in our construction
        of $R\Gamma_\Wc (X, \ZZ(n))$ in \cite{Beshenov-Weil-etale-1} decompose into
        the corresponding direct sum over $i = 1,\ldots,r$, and
        \eqref{eqn:RGamma-Wc-and-disjoint-unions} follows.

        After tensoring with $\RR$, we obtain a commutative diagram

        \begin{equation*}
            \begin{tikzcd}[column sep=1.25em]
                \bigoplus_i \left(\!\!\!\begin{array}{c} R\Gamma_c (G_\RR, X_i (\CC), \RR (n)) [-2] \\ \oplus \\ R\Gamma_c (G_\RR, X_i (\CC), \RR (n)) [-1] \end{array}\!\!\!\right) \ar{d}{\bigoplus_i \Reg_{X_i,n}^\vee [-1] \oplus id}[swap]{\cong} \ar{r}{\cong} & \begin{array}{c} R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-2] \\ \oplus \\ R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1] \end{array} \ar{d}{\Reg_{X,n}^\vee [-1] \oplus id}[swap]{\cong} \\
                \bigoplus_i \left(\!\!\!\begin{array}{c} \RHom (R\Gamma (X_{i,\et}, \ZZ^c (n)), \RR) [-1] \\ \oplus \\ R\Gamma_c (G_\RR, X_i (\CC), \RR (n)) [-1] \end{array}\!\!\!\right) \ar{d}{\text{split}}[swap]{\cong} \ar{r}{\cong} & \begin{array}{c} \RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR) [-1] \\ \oplus \\ R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1] \end{array} \ar{d}{\text{split}}[swap]{\cong} \\
                \bigoplus_i R\Gamma_\Wc (X_i, \RR (n)) \ar{r}{\cong} & R\Gamma_\Wc (X, \RR (n))
            \end{tikzcd}
        \end{equation*}
        Taking the determinants, we obtain \eqref{eqn:lambda-and-disjoint-unions}.
    \end{proof}
\end{lemma}

\begin{lemma}
    \label{lemma:lambda-and-closed-open-decompositions}
    Let $n < 0$ and let $Z \not\hookrightarrow X \hookleftarrow U$ be a
    closed-open decomposition of arithmetic schemes, such that the conjectures
    \begin{gather*}
        \mathbf{L}^c (U_\et,n), ~ \mathbf{L}^c (X_\et,n), ~ \mathbf{L}^c (Z_\et,n),\\
        \mathbf{B} (U,n), ~ \mathbf{B} (X,n), ~ \mathbf{B} (Z_\et,n)
    \end{gather*}
    hold (it suffices to assume two of the three conjectures thanks to Lemmas
    \ref{lemma:compatibility-of-Lc(X,n)} and \ref{lemma:compatibility-of-B(X,n)}).
    Then there is an isomorphism of determinants
    \begin{equation}
        \label{eqn:isomorphism-of-det-RGamma-Wc-for-closed-open-decompositions}
        \det_\ZZ R\Gamma_\Wc (U, \ZZ(n)) \otimes
        \det_\ZZ R\Gamma_\Wc (Z, \ZZ(n)) \cong
        \det_\ZZ R\Gamma_\Wc (X, \ZZ(n))
    \end{equation}
    making the following diagram commute up to signs:
    \begin{equation}
        \label{eqn:lambda-and-closed-open-decompositions}
        \begin{tikzcd}
            \RR \otimes_\RR \RR \ar{r}{x\otimes y \mapsto xy}\ar{d}{\lambda_{U,n} \otimes \lambda_{Z,n}}[swap]{\cong} & \RR\ar{d}{\lambda_{X,n}}[swap]{\cong} \\
            \begin{array}{c} (\det_\ZZ R\Gamma_\Wc (U, \ZZ(n)))\otimes \RR \\ \otimes_\RR \\ (\det_\ZZ R\Gamma_\Wc (Z, \ZZ(n)))\otimes \RR \end{array} \ar{r}{\cong} & (\det_\ZZ R\Gamma_\Wc (X, \ZZ(n))) \otimes \RR
        \end{tikzcd}
    \end{equation}

    \begin{proof}
        A closed-open decomposition $Z \not\hookrightarrow X \hookleftarrow U$
        induces the distinguished triangles
        \begin{equation*}
            \begin{tikzcd}[row sep=0pt,column sep=1em,font=\small]
                R\Gamma (Z_\et, \ZZ^c (n)) \ar{r} & R\Gamma (X_\et, \ZZ^c (n)) \ar{r} & R\Gamma (U_\et, \ZZ^c (n)) \ar{r} & {[1]} \\
                R\Gamma_c (U_\et, \ZZ (n)) \ar{r} & R\Gamma_c (X_\et, \ZZ (n)) \ar{r} & R\Gamma_c (Z_\et, \ZZ (n)) \ar{r} & {[1]} \\
                R\Gamma_c (G_\RR, U (\CC), \RR (n)) \ar{r} & R\Gamma_c (G_\RR, X (\CC), \RR (n)) \ar{r} & R\Gamma_c (G_\RR, Z (\CC), \RR (n)) \ar{r} & {[1]}
            \end{tikzcd}
        \end{equation*}
        The first triangle is \cite[Corollary~7.2]{Geisser-2010} and it means that
        $R\Gamma (-, \ZZ^c (n))$ behaves like Borel--Moore homology. The following two
        are the usual triangles for cohomology with compact support. These fit together
        in a commutative diagram shown in
        Figure~\ref{fig:RGamma-Wc-and-closed-open-decompositions}.
        Figure~\ref{fig:RGamma-Wc-and-closed-open-decompositions-otimes-Q} shows the
        same diagram tensored with $\RR$.

        In this diagram we start from the morphism of triangles $(\alpha_{U,n},
            \alpha_{X,n}, \alpha_{Z,n})$ and then take the corresponding cones $R\Gamma_\fg
            (-, \ZZ(n))$. By \cite[Proposition~5.6]{Beshenov-Weil-etale-1}, these cones are
        defined up to a \emph{unique} isomorphism in the derived category $\mathbf{D}
            (\ZZ)$, and the same argument shows that the induced morphisms of complexes
        \begin{equation}
            \label{eqn:triangle-RGamma-fg}
            R\Gamma_\fg (U, \ZZ(n)) \to
            R\Gamma_\fg (X, \ZZ(n)) \to
            R\Gamma_\fg (Z, \ZZ(n)) \to
            R\Gamma_\fg (U, \ZZ(n)) [1]
        \end{equation}
        are also well-defined
        (see \cite[Corollary~A.3]{Beshenov-Weil-etale-1}). A priori,
        \eqref{eqn:triangle-RGamma-fg} need not be a distinguished triangle, but we
        claim that it induces a long exact sequence in cohomology.

        To this end, note that tensoring the diagram with $\ZZ/m\ZZ$ gives us an
        isomorphism
        \begin{equation*}
            \begin{tikzcd}[column sep=1em,font=\small]
                R\Gamma_c (U_\et, \ZZ/m\ZZ (n)) \ar{r}\ar{d}{\cong} & R\Gamma_c (X_\et, \ZZ/m\ZZ (n)) \ar{r}\ar{d}{\cong} & R\Gamma_c (Z_\et, \ZZ/m\ZZ (n)) \ar{r}\ar{d}{\cong} & {[1]}\ar{d}{\cong} \\
                \begin{array}{c} R\Gamma_\fg (U, \ZZ (n)) \\ \otimes^\mathbf{L} \\ \ZZ/m\ZZ \end{array} \ar{r} & \begin{array}{c} R\Gamma_\fg (X, \ZZ (n)) \\ \otimes^\mathbf{L} \\ \ZZ/m\ZZ \end{array} \ar{r} & \begin{array}{c} R\Gamma_\fg (Z, \ZZ (n)) \\ \otimes^\mathbf{L} \\ \ZZ/m\ZZ \end{array} \ar{r} & {[1]}
            \end{tikzcd}
        \end{equation*}
        More generally, for each prime $p$ we can take the corresponding derived
        $p$-adic completions (see \cite{Bhatt-Scholze-2015} and
        \cite[Tag~091N]{Stacks-project})
        \begin{equation*}
            R\Gamma_\fg (-, \ZZ(n))^\wedge_p \dfn
            R\varprojlim_k (R\Gamma_\fg (-, \ZZ(n)) \otimes^\mathbf{L} \ZZ/p^k\ZZ),
        \end{equation*}
        which give us a distinguished triangle for each prime $p$
        \begin{equation*}
            R\Gamma_\fg (U, \ZZ(n))^\wedge_p \to
            R\Gamma_\fg (X, \ZZ(n))^\wedge_p \to
            R\Gamma_\fg (Z, \ZZ(n))^\wedge_p \to
            R\Gamma_\fg (U, \ZZ(n))^\wedge_p [1].
        \end{equation*}
        At the level of cohomology, there are natural isomorphisms
        \cite[Tag~0A06]{Stacks-project}
        \begin{equation*}
            H^i (R\Gamma_\fg (-, \ZZ(n))^\wedge_p) \cong
            H^i_\fg (-, \ZZ(n)) \otimes \ZZ_p.
        \end{equation*}
        In particular, for each $p$ there is a long exact sequence of cohomology groups
        \begin{multline*}
            \cdots \to H^i_\fg (U, \ZZ(n)) \otimes \ZZ_p \to
            H^i_\fg (X, \ZZ(n)) \otimes \ZZ_p \to
            H^i_\fg (Z, \ZZ(n)) \otimes \ZZ_p \\
            \to H^{i+1}_\fg (U, \ZZ(n)) \otimes \ZZ_p \to \cdots
        \end{multline*}
        induced by \eqref{eqn:triangle-RGamma-fg}. By finite generation of
        $H^i_\fg (-, \ZZ(n))$ and flatness of $\ZZ_p$ this implies that the sequence
        \begin{equation}
            \label{eqn:RGamma-fg-long-exact-sequence}
            \cdots \to H^i_\fg (U, \ZZ(n)) \to
            H^i_\fg (X, \ZZ(n)) \to
            H^i_\fg (Z, \ZZ(n)) \to
            H^{i+1}_\fg (U, \ZZ(n)) \to \cdots
        \end{equation}
        is exact.

        Now we consider the diagram
        \begin{equation*}
            \begin{tikzcd}[column sep=0.75em,font=\small]
                \tau_{\le m} R\Gamma_c (G_\RR, U (\CC), \ZZ (n))[-1] \ar{r}\ar{d} & R\Gamma_\Wc (U, \ZZ (n))\ar{d}\ar{r} & \tau_{\le m} R\Gamma_\fg (U,\ZZ(n)) \ar{d}\ar{r} & {[0]} \ar{d} \\
                \tau_{\le m} R\Gamma_c (G_\RR, X (\CC), \ZZ (n))[-1] \ar{r}\ar{d} & R\Gamma_\Wc (X, \ZZ (n))\ar{d}\ar{r} & \tau_{\le m} R\Gamma_\fg (X,\ZZ(n)) \ar{d}\ar{r} & {[0]} \ar{d} \\
                \tau_{\le m} R\Gamma_c (G_\RR, Z (\CC), \ZZ (n))[-1] \ar{r}\ar{d} & R\Gamma_\Wc (Z, \ZZ (n))\ar{d}\ar{r} & \tau_{\le m} R\Gamma_\fg (Z,\ZZ(n)) \ar{d}\ar{r} & {[0]} \ar{d} \\
                \tau_{\le m} R\Gamma_c (G_\RR, U (\CC), \ZZ (n)) \ar{r} & R\Gamma_\Wc (U, \ZZ (n)) [1] \ar{r} & \tau_{\le m} R\Gamma_\fg (U,\ZZ(n)) [1] \ar{r} & {[1]}
            \end{tikzcd}
        \end{equation*}
        Here we took truncations for $m$ big enough, as in the proof of
        Lemma~\ref{lemma:determinant-of-RGamma-Wc-well-defined}. There are canonical
        isomorphisms
        \begin{align*}
            \notag \det_\ZZ R\Gamma_\Wc (U, \ZZ(n))                    & \cong \begin{array}{c} \det_\ZZ (\tau_{\le m} R\Gamma_c (G_\RR, U (\CC), \ZZ (n)) [-1]) \\ \otimes \\ \det_\ZZ (\tau_{\le m} R\Gamma_\fg (U, \ZZ(n))), \end{array}  \\
            \\
            \notag \det_\ZZ R\Gamma_\Wc (X, \ZZ(n))                    & \cong \begin{array}{c} \det_\ZZ (\tau_{\le m} R\Gamma_c (G_\RR, X (\CC), \ZZ (n)) [-1]) \\ \otimes \\ \det_\ZZ (\tau_{\le m} R\Gamma_\fg (X, \ZZ(n))), \end{array}  \\
            \\
            \notag \det_\ZZ R\Gamma_\Wc (Z, \ZZ(n))                    & \cong \begin{array}{c} \det_\ZZ (\tau_{\le m} R\Gamma_c (G_\RR, Z (\CC), \ZZ (n)) [-1]) \\ \otimes \\ \det_\ZZ (\tau_{\le m} R\Gamma_\fg (Z, \ZZ(n))), \end{array}  \\
            \\
            \det_\ZZ (\tau_{\le m} R\Gamma_c (G_\RR, X (\CC), \ZZ(n))) & \cong \begin{array}{c} \det_\ZZ (\tau_{\le m} R\Gamma_c (G_\RR, U (\CC), \ZZ (n))) \\ \otimes \\ \det_\ZZ (\tau_{\le m} R\Gamma_c (G_\RR, Z (\CC), \ZZ(n))), \end{array}               \\
            \\
            \det_\ZZ (\tau_{\le m} R\Gamma_\fg (X, \ZZ(n)))            & \cong \begin{array}{c} \det_\ZZ (\tau_{\le m} R\Gamma_\fg (U, \ZZ (n))) \\ \otimes \\ \det_\ZZ (\tau_{\le m} R\Gamma_\fg (Z, \ZZ(n))).\end{array}
        \end{align*}
        The first four isomorphisms arise from the corresponding distinguished
        triangles, while the last isomorphism comes from the long exact sequence
        \eqref{eqn:RGamma-fg-long-exact-sequence}, which gives an isomorphism
        \begin{equation}\small
            \bigotimes_{i \le m}
            \Bigl(\det_\ZZ H^i_\fg (U, \ZZ(n))^{(-1)^i} \otimes
            \det_\ZZ H^i_\fg (X, \ZZ(n))^{(-1)^{i+1}} \otimes
            \det_\ZZ H^i_\fg (Z, \ZZ(n))^{(-1)^i}\Bigr) \cong \ZZ.
        \end{equation}
        We can rearrange the terms
        (at the expense of introducing a $\pm 1$ sign),
        to obtain
        \begin{multline*}
            \det_\ZZ (\tau_{\le m} R\Gamma_\fg (X, \ZZ(n))) \cong
            \bigotimes_{i \le m} \det_\ZZ H^i_\fg (X, \ZZ(n)) \cong \\
            \bigotimes_{i \le m} \det_\ZZ H^i_\fg (U, \ZZ(n)) \otimes
            \bigotimes_{i \le m} \det_\ZZ H^i_\fg (Z, \ZZ(n)) \cong \\
            \det_\ZZ (\tau_{\le m} R\Gamma_\fg (U, \ZZ(n))) \otimes
            \det_\ZZ (\tau_{\le m} R\Gamma_\fg (Z, \ZZ(n))).
        \end{multline*}

        All this gives us the desired isomorphism of integral determinants
        \eqref{eqn:isomorphism-of-det-RGamma-Wc-for-closed-open-decompositions}.

        Let us now consider the diagram with distinguished rows in
        Figure~\ref{fig:Regulators-and-closed-open-decompositions}. Here the three
        squares with the regulators involved commute thanks to
        Lemma~\ref{lemma:compatibility-of-B(X,n)}. Taking the determinants, we obtain
        \eqref{eqn:lambda-and-closed-open-decompositions}, by the compatibility with
        distinguished triangles.
    \end{proof}
\end{lemma}

\begin{remark}
    Morally, we expect that a closed-open decomposition induces a distinguished
    triangle of the form
    \begin{equation}
        \label{eqn:closed-open-decompositions-hypothetical-RGamma-triangle}
        R\Gamma_\Wc (U, \ZZ(n)) \to
        R\Gamma_\Wc (X, \ZZ(n)) \to
        R\Gamma_\Wc (Z, \ZZ(n)) \to [1].
    \end{equation}
    However, $R\Gamma_\Wc (X, \ZZ(n))$ is defined in \cite{Beshenov-Weil-etale-1}
    as a mapping fber of a morphism in $\mathbf{D} (\ZZ)$, so it is not quite
    functorial.

    We recall that in the usual derived (1-)category $\mathbf{D} (\mathcal{A})$,
    taking naively a ``cone of a morphism of distinguished triangles'' $$\begin{tikzpicture}[ampersand replacement=\&]
            \matrix(m)[matrix of math nodes, row sep=1.5em, column sep=1.5em,
                text height=1.5ex, text depth=0.25ex]{
                A^\bullet \& B^\bullet \& C^\bullet \& A^\bullet[1]                                                          \\
                A^{\bullet\prime} \& B^{\bullet\prime} \& C^{\bullet\prime} \& A^{\bullet\prime} [1]                         \\
                A^{\bullet\prime\prime} \& B^{\bullet\prime\prime} \& C^{\bullet\prime\prime} \& A^{\bullet\prime\prime} [1] \\
                A^\bullet [1]  \& B^\bullet [1]  \& C^\bullet [1]  \& A^\bullet [2]\\};

            \path[->] (m-1-1) edge (m-1-2);
            \path[->] (m-1-2) edge (m-1-3);
            \path[->] (m-1-3) edge (m-1-4);

            \path[->] (m-2-1) edge (m-2-2);
            \path[->] (m-2-2) edge (m-2-3);
            \path[->] (m-2-3) edge (m-2-4);

            \path[dashed,->] (m-3-1) edge (m-3-2);
            \path[dashed,->] (m-3-2) edge (m-3-3);
            \path[dashed,->] (m-3-3) edge (m-3-4);

            \path[->] (m-4-1) edge (m-4-2);
            \path[->] (m-4-2) edge (m-4-3);
            \path[->] (m-4-3) edge (m-4-4);

            \path[->] (m-1-1) edge (m-2-1);
            \path[->] (m-1-2) edge (m-2-2);
            \path[->] (m-1-3) edge (m-2-3);
            \path[->] (m-1-4) edge (m-2-4);

            \path[dashed,->] (m-2-1) edge (m-3-1);
            \path[dashed,->] (m-2-2) edge (m-3-2);
            \path[dashed,->] (m-2-3) edge (m-3-3);
            \path[dashed,->] (m-2-4) edge (m-3-4);

            \path[dashed,->] (m-3-1) edge (m-4-1);
            \path[dashed,->] (m-3-2) edge (m-4-2);
            \path[dashed,->] (m-3-3) edge (m-4-3);
            \path[dashed,->] (m-3-4) edge (m-4-4);

            \node[font=\small] at ($(m-3-3)!.5!(m-4-4)$) {($-$)};
        \end{tikzpicture}$$
    usually \emph{does not} yield a distinguished triangle
    $A^{\bullet\prime\prime} \to B^{\bullet\prime\prime} \to C^{\bullet\prime\prime} \to A^{\bullet\prime\prime} [1]$.
    For a thorough discussion of this problem, see \cite{Neeman-1991}.

    For lack of a better definition for $R\Gamma_\Wc (X, \ZZ(n))$, we constructed
    the isomorphism
    \eqref{eqn:isomorphism-of-det-RGamma-Wc-for-closed-open-decompositions} ad hoc,
    without the hypothetical triangle
    \eqref{eqn:closed-open-decompositions-hypothetical-RGamma-triangle}.
\end{remark}

\begin{landscape}
    \begin{figure}
        \begin{equation*}
            \begin{tikzcd}[font=\small]
                &[2em] &[-2.5em] &[-2.5em] &[-2.5em] R\Gamma_\Wc (U, \ZZ (n))\ar{dl} &[-2.5em] \\
                \RHom (R\Gamma (U_\et, \ZZ^c (n)), \QQ[-2]) \ar{r}{\alpha_{U,n}}\ar{dd} & R\Gamma_c (U_\et, \ZZ (n)) \ar{dr}{u_\infty} \ar{dd} \ar{rr} & & R\Gamma_\fg (U, \ZZ (n))\ar{dl}[swap]{i_\infty} \ar[dashed]{dd}{\exists!} \ar[crossing over]{rr} & & \cdots [-1] \ar{dd} \\
                & & R\Gamma_c (G_\RR, U (\CC), \ZZ (n)) & & R\Gamma_\Wc (X, \ZZ (n))\ar{dl} \\
                \RHom(R\Gamma (X_\et, \ZZ^c (n)), \QQ[-2]) \ar{r}{\alpha_{X,n}}\ar{dd} & R\Gamma_c (X_\et, \ZZ (n)) \ar{dr}{u_\infty} \ar{dd} \ar{rr} & & R\Gamma_\fg (X, \ZZ (n)) \ar{dl}[swap]{i_\infty} \ar[dashed]{dd}{\exists!} \ar[crossing over]{rr} & & \cdots [-1] \ar{dd} \\
                & & R\Gamma_c (G_\RR, X (\CC), \ZZ (n)) \ar[<-,near end,crossing over]{uu} & & R\Gamma_\Wc (Z, \ZZ (n))\ar{dl} \\
                \RHom(R\Gamma (Z_\et, \ZZ^c (n)), \QQ[-2]) \ar{r}{\alpha_{Z,n}}\ar{dd} & R\Gamma_c (Z_\et, \ZZ (n)) \ar{dr}{u_\infty} \ar{dd} \ar{rr} & & R\Gamma_\fg (Z, \ZZ (n)) \ar{dl}[swap]{i_\infty} \ar[dashed]{dd}{\exists!} \ar[crossing over]{rr} & & \cdots [-1] \ar{dd} \\
                & & R\Gamma_c (G_\RR, Z (\CC), \ZZ (n)) \ar[<-,near end,crossing over]{uu} & & R\Gamma_\Wc (U, \ZZ (n)) [1]\ar{dl} \\
                \RHom(R\Gamma (U_\et, \ZZ^c (n)), \QQ[-1]) \ar{r}{\alpha_{U,n} [1]} & R\Gamma_c (U_\et, \ZZ (n)) [1] \ar{dr}{u_\infty} \ar{rr} & & R\Gamma_\fg (U, \ZZ (n)) [1] \ar{dl}[swap]{i_\infty [1]} \ar{rr} & & \cdots [0] \\
                & & R\Gamma_c (G_\RR, U (\CC), \ZZ (n)) [1] \ar[<-,near end,crossing over]{uu} \\
            \end{tikzcd}
        \end{equation*}

        \caption{Diagram induced by a closed-open decomposition
            $Z \not\hookrightarrow X \hookleftarrow U$.}
        \label{fig:RGamma-Wc-and-closed-open-decompositions}
    \end{figure}
\end{landscape}

\begin{landscape}
    \begin{figure}
        \begin{equation*}
            \begin{tikzcd}[column sep=0.65em, font=\small]
                &[2em] &[-2.5em] &[-2.5em] &[-2.5em] R\Gamma_\Wc (U, \RR (n))\ar{dl}\ar{dd} &[-2.5em] \\
                \RHom (R\Gamma (U_\et, \ZZ^c (n)), \RR[-2]) \ar{r}\ar{dd} & 0 \ar{dr} \ar{dd} \ar{rr} & & R\Gamma_\fg (U, \RR (n))\ar{dl}[swap]{0} \ar[dashed]{dd}{\exists!} \ar[crossing over,near start]{rr}{\cong} & & \RHom (R\Gamma (U_\et, \ZZ^c (n)), \RR[-1]) \ar{dd} \\
                & & R\Gamma_c (G_\RR, U (\CC), \RR (n)) & & R\Gamma_\Wc (X, \RR (n))\ar{dl}\ar{dd} \\
                \RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR[-2]) \ar{r}\ar{dd} & 0 \ar{dr} \ar{dd} \ar{rr} & & R\Gamma_\fg (X, \RR (n)) \ar{dl}[swap]{0} \ar[dashed]{dd}{\exists!} \ar[crossing over,near start]{rr}{\cong} & & \RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR[-1]) \ar{dd} \\
                & & R\Gamma_c (G_\RR, X (\CC), \RR (n)) \ar[<-,near end,crossing over]{uu} & & R\Gamma_\Wc (Z, \RR (n))\ar{dl}\ar{dd} \\
                \RHom (R\Gamma (Z_\et, \ZZ^c (n)), \RR[-2]) \ar{r}\ar{dd} & 0 \ar{dr} \ar{dd} \ar{rr} & & R\Gamma_\fg (Z, \RR (n)) \ar{dl}[swap]{0} \ar[dashed]{dd}{\exists!} \ar[crossing over,near start]{rr}{\cong} & & \RHom (R\Gamma (Z_\et, \ZZ^c (n)), \RR[-1]) \ar{dd} \\
                & & R\Gamma_c (G_\RR, Z (\CC), \RR (n)) \ar[<-,near end,crossing over]{uu} & & R\Gamma_\Wc (U, \RR (n)) [1]\ar{dl} \\
                \RHom (R\Gamma (U_\et, \ZZ^c (n)), \RR[-1]) \ar{r} & 0 \ar{dr} \ar{rr} & & R\Gamma_\fg (U, \RR (n)) [1] \ar{dl}[swap]{0} \ar{rr}{\cong} & & \RHom (R\Gamma (U_\et, \ZZ^c (n)), \RR) \\
                & & R\Gamma_c (G_\RR, U (\CC), \RR (n)) [1] \ar[<-,near end,crossing over]{uu} \\
            \end{tikzcd}
        \end{equation*}

        \caption{Diagram induced by a closed-open decomposition
            $Z \not\hookrightarrow X \hookleftarrow U$, tensored with $\RR$.}
        \label{fig:RGamma-Wc-and-closed-open-decompositions-otimes-Q}
    \end{figure}
\end{landscape}

\begin{landscape}
    \begin{figure}
        \begin{equation*}
            \begin{tikzcd}[column sep=0.65em, font=\small]
                \begin{array}{c} R\Gamma_c (G_\RR, U (\CC), \RR (n)) [-2] \\ \oplus \\ R\Gamma_c (G_\RR, U (\CC), \RR (n)) [-1] \end{array} \ar{d}{\Reg_{U,n}^\vee [-1] \oplus id}[swap]{\cong} \ar{r} & \begin{array}{c} R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-2] \\ \oplus \\ R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1] \end{array} \ar{d}{\Reg_{X,n}^\vee [-1] \oplus id}[swap]{\cong} \ar{r} & \begin{array}{c} R\Gamma_c (G_\RR, Z (\CC), \RR (n)) [-2] \\ \oplus \\ R\Gamma_c (G_\RR, Z (\CC), \RR (n)) [-1] \end{array} \ar{d}{\Reg_{Z,n}^\vee [-1] \oplus id}[swap]{\cong} \ar{r} & \begin{array}{c} R\Gamma_c (G_\RR, U (\CC), \RR (n)) [-1] \\ \oplus \\ R\Gamma_c (G_\RR, U (\CC), \RR (n)) \end{array} \ar{d}{\Reg_{U,n}^\vee \oplus id}[swap]{\cong} \\
                \begin{array}{c} \RHom (R\Gamma (U_\et, \ZZ^c (n)), \RR) [-1] \\ \oplus \\ R\Gamma_c (G_\RR, U (\CC), \RR (n)) [-1] \end{array} \ar{d}{\text{split}}[swap]{\cong} \ar{r} & \begin{array}{c} \RHom (R\Gamma (X_\et, \ZZ^c (n)), \RR) [-1] \\ \oplus \\ R\Gamma_c (G_\RR, X (\CC), \RR (n)) [-1] \end{array} \ar{d}{\text{split}}[swap]{\cong} \ar{r} & \begin{array}{c} \RHom (R\Gamma (Z_\et, \ZZ^c (n)), \RR) [-1] \\ \oplus \\ R\Gamma_c (G_\RR, Z (\CC), \RR (n)) [-1] \end{array} \ar{d}{\text{split}}[swap]{\cong} \ar{r} & \begin{array}{c} \RHom (R\Gamma (U_\et, \ZZ^c (n)), \RR) \\ \oplus \\ R\Gamma_c (G_\RR, U (\CC), \RR (n)) \end{array} \ar{d}{\text{split}}[swap]{\cong} \\
                R\Gamma_\Wc (U, \RR (n)) \ar{r} & R\Gamma_\Wc (X, \RR (n)) \ar{r} & R\Gamma_\Wc (Z, \RR (n)) \ar{r} & R\Gamma_\Wc (U, \RR (n)) [1]
            \end{tikzcd}
        \end{equation*}

        \caption{Diagram induced by a closed-open decomposition
            $Z \not\hookrightarrow X \hookleftarrow U$}
        \label{fig:Regulators-and-closed-open-decompositions}
    \end{figure}
\end{landscape}

\begin{lemma}
    \label{lemma:lambda-and-affine-bundles}
    For $n < 0$ and $r \ge 0$, let $X$ be an arithmetic scheme satisfying
    $\mathbf{L}^c (X_\et,n-r)$ and $\mathbf{B} (X,n-r)$. Then there is a natural
    quasi-isomorphism of complexes
    \begin{equation}
        \label{eqn:RGamma-Wc-and-affine-bundles}
        R\Gamma_\Wc (\AA^r_X, \ZZ (n)) \cong R\Gamma_\Wc (X, \ZZ (n-r)) [-2r],
    \end{equation}
    which after passing to the determinants makes the following diagram commute:
    \begin{equation}
        \label{eqn:lambda-and-affine-bundles}
        \begin{tikzcd}[column sep=0.5em]
            & \RR\ar{dl}{\cong}[swap]{\lambda_{\AA^r_X,n}}\ar{dr}{\lambda_{X,n-r}}[swap]{\cong} \\
            (\det_\ZZ R\Gamma_\Wc (\AA^r_X, \ZZ (n)))\otimes \RR \ar{rr}{\cong} & & (\det_\ZZ R\Gamma_\Wc (X, \ZZ (n-r)))\otimes \RR
        \end{tikzcd}
    \end{equation}

    \begin{proof}
        We refer to Figure~\ref{fig:RGamma-Wc-and-affine-bundles}, which shows how the flat
        morphism $p\colon \AA^r_X \to X$ induces the desired quasi-isomorphism
        \eqref{eqn:RGamma-Wc-and-affine-bundles}. It all boils down to the homotopy
        property of motivic cohomology, namely the fact that $p$ induces a
        quasi-isomorphism
        \begin{equation*}
            p^*\colon R\Gamma (X_\et, \ZZ^c (n-r)) [2r] \xrightarrow{\cong}
            R\Gamma (\AA^r_{X,\et}, \ZZ^c (n));
        \end{equation*}
        see, e.g. \cite[Lemma~5.11]{Morin-2014}. After passing to real coefficients, we
        obtain the following diagram:
        \begin{equation*}
            \begin{tikzcd}[column sep=1em, font=\small]
                \begin{array}{c} R\Gamma_c (G_\RR, \AA^r_X (\CC), \RR (n)) [-2] \\ \oplus \\ R\Gamma_c (G_\RR, \AA^r_X (\CC), \RR (n)) [-1] \end{array} \ar{d}{\Reg_{\AA^r_X,n}^\vee [-1] \oplus id}[swap]{\cong} \ar{r}{\cong} & \begin{array}{c} R\Gamma_c (G_\RR, X (\CC), \RR (n-r)) [-2] [-2r] \\ \oplus \\ R\Gamma_c (G_\RR, X (\CC), \RR (n-r)) [-1] [-2r] \end{array} \ar{d}{\Reg_{X,n-r}^\vee [-1] [-2r] \oplus id}[swap]{\cong} \\
                \begin{array}{c} \RHom (R\Gamma (\AA^r_{X,\et}, \ZZ^c (n)), \RR) [-1] \\ \oplus \\ R\Gamma_c (G_\RR, \AA^r_X (\CC), \RR (n)) [-1] \end{array} \ar{d}{\text{split}}[swap]{\cong} \ar{r}{\cong} & \begin{array}{c} \RHom (R\Gamma (X_\et, \ZZ^c (n-r)) [2r], \RR) [-1] \\ \oplus \\ R\Gamma_c (G_\RR, X (\CC), \RR (n-r)) [-1] [-2r] \end{array} \ar{d}{\text{split}}[swap]{\cong} \\
                R\Gamma_\Wc (\AA^r_X, \RR (n)) \ar{r}{\cong} & R\Gamma_\Wc (X, \RR (n-r)) [-2r]
            \end{tikzcd}
        \end{equation*}
        Here the first square is commutative due to the compatibility of the regulator
        with affine bundles (Lemma~\ref{lemma:compatibility-of-B(X,n)}), and the second
        square commutes because the quasi-isomorphism
        \eqref{eqn:RGamma-Wc-and-affine-bundles} gives compatible splittings (see again
        Figure~\ref{fig:RGamma-Wc-and-affine-bundles}. Taking the determinants, we
        obtain the desired commutative diagram \eqref{eqn:lambda-and-affine-bundles}.
    \end{proof}
\end{lemma}

\begin{landscape}
    \begin{figure}
        \begin{equation*}
            \begin{tikzcd}[font=\small,column sep=0.7em]
                &[0.5em] &[-2.75em] &[-2.75em] &[-2.75em] R\Gamma_\Wc (\AA^r_X, \ZZ (n))\ar{dl}\ar[dashed,near start]{dd}{\cong} &[-2.75em] \\
                \RHom (R\Gamma (\AA^r_{X,\et}, \ZZ^c (n)), \QQ[-2]) \ar{r}{\alpha_{\AA^r_X,n}}\ar{dd}{(p^*)^\vee}[swap]{\cong} & R\Gamma_c (\AA^r_{X,\et}, \ZZ (n)) \ar{dr}{u_\infty} \ar{dd}{p_*}[swap]{\cong} \ar{rr} & & R\Gamma_\fg (\AA^r_X, \ZZ (n))\ar{dl}[swap]{i_\infty} \ar[dashed]{dd}{\cong} \ar[crossing over]{rr} & & \cdots [-1] \ar{dd}{\cong} \\
                & & R\Gamma_c (G_\RR, \AA^r_X (\CC), \ZZ (n)) & & R\Gamma_\Wc (X, \ZZ (n-r)) [-2r]\ar{dl} \\
                \RHom (R\Gamma (X_\et, \ZZ^c (n-r)) [2r], \QQ[-2]) \ar[outer sep=0.75em]{r}{\alpha_{X,n-r} [-2r]} & R\Gamma_c (X_\et, \ZZ (n-r)) [-2r] \ar{dr}{u_\infty} \ar{rr} & & R\Gamma_\fg (X, \ZZ (n-r)) [-2r] \ar{dl}[swap]{i_\infty} \ar{rr} & & \cdots [-1] \\
                & & R\Gamma_c (G_\RR, X (\CC), \ZZ (n-r)) [-2r] \ar[<-,near end,crossing over]{uu}[swap]{p_*}{\cong}\\
                & & & & R\Gamma_\Wc (\AA^r_X, \RR (n))\ar{dl}\ar[dashed,near start]{dd}{\cong} & \\
                \RHom (R\Gamma (\AA^r_{X,\et}, \ZZ^c (n)), \RR[-2]) \ar{r}\ar{dd}{(p^*)^\vee}[swap]{\cong} & 0 \ar{dr} \ar{dd} \ar{rr} & & R\Gamma_\fg (\AA^r_X, \RR (n))\ar{dl}[swap]{0} \ar[dashed]{dd}{\cong} \ar[crossing over]{rr} & & \cdots [-1] \ar{dd}{\cong} \\
                & & R\Gamma_c (G_\RR, \AA^r_X (\CC), \RR (n)) & & R\Gamma_\Wc (X, \RR (n-r)) [-2r]\ar{dl} \\
                \RHom (R\Gamma (X_\et, \ZZ^c (n-r)) [2r], \RR[-2]) \ar{r} & 0 \ar{dr} \ar{rr} & & R\Gamma_\fg (X, \RR (n-r)) [-2r] \ar{dl}[swap]{0} \ar{rr} & & \cdots [-1] \\
                & & R\Gamma_c (G_\RR, X (\CC), \RR (n-r)) [-2r] \ar[<-,near end,crossing over]{uu}[swap]{p_*}{\cong}
            \end{tikzcd}
        \end{equation*}

        \caption{Isomorphism
            $R\Gamma_\Wc (\AA^r_X, \ZZ (n)) \cong R\Gamma_\Wc (X, \ZZ (n-r)) [-2r]$
            and its splitting after tensoring with $\RR$.}
        \label{fig:RGamma-Wc-and-affine-bundles}
    \end{figure}
\end{landscape}

\begin{theorem}
    \label{thm:compatibility-of-C(X,n)}
    For an arithmetic scheme $X$ and $n < 0$, assume $\mathbf{L}^c (X_\et,n)$,
    $\mathbf{B} (X,n)$, and the meromorphic continuation of $\zeta (X,s)$ around
    $s = n$.

    \begin{enumerate}
        \item[1)] If $X = \coprod_{1 \le i \le r} X_i$ is a finite disjoint union of
              arithmetic schemes, then
              \begin{equation*}
                  \mathbf{C} (X,n) \iff \mathbf{C} (X_i,n)\text{ for all }i.
              \end{equation*}

        \item[2)] For a closed-open decomposition
              $Z \not\hookrightarrow X \hookleftarrow U$, if two of three conjectures
              \begin{equation*}
                  \mathbf{C} (X,n), \quad
                  \mathbf{C} (Z,n), \quad
                  \mathbf{C} (U,n)
              \end{equation*}
              are true, then the third is also true.

        \item[3)] For any $r \ge 0$, one has
              \begin{equation*}
                  \mathbf{C} (\AA^r_X, n) \iff \mathbf{C} (X, n-r).
              \end{equation*}
    \end{enumerate}

    \begin{proof}
        Follows from Lemmas
        \ref{lemma:lambda-and-disjoint-unions},
        \ref{lemma:lambda-and-closed-open-decompositions},
        \ref{lemma:lambda-and-affine-bundles},
        together with the corresponding identities for the zeta functions
        \eqref{eqn:zeta-function-for-disjoint-unions},
        \eqref{eqn:zeta-function-for-closed-open-decompositions},
        \eqref{eqn:zeta-function-for-affine-space}.
    \end{proof}
\end{theorem}

The following is a special case of compatibility with closed-open
decompositions.

\begin{lemma}
    \label{lemma:compatibility-for-Xred}
    For an arithmetic scheme $X$ and $n < 0$, the conjectures
    \begin{equation*}
        \mathbf{L}^c (X_\et,n), ~
        \mathbf{B} (X,n), ~
        \mathbf{VO} (X,n), ~
        \mathbf{C} (X,n)
    \end{equation*}
    are equivalent to
    \begin{equation*}
        \mathbf{L}^c (X_\text{\itshape red,\'{e}t},n), ~
        \mathbf{B} (X_\red,n), ~
        \mathbf{VO} (X_\red,n), ~
        \mathbf{C} (X_\red,n)
    \end{equation*}
    respectively.

    \begin{proof}
        Apply Lemma~\ref{lemma:compatibility-of-Lc(X,n)},
        Lemma~\ref{lemma:compatibility-of-B(X,n)},
        Proposition~\ref{prop:compatibility-of-VO(X,n)}, and
        Theorem~\ref{thm:compatibility-of-C(X,n)} to the canonical closed embedding
        $X_\red \hookrightarrow X$.
    \end{proof}
\end{lemma}

The above lemma can be proved directly, by going through the construction of
Weil-\'{e}tale cohomology in \cite{Beshenov-Weil-etale-1} and the statements of
the conjectures. In particular, $$R\Gamma_\Wc (X, \ZZ(n)) \cong R\Gamma_\Wc
    (X_\red, \ZZ(n)).$$ It is important to note that the cycle complexes do not
distinguish $X$ from $X_\red$, and neither does the zeta function: $\zeta (X,s)
    = \zeta (X_\red,s)$.

\begin{remark}
    If $X/\FF_q$ is a variety over a finite field, then the proof of
    Theorem~\ref{thm:compatibility-of-C(X,n)} simplifies drastically: we can work
    with the formula \eqref{eqn:special-value-for-X/Fq} and the following
    properties of motivic cohomology:
    \begin{enumerate}
        \item[1)] $R\Gamma (\coprod_i X_{i,\et}, \ZZ^c (n)) \cong
                  \bigoplus_i R\Gamma (X_{i,\et}, \ZZ^c (n))$;

        \item[2)] triangles associated to closed-open decompositions
              \begin{equation*}
                  R\Gamma (Z_\et, \ZZ^c (n)) \to
                  R\Gamma (X_\et, \ZZ^c (n)) \to
                  R\Gamma (U_\et, \ZZ^c (n)) \to
                  R\Gamma (Z_\et, \ZZ^c (n))[1]
              \end{equation*}

        \item[3)] homotopy invariance
              $R\Gamma (X_\et, \ZZ^c (n-r)) [2r] \cong
                  R\Gamma (\AA^r_{X,\et}, \ZZ^c (n))$.
    \end{enumerate}
    In this case, no regulators are involved, so we do not need the technical
    lemmas \ref{lemma:lambda-and-disjoint-unions},
    \ref{lemma:lambda-and-closed-open-decompositions},
    \ref{lemma:lambda-and-affine-bundles}.
\end{remark}

If we consider the projective space $\PP_X^r = \PP_\ZZ^r \times X$, we have a
formula for the zeta function
\begin{equation}
    \label{eqn:zeta-function-for-projective-space}
    \zeta (\PP_X^r, s) = \prod_{0 \le i \le r} \zeta (X, s-i).
\end{equation}
Our special value conjecture satisfies the corresponding compatibility.

\begin{corollary}[Projective bundles]
    Let $X$ be an arithmetic scheme, $n < 0$, and $r \ge 0$.
    For all $0 \le i \le r$ assume Conjectures $\mathbf{L}^c (X_\et,n-i)$,
    $\mathbf{B} (X,n-i)$, and the meromorphic continuation of $\zeta (X,s)$ around
    $s = n-i$. Then
    \begin{equation*}
        \mathbf{C} (X,n-i)\text{ for }0 \le i \le r \Longrightarrow
        \mathbf{C} (\PP_X^r, n).
    \end{equation*}

    \begin{proof}
        Applied to the closed-open decomposition
        $\PP_X^{r-1} \not\hookrightarrow \PP_X^r \hookleftarrow \AA_X^r$,
        Theorem~\ref{thm:compatibility-of-C(X,n)} gives
        \begin{equation*}
            \mathbf{C} (X, n-r) \text{ and } \mathbf{C} (\PP_X^{r-1}, n)
            \Longrightarrow
            \mathbf{C} (\AA_X^r, n) \text{ and } \mathbf{C} (\PP_X^{r-1}, n)
            \Longrightarrow
            \mathbf{C} (\PP_X^r,n).
        \end{equation*}
        The assertion follows by induction on $r$. (The same inductive argument proves
        the identity \eqref{eqn:zeta-function-for-projective-space} from
        \eqref{eqn:zeta-function-for-affine-space}.)
    \end{proof}
\end{corollary}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 7
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Unconditional results}
\label{sec:unconditional-results}

Now we apply Theorem~\ref{thm:compatibility-of-C(X,n)} to prove the main
theorem stated in the introduction: the validity of $\mathbf{VO} (X,n)$ and
$\mathbf{C} (X,n)$ for all $n < 0$ for cellular schemes over certain
one-dimensional bases. In fact, we will construct an even larger class of
schemes $\mathcal{C} (\ZZ)$ whose elements satisfy the conjectures. This
approach is motivated by \cite[\S 5]{Morin-2014}.

\begin{definition}
    Let $\mathcal{C} (\ZZ)$ be the full subcategory of the category of arithmetic
    schemes generated by the following objects:
    \begin{itemize}
        \item the empty scheme $\emptyset$,
        \item $\Spec \FF_q$ for each finite field,
        \item $\Spec \mathcal{O}_F$ for an abelian number field $F/\QQ$,
        \item curves over finite fields $C/\FF_q$,
    \end{itemize}
    and the following operations.
    \begin{enumerate}
        \item[$\mathcal{C}0)$] $X$ is in $\mathcal{C} (\ZZ)$ if and only if $X_\red$
              is in $\mathcal{C} (\ZZ)$.

        \item[$\mathcal{C}1)$] A finite disjoint union $\coprod_{1 \le i \le r} X_i$
              is in $\mathcal{C} (\ZZ)$ if and only if each $X_i$ is in
              $\mathcal{C} (\ZZ)$.

        \item[$\mathcal{C}2)$] Let $Z \not\hookrightarrow X \hookleftarrow U$ be a
              closed-open decomposition such that $Z_{\red,\CC}$, $X_{\red,\CC}$,
              $U_{\red,\CC}$ are smooth and quasi-projective. If two of the three schemes
              $Z,X,U$ lie in $\mathcal{C} (\ZZ)$, then the third also lies in
              $\mathcal{C} (\ZZ)$.

        \item[$\mathcal{C}3)$] If $X$ lies in $\mathcal{C} (\ZZ)$, then the affine
              space $\AA^r_X$ for each $r \ge 0$ also lies in $\mathcal{C} (\ZZ)$.
    \end{enumerate}
\end{definition}

Recall that the condition that $X_{\red,\CC}$ is smooth and quasi-projective is
necessary to ensure that the regulator morphism exists (see
Remark~\ref{rmk:regulator-is-defined-for-XC-smooth-quasi-proj}).

\begin{proposition}
    \label{prop:C(X,n)-holds-for-C(Z)}
    Conjectures $\mathbf{VO} (X,n)$ and $\mathbf{C} (X,n)$ hold for any
    $X \in \mathcal{C} (\ZZ)$ and $n < 0$.

    \begin{proof}
        Finite fields satisfy $\mathbf{C} (X,n)$ by
        Example~\ref{example:C(X,n)-for-Spec-Fq}.

        If $X = \Spec \mathcal{O}_F$ for a number field $F/\QQ$, then
        Conjecture~$\mathbf{C} (X,n)$ is equivalent to the conjecture of Flach and
        Morin \cite[Conjecture~5.12]{Flach-Morin-2018}, which holds unconditionally for
        abelian $F/\QQ$, via reduction to the Tamagawa number conjecture; see \cite[\S
            5.8.3]{Flach-Morin-2018}, in particular [ibid., Proposition~5.35]. The
        condition $\mathbf{VO} (X,n)$ is also true in this case (see
        Example~\ref{example:VO(X,n)-for-number-rings}).

        If $X = C/\FF_q$ is a curve over a finite field, then $\mathbf{C} (X,n)$ holds
        thanks to Theorem~\ref{thm:C(X,n)-over-finite-fields}. Conjecture~$\mathbf{L}^c
            (X_\et,n)$ is known for curves and essentially goes back to Soul\'{e}; see, for
        example, \cite[Proposition~4.3]{Geisser-2017}.

        Finally, the fact that the Conjectures $\mathbf{L}^c (X_\et,n)$, $\mathbf{B}
            (X,n)$, $\mathbf{VO} (X,n)$, $\mathbf{C} (X,n)$ are closed under the operations
        $\mathcal{C}0)$--$\mathcal{C}3)$ is Lemma~\ref{lemma:compatibility-for-Xred},
        Lemma~\ref{lemma:compatibility-of-Lc(X,n)},
        Lemma~\ref{lemma:compatibility-of-B(X,n)},
        Proposition~\ref{prop:compatibility-of-VO(X,n)}, and
        Theorem~\ref{thm:compatibility-of-C(X,n)} respectively.
    \end{proof}
\end{proposition}

\begin{lemma}
    Any zero-dimensional arithmetic scheme $X$ is in $\mathcal{C} (\ZZ)$.

    \begin{proof}
        Since $X$ is a Noetherian scheme of dimension $0$, it is a finite disjoint
        union of $\Spec A_i$ for some Artinian local rings $A_i$. Thanks to
        $\mathcal{C}1)$, we can assume that $X = \Spec A$, and thanks to
        $\mathcal{C}0)$, we can assume that $X$ is reduced. But then $A = k$ is a
        field. Since $X$ is a scheme of finite type over $\Spec \ZZ$, we conclude
        that $X = \Spec \FF_q \in \mathcal{C} (\ZZ)$.
    \end{proof}
\end{lemma}

\begin{proposition}
    \label{prop:particular-cases-1-dim-base}
    Let $B$ be a one-dimensional arithmetic scheme. Suppose that each of the
    generic points $\eta \in B$ satisfies one of the following properties:
    \begin{enumerate}
        \item[a)] $\fchar \kappa (\eta) = p > 0$;

        \item[b)] $\fchar \kappa (\eta) = 0$, and $\kappa (\eta)/\QQ$ is an abelian number
              field.
    \end{enumerate}
    Then $B \in \mathcal{C} (\ZZ)$.

    \begin{proof}
        We verify that such a scheme can be obtained from $\Spec \mathcal{O}_F$ for
        an abelian number field $F/\QQ$, or a curve over a finite field $C/\FF_q$,
        using the operations $\mathcal{C}0)$, $\mathcal{C}1)$, $\mathcal{C}2)$ which
        appear in the definition of $\mathcal{C} (\ZZ)$.

        Thanks to $\mathcal{C}0)$, we can assume that $B$ is reduced. Consider the
        normalization $\nu\colon B' \to B$. This is a birational morphism, so there
        exist open dense subsets $U' \subseteq B'$ and $U \subseteq B$ such that
        $\left.\nu\right|_{U'}\colon U' \xrightarrow{\cong} U$. Now $B\setminus U$ is
        zero-dimensional, and therefore $B\setminus U \in \mathcal{C} (\ZZ)$ by the
        previous lemma. Thanks to $\mathcal{C}2)$, it suffices to check that $U' \in
            \mathcal{C} (\ZZ)$, and this would imply $B \in \mathcal{C} (\ZZ)$.

        Now $U'$ is a finite disjoint union of normal integral schemes, so according to
        $\mathcal{C}1)$ we can assume that $U'$ is integral. Consider the generic point
        $\eta \in U'$ and the residue field $F = \kappa (\eta)$. There are two cases to
        consider.

        \begin{enumerate}
            \item[a)] If $\fchar F = p > 0$, then $U'$ is a curve over a finite field, so it lies
                  in $\mathcal{C} (\ZZ)$.

            \item[b)] If $\fchar F = 0$, then by our assumptions, $F/\QQ$ is an abelian number
                  field.

                  We note that if $V' \subseteq U'$ is an affine open neighborhood of $\eta$,
                  then $U'\setminus V' \in \mathcal{C} (\ZZ)$ by the previous lemma. Therefore,
                  we can assume without loss of generality that $U'$ is affine.

                  We have $U' = \Spec \mathcal{O}$, where $\mathcal{O}$ is a finitely generated
                  integrally closed domain. This means that $\mathcal{O}_F \subseteq \mathcal{O}
                      = \mathcal{O}_{F,S}$ for a finite set of places $S$. Now $U' = \Spec
                      \mathcal{O}_F \setminus S$, and $S \in \mathcal{C} (\ZZ)$, so everything
                  reduces to the case of $U' = \Spec \mathcal{O}_F$, which is in $\mathcal{C}
                      (\ZZ)$. \qedhere
        \end{enumerate}
    \end{proof}
\end{proposition}

\begin{remark}
    Schemes like the above were considered by Jordan and Poonen in
    \cite{Jordan-Poonen-2020}, where the authors write down a special value
    formula for $s = 1$ that generalizes the classical class number
    formula. Namely, they consider the case where $B$ is reduced and affine,
    but without requiring $\kappa (\eta)/\QQ$ to be abelian.
\end{remark}

\begin{example}
    If $B = \Spec \mathcal{O}$ for a nonmaximal order
    $\mathcal{O} \subset \mathcal{O}_F$, where $F/\QQ$ is an abelian number field,
    then our formalism gives a cohomological interpretation of the special values
    of $\zeta_\mathcal{O} (s)$ at $s = n < 0$. This already seems to be a new
    result.
\end{example}

\begin{definition}
    \label{dfn:B-cellular-scheme}
    Let $X \to B$ be a $B$-scheme. We say that $X$ is \textbf{$B$-cellular} if it
    admits a filtration by closed subschemes
    \begin{equation}
        \label{eqn:cellular-decomposition}
        X = Z_N \supseteq Z_{N-1} \supseteq \cdots \supseteq Z_0 \supseteq Z_{-1} = \emptyset
    \end{equation}
    such that $Z_i\setminus Z_{i-1} \cong \coprod_j \AA^{r_{i_j}}_B$ is a finite
    union of affine $B$-spaces.
\end{definition}

For example, projective spaces $\PP^r_B$ and, in general, Grassmannians $\Gr
    (k,\ell)_B$ are cellular. Many interesting examples of cellular schemes arise
from actions of algebraic groups on varieties and the Bia\l{}ynicki-Birula
theorem; see \cite{Wendt-2010} and \cite{Brosnan-2005}.

\begin{proposition}
    \label{prop:cellular-schemes-in-C(Z)}
    Let $X$ be a $B$-cellular arithmetic scheme, where $B \in \mathcal{C} (\ZZ)$,
    and $X_{\red,\CC}$ is smooth and quasi-projective. Then
    $X \in \mathcal{C} (\ZZ)$.

    \begin{proof}
        Considering the corresponding cellular decomposition
        \eqref{eqn:cellular-decomposition}, we pass to open complements
        $U_i = X\setminus Z_i$ to obtain a filtration
        \begin{equation*}
            X = U_{-1} \supseteq U_1 \supseteq \cdots \supseteq U_{N-1} \supseteq U_N = \emptyset,
        \end{equation*}
        where $U_{i,\CC}$ are smooth and quasi-projective, being \emph{open}
        subvarieties in $X_\CC$. Now we have closed-open decompositions
        $\coprod_j \AA^{r_{i,j}}_B \not\hookrightarrow U_i \hookleftarrow U_{i+1}$,
        and the claim follows by induction on the length of the cellular
        decomposition, using operations $\mathcal{C}1)$--$\mathcal{C}3)$.
    \end{proof}
\end{proposition}

As a corollary of the above, we obtain the following result, stated in the
introduction.

\begin{theorem}
    \label{thm:main-result}
    Let $B$ be a one-dimensional arithmetic scheme satisfying the assumptions of
    Proposition~\ref{prop:particular-cases-1-dim-base}. If $X$ is a $B$-cellular
    arithmetic scheme with smooth and quasi-projective fiber $X_{\red,\CC}$, then
    Conjectures $\mathbf{VO} (X,n)$ and $\mathbf{C} (X,n)$ hold unconditionally
    for any $n < 0$.

    \begin{proof}
        Follows from Propositions
        \ref{prop:C(X,n)-holds-for-C(Z)},
        \ref{prop:particular-cases-1-dim-base},
        \ref{prop:cellular-schemes-in-C(Z)}.
    \end{proof}
\end{theorem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 8 / Appendix A
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Appendix A: Determinants of complexes}
\label{app:determinants}

Here we give a brief overview of the determinants of complexes. The original
construction goes back to Knudsen and Mumford \cite{Knudsen-Mumford-1976}, and
useful expositions can be found in
\cite[Appendix~A]{Gelfand-Kapranov-Zelevinsky-1994} and \cite[\S
    2.1]{Kato-1993}.

For our purposes, let $R$ be an integral domain.

\begin{definition}
    Denote by $\mathcal{P}_\is (R)$ the category of
    \textbf{graded invertible $R$-modules}. It has as objects $(L,r)$, where $L$
    is an invertible $R$-module (i.e. projective of rank $1$) and $r \in \ZZ$. The
    morphisms in this category are given by
    \begin{equation*}
        \Hom_{\mathcal{P}_\is (R)} ((L,r), (M,s)) = \begin{cases}
            \Isom_R (L,M), & r = s,   \\
            \emptyset,     & r \ne s.
        \end{cases}
    \end{equation*}
    This category is equipped with tensor products $$(L,r) \otimes_R (M,s) =
        (L\otimes_R M, r + s)$$ with (graded) commutativity isomorphisms
    \begin{equation*}
        (L,r) \otimes_R (M,s)
        \xrightarrow{\cong}
        (M,s) \otimes_R (L,r),
        \quad
        \ell \otimes m \mapsto (-1)^{rs}\,m\otimes \ell.
    \end{equation*}
\end{definition}

The unit object with respect to this product is $(R,0)$, and for each $(L,r)
    \in \mathcal{P}_\is (R)$ the inverse is given by $(L^{-1}, -r)$ where $L^{-1} =
    \iHom_R (L,R)$. The canonical evaluation morphism $L \otimes_R \iHom_R (L,R)
    \to R$ induces an isomorphism \begin{equation*}
    (L,r) \otimes_R (L^{-1}, -r) \cong (R,0).
\end{equation*}

\begin{definition}
    Denote by $\mathcal{C}_\is (R)$ the category whose objects are finitely
    generated projective $R$-modules and whose morphisms are isomorphisms.
    For $A \in \mathcal{C}_\is (R)$ we define the corresponding determinant by
    \begin{equation}
        \label{eqn:det-proj-module}
        \det_R (A) = \Bigl(\bigwedge^{\rk_R A}_R A, \rk_R A\Bigr) \in \mathcal{P}_\is (R).
    \end{equation}
    Here $\rk_R A$ is the rank of $A$, so that the top exterior power
    $\bigwedge^{\rk_R A}_R A$ is an invertible $R$-module.
\end{definition}

This yields a functor $\det_R\colon \mathcal{C}_\is (R) \to \mathcal{P}_\is
    (R)$. For $(L,r) \in \mathcal{P}_\is (R)$ we usually forget about $r$ and treat
the determinant as an invertible $R$-module.

The main result of \cite[Chapter~I]{Knudsen-Mumford-1976} is that this
construction can be generalized to complexes and morphisms in the derived
category.

\begin{definition}
    Let $\mathbf{D} (R)$ be the derived category of the category of
    $R$-modules. Recall that a complex $A^\bullet$ is \textbf{perfect} if it is
    quasi-isomorphic to a bounded complex of finitely generated projective
    $R$-modules. We denote by $\Parf_\is (R)$ the subcategory of $\mathbf{D} (R)$
    whose objects consist of perfect complexes, and whose morphisms are
    quasi-isomorphisms of complexes.
\end{definition}

\begin{theorem}[Knudsen--Mumford]
    The determinant \eqref{eqn:det-proj-module} extends to perfect complexes of
    $R$-modules as a functor
    \begin{equation*}
        \det_R\colon \Parf_\is (R) \to \mathcal{P}_\is (R),
    \end{equation*}
    satisfying the following properties.

    \begin{itemize}
        \item $\det_R (0) = (R,0)$.

        \item For a distinguished triangle of complexes in $\Parf_\is (R)$
              \begin{equation*}
                  A^\bullet \xrightarrow{u}
                  B^\bullet \xrightarrow{v}
                  C^\bullet \xrightarrow{w} A^\bullet [1]
              \end{equation*}
              there is a canonical isomorphism
              \begin{equation*}
                  i_R (u,v,w)\colon \det_R A^\bullet \otimes_R \det_R C^\bullet
                  \xrightarrow{\cong} \det_R B^\bullet.
              \end{equation*}

        \item In particular, there exist canonical isomorphisms
              \begin{equation*}
                  \det_R (A^\bullet \oplus B^\bullet) \cong
                  \det_R (A^\bullet) \otimes_R \det_R (B^\bullet).
              \end{equation*}

        \item For the triangles
              \begin{equation*}
                  \begin{tikzcd}[column sep=1em]
                      A^\bullet \ar{r}{id} & A^\bullet \ar{r} & 0^\bullet \ar{r} & A^\bullet [1] \\[-2em]
                      0^\bullet \ar{r} & A^\bullet \ar{r}{id} & A^\bullet \ar{r} & 0^\bullet [1]
                  \end{tikzcd}
              \end{equation*}
              the isomorphism $i_R$ comes from the canonical isomorphism $\det_R A^\bullet
                  \otimes_R (R,0) \cong \det_R A^\bullet$.

        \item For an isomorphism of distinguished triangles
              \begin{equation*}
                  \begin{tikzcd}
                      A^\bullet \ar{r}{u}\ar{d}{f}[swap]{\cong} & B^\bullet \ar{r}{v}\ar{d}{g}[swap]{\cong} & C^\bullet \ar{r}{w}\ar{d}{h}[swap]{\cong} & A^\bullet [1]\ar{d}{f[1]}[swap]{\cong} \\
                      A'^\bullet \ar{r}{u'} & B'^\bullet \ar{r}{v'} & C'^\bullet \ar{r}{w'} & A'^\bullet [1]
                  \end{tikzcd}
              \end{equation*}
              the diagram
              \begin{equation*}
                  \begin{tikzcd}[column sep=5em]
                      \det_R A^\bullet \otimes_R \det_R C^\bullet \ar{r}{i_R (u,v,w)}[swap]{\cong}\ar{d}{\det_R (f) \otimes \det_R (h)}[swap]{\cong} & \det_R B^\bullet\ar{d}{\det_R (g)}[swap]{\cong} \\
                      \det_R A'^\bullet \otimes_R \det_R C'^\bullet \ar{r}{i_R (u',v',w')}[swap]{\cong} & \det_R B'^\bullet
                  \end{tikzcd}
              \end{equation*}
              is commutative.

        \item The determinant is compatible with base change: given a ring homomorphism $R\to
                  S$, there is a natural isomorphism
              \begin{equation*}
                  \det_S (A^\bullet \otimes_R^\mathbf{L} S)
                  \xrightarrow{\cong}
                  (\det_R A^\bullet) \otimes_R S.
              \end{equation*}
              Moreover, this isomorphism is compatible with $i_R$ and $i_S$.

        \item If $A^\bullet$ is a bounded complex where each object $A^i$ is perfect (i.e.
              admits a finite length resolution by finitely generated projective
              $R$-modules), then $$\det_R A^\bullet \cong \bigotimes_{i\in \ZZ} (\det_R
                  A^i)^{(-1)^i}.$$ If each $A^i$ is already a finitely generated projective
              $R$-module, then $\det_R A^i$ in the above formula is given by
              \eqref{eqn:det-proj-module}.

        \item If the cohomology modules $H^i (A^\bullet)$ are perfect, then
              \begin{equation}
                  \label{eqn:det-in-terms-of-cohomology}
                  \det_R A^\bullet \cong
                  \bigotimes_{i\in \ZZ} (\det_R H^i (A^\bullet))^{(-1)^i}.
              \end{equation}
    \end{itemize}
\end{theorem}

We refer the reader to \cite{Knudsen-Mumford-1976} for the actual construction
and proofs. Here is a particularly simple case of interest.

\begin{lemma}
    \label{lemma:determinant-for-torsion-cohomology}
    When $R = \ZZ$ and cohomology groups $H^i (A^\bullet)$ are finite,
    we have the following.

    \begin{enumerate}
        \item[1)] If $A$ is a finite abelian group, then
              \begin{equation*}
                  (\det_\ZZ A) \subset (\det_\ZZ A) \otimes \QQ
                  \cong \det_\QQ (A \otimes \QQ) = \det_\QQ (0) \cong \QQ
              \end{equation*}
              corresponds to the fractional ideal $\frac{1}{\# A} \ZZ \subset \QQ$.

        \item[2)] In general, let $A^\bullet$ be a perfect complex of abelian groups
              such that the cohomology groups $H^i (A^\bullet)$ are all finite. Then
              $\det_\ZZ A^\bullet$ corresponds to the fractional ideal
              $\frac{1}{m}\,\ZZ \subset \QQ$, where
              \begin{equation*}
                  m = \prod_{i\in \ZZ} |H^i (A^\bullet)|^{(-1)^i}.
              \end{equation*}
    \end{enumerate}

    \begin{proof}
        Since $\det_\ZZ (A\oplus B) \cong \det_\ZZ A \otimes \det_\ZZ B$, in
        part 1) it suffices to consider the case of a cyclic group
        $A = \ZZ/m\ZZ$. Using the resolution
        \begin{equation*}
            \ZZ/m\ZZ [0] \cong \Bigl[
            \mathop{m\ZZ}_{\text{deg.\,}-1} \hookrightarrow
            \mathop{\ZZ}_{\text{deg.\,}0}
            \Bigr],
        \end{equation*}
        we calculate $$\det_\ZZ (\ZZ/m\ZZ) \cong \ZZ \otimes (m\ZZ)^{-1} \cong
            (m\ZZ)^{-1},$$ which corresponds to $\frac{1}{m}\,\ZZ$ in $\QQ$. Part 2)
        follows directly from 1) and \eqref{eqn:det-in-terms-of-cohomology}.
    \end{proof}
\end{lemma}

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