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Central extensions and the Riemann-Roch theorem on algebraic surfaces D. V. Osipovabc a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia b National Research University Higher School of Economics, Moscow, Russia c National University of Science and Technology «MISiS», Moscow, Russia
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     § 1. IntroductionIn this paper we consider locally free sheaves, adeles and the Riemann-Roch theorem on algebraic surfaces. But first we briefly recall the well-known case of algebraic curves. Recall that two vector subspaces $A$ and $B$ of a vector space $V$ over a field $k$ are commensurable (see [20]), that is, $A \sim B$, if and only if For such $A$ and $B$ we denote their relative dimension byWe fix a $k$-vector subspace $K$ in $V$ and any $k$-vector subspaces $D_1$ and $D_2$ in $V$ such that $D_1 \sim D_2$ and
$$
\begin{equation*}
H^0(D_i)=D_i \cap K \quad\text{and}\quad H^1(D_i)=V/ (D_i+K)
\end{equation*}
\notag
$$
are finite-dimensional $k$-vector spaces for $i \in \{1,2\}$. For $i \in \{ 1,2 \}$ we denote Then we have (see, for instance, [6], § 14.14, Exercises 14.59–14.61) ‘the abstract Riemann-Roch theorem’: Let $n \geqslant 1$ be an integer. For a smooth projective curve $S$ over $k$ we consider
$$
\begin{equation*}
V=\mathbb{A}_S^n, \quad\text{where } \mathbb{A}_S=\sideset{}{'}\prod_{p \in S} K_p;
\end{equation*}
\notag
$$
$\mathbb{A}_S$ is the space of adeles of the curve $S$, and $K_p$ is the field of fractions of the completion $\widehat{\mathcal{O}}_p$ of the local ring $\mathcal{O}_p$ at a (closed) point $p \in S$. We consider ${K = k(S)^n}$, where $k(S)$ is the field of rational functions on $S$. Now we consider a locally free sheaf $\mathcal{F}$ of $\mathcal{O}_S$-modules of rank $n$ on $S$. The stalk of $\mathcal{F}$ at the generic point $\operatorname{Spec} k(S)$ of $S$ is a $k(S)$-vector space. We fix a basis $e_0$ of this vector space. For any (closed) point $p \in S$, the completion of the stalk $\mathcal{F}_p$ of $\mathcal{F}$ at $p$ is a free $\widehat{\mathcal{O}}_p$-module. We fix a basis $e_p$ of this module. For all (closed) points $p \in S$, we consider the transition matrices $\gamma_{01,p} \in \mathrm{GL}_n(K_p)$ defined by the equality $e_0 = \gamma_{01,p} e_p$, which is calculated in the $K_p$-vector space $\mathcal{F}_p \otimes_{\mathcal{O}_p} K_p$. The element given by the collection of matrices
$$
\begin{equation*}
\gamma_{01, \mathcal F}=\prod_{p \in S} \gamma_{01, p}\in \mathrm{GL}_n\biggl(\prod_{p \in S} K_p\biggr)
\end{equation*}
\notag
$$
belongs to the subgroup $\mathrm{GL}_n(\mathbb{A}_S)$. We note that the chosen fixed basis $e_0$ gives an embedding of $\mathcal{F}$ into a constant sheaf $K$ on $S$. Therefore, we can associate with $\mathcal{F}$ the $k$-vector subspace $D_{\mathcal{F}}$ in $V$ explicitly given as where $D=\bigl(\prod_{p \in S} \widehat{\mathcal O}_p\bigr)^n$ is a $k$-vector subspace of $V$.Note that $D_{\mathcal{F}} \sim D$. Now, applying (1.1) to the $k$-vector subspaces $D_{\mathcal{F}}$ and $D$ and using the adelic complex for $\mathcal{F}$ on $S$, which gives the equalities
$$
\begin{equation*}
H^i(D_{\mathcal F})=H^i(S, \mathcal F) \quad\text{and}\quad H^i(D)=H^i(S, \mathcal O_S^n)
\end{equation*}
\notag
$$
for $i\in\{0,1\}$, we obtain the Riemann-Roch theorem for $\mathcal{F}$ on $S$:
$$
\begin{equation}
\chi(\mathcal F) -n \chi(\mathcal O_S)=c_1(\mathcal F),
\end{equation}
\tag{1.2}
$$
where $\chi (\mathcal{F})$ is the Euler characteristic of the sheaf $\mathcal{F}$ on $S$, and the first Chern number $c_1(\mathcal{F})$ is obtained from the homomorphism of groups:
$$
\begin{equation}
\deg\colon \mathrm{GL}_n(\mathbb{A}_S)\to \mathbb Z, \qquad a \mapsto [a D\mid D],
\end{equation}
\tag{1.3}
$$
such that $c_1(\mathcal{F}) = \deg(\gamma_{01, \mathcal{F}})$. This can be called the local (or adelic) decomposition for the difference of the Euler characteristics of the sheaves $\mathcal{F}$ and $\mathcal{O}_S^n$. Now let $X$ be a smooth projective algebraic surface over $k$. Then the Parshin-Beilinson adelic ring $\mathbb{A}_X$ of $X$ is defined (see [14], [1], [5], [8], [16] and § 2 below). We have
$$
\begin{equation*}
\mathbb{A}_X=\sideset{}{'}\prod_{x \in C} K_{x,C} \subset\prod_{x \in C} K_{x,C},
\end{equation*}
\notag
$$
where the (‘two-dimensional’) adelic product is taken over all pairs $x \in C$, where $C$ is an irreducible curve on $X$, $x$ is a point on $C$ and the Artinian ring $K_{x,C}$ is a finite direct product of two-dimensional local fields such that this product consists of one field, provided that $x$ is smooth on $C$. (Here $x$ is an ordinary closed point.) Every two-dimensional local field which appears here is isomorphic to the field of iterated Laurent series $k'((u))((t))$, where the field $k'$ is a finite extension of $k$. In addition, instead of the homomorphism (1.3), now we have the canonical central extension:
$$
\begin{equation*}
0 \to \mathbb Z \to \widetilde{\mathrm{GL}_n(\mathbb{A}_X)} \to \mathrm{GL}_n(\mathbb{A}_X) \to 1
\end{equation*}
\notag
$$
(see more in § 2.1 below). The goal of this paper is to connect this central extension with the Riemann-Roch theorem for a locally free sheaf of $\mathcal{O}_X$-modules of rank $n$ on $X$ using the transition matrices for this sheaf, where these transition matrices are obtained from the bases of the completions of the stalks of the sheaf at scheme points of $X$. From this central extension one obtains canonically another central extension $\widehat{\mathrm{GL}_n(\mathbb{A}_X)}$ (see § 2.2) of $\mathrm{GL}_n(\mathbb{A}_X)$ by $\mathbb{Z}$ such that from the transition matrices ${\alpha_{ij} \in \mathrm{GL}_n(\mathbb{A}_X)}$ ($i$ and $j$, $i \ne j$, are from $\{0,1,2\}$) for a rank $n$ locally free sheaf of $\mathcal{O}_X$-modules on $X$ and from the central extension $\widehat{\mathrm{GL}_n(\mathbb{A}_X)}$ it is possible to obtain the second Chern number $c_2$ of this sheaf. This was done in [11]; see also Remark 9. In addition, the transition matrices $\alpha_{ij}$ are analogues for a locally free sheaf of $\mathcal{O}_X$-modules on the surface $X$ of the transition matrix $\gamma_{01}$ for a locally free sheaf of $\mathcal{O}_S$-modules on the curve $S$; see the reasoning above. (We have omitted here the indication of the sheaf in the notation for transition matrices.) To solve the above tasks we use the canonical lifts of the transition matrices $\alpha_{ij}$ of a sheaf from the group $\mathrm{GL}_n(\mathbb{A}_X)$ to the groups $\widetilde{\mathrm{GL}_n(\mathbb{A}_X)}$ and $\widehat{\mathrm{GL}_n(\mathbb{A}_X)}$. Thus we obtain the local (or adelic) decomposition for the difference of the Euler characteristics of a rank $n$ locally free sheaf of $\mathcal{O}_X$-modules and the sheaf $\mathcal{O}_X^n$; see Remark 10. We can say also that the main ingredient of this paper is the calculation of the integer $\widetilde{\alpha_{02}} \cdot \widetilde{\alpha_{21}} \cdot \widetilde{\alpha_{10}}$, where $\widetilde{\alpha_{ij}}$ is the canonical lift of $\alpha_{ij}$ from $\mathrm{GL}_n\mkern-1mu(\mkern-1mu\mathbb{A}_X\mkern-2mu)$ to $\widetilde{\mathrm{GL}_n\mkern-1mu(\mkern-1mu\mathbb{A}_X\mkern-2mu)}$. This integer does not depend on the choice of the transition matrices $\alpha_{ij}$. We do this calculation in two ways. The first way leads to Theorem 1 and uses adelic complexes for rank $n$ locally free sheaves of $\mathcal{O}_X$-modules on $X$: see Proposition 1. For the second way we assume that the ground field $k$ is perfect, and we use the ‘self-duality’ of the adelic space $\mathbb{A}_X$, which is based on the reciprocity laws on $X$ for the residues of differential two-forms on two-dimensional local fields that were introduced and studied in [14]. This other way leads in Theorem 2 to an answer, which also uses other invariants of a sheaf and $X$. The comparison of these two answers (after Theorems 1 and 2) gives the Riemann-Roch theorem for a rank $n$ locally free sheaf of $\mathcal{O}_X$-modules on $X$ (without the Noether formula). We note that the relation of the Riemann-Roch theorem on an algebraic surface to local constructions was also discussed in [2], [4] and [17], but without using the adelic ring on a surface. The paper is organized as follows. In § 2.1 we very briefly remind ourselves about adeles on algebraic surfaces and also recall the construction of the central extension $\widetilde{\mathrm{GL}_n(\mathbb{A}_X)}$ of the group $\mathrm{GL}_n(\mathbb{A}_X)$ by $\mathbb{Z}$. In § 2.2 we recall the construction of the central extension $\widehat{\mathrm{GL}_n(\mathbb{A}_X)}$ of the group $\mathrm{GL}_n(\mathbb{A}_X)$ by $\mathbb{Z}$. This central extension is obtained from $\widetilde{\mathrm{GL}_n(\mathbb{A}_X)}$. In Remark 2 we compare these central extensions from the point of view of group cohomology. In § 3 we construct special elements in $\mathbb{Z}$-torsors. These elements are related to certain special $k$-vector subspaces and subrings of $\mathbb{A}_X$. These subspaces and subrings are connected with points and irreducible curves on $X$. In § 4 we discuss various formulae (formula (4.1) and Proposition 2) for the intersection index of divisors on $X$ related to the special elements constructed in § 3 and to the central extension $\widetilde{\mathrm{GL}_1(\mathbb{A}_X)}$. In § 5 we construct canonical splittings of the central extension $\widetilde{\mathrm{GL}_n(\mathbb{A}_X)}$ over special subgroups of the group $\mathrm{GL}_n(\mathbb{A}_X)$. We discuss the properties of these splittings. In Remark 7 we recall the corresponding splittings and their properties for the central extension $\widehat{\mathrm{GL}_n(\mathbb{A}_X)}$ from [11]. In § 6 we introduce the transition matrices $\alpha_{ij} \in \mathrm{GL}_n(\mathbb{A}_X)$ for a rank $n$ locally free sheaf $\mathcal{E}$ of $\mathcal{O}_X$-modules on $X$ and the canonical lifts $\widetilde{\alpha_{ij}}$ of $\alpha_{ij}$ to $\widetilde{\mathrm{GL}_n(\mathbb{A}_X)}$ by means of the canonical splittings from § 5. In § 7 we calculate the integer
$$
\begin{equation*}
f_{\mathcal E}=\widetilde{\alpha_{02}} \cdot \widetilde{\alpha_{21}} \cdot \widetilde{\alpha_{10}} =(\chi(\mathcal E)-n \chi(\mathcal O_X))-2 \mathrm{ch}_2(\mathcal E)
\end{equation*}
\notag
$$
in the first way, where the rational number $\mathrm{ch}_2(\mathcal{E})$ equals $\frac{1}{2}c_1(\mathcal{E})^2 -c_2(\mathcal{E})$. In § 8, when the field $k$ is perfect, we calculate the integer
$$
\begin{equation*}
f_{\mathcal E}=-\frac{1}{2} K \cdot c_1(\mathcal E)-\mathrm{ch}_2 (\mathcal E)
\end{equation*}
\notag
$$
in the second way, where $K \simeq \mathcal{O}_X(\omega)$, $\omega \in \Omega^2_{k(X)/k}$ and $\omega \ne 0$. We derive the Riemann-Roch theorem for $\mathcal{E}$. I am grateful to A. N. Parshin for some discussions and comments. The original motivation for this paper was a discussion with him that the adelic technique on an algebraic surface should imply the Riemann-Roch theorem. § 2. Constructions of central extensions of $\mathrm{GL}_n( {\mathbb{A}}_X)$2.1. The adelic ring on a surface and a first central extensionAs already mentioned in § 1, throughout the article $X$ is a smooth projective algebraic surface over a field $k$. But for the constructions in § 2 it is not important that $X$ is projective. We recall (see, for example, the survey [8] and also [11], § 2.1) that the adelic ring of $X$ is
$$
\begin{equation*}
\mathbb{A}_X=\mathbb{A}_X=\sideset{}{'}\prod_{x \in C} K_{x,C} \subset\prod_{x \in C} K_{x,C}
\end{equation*}
\notag
$$
and every ring $K_{x,C}= \prod_i K_i$ is the finite direct product of two-dimensional local fields $K_i$ such that every two-dimensional local field $K_i$ corresponds to a formal branch of $C$ at $x$. We define a subring $\mathbb{A}_{12} \subset \mathbb{A}_X$:
$$
\begin{equation*}
\mathbb{A}_{12}=\mathbb{A}_X \cap\prod_{x \in C} \mathcal O_{K_{x,C}} \subset\prod_{x \in C} K_{x,C},
\end{equation*}
\notag
$$
where $\mathcal{O}_{K_{x,C}} = \prod_i \mathcal{O}_{K_i}$ is the finite direct product of the discrete valuation rings $\mathcal{O}_{K_i}$ of $K_i$. (If $K_i$ is isomorphic to $k_i(\mkern-1mu(u)\mkern-1mu)((t))$, then $\mathcal{O}_{K_i}$ is isomorphic to $k_i(\mkern-1mu(\mkern-1mu u\mkern-1mu)\mkern-1mu)[[t]]$.) For any locally linearly compact $k$-vector space (or, in other words, a Tate vector space) $U$ we recall the canonical construction of the $\mathbb{Z}$-torsor $\operatorname{Dim}(U)$ from [7]. As a set, $\operatorname{Dim}(U)$ consists of all maps $d$ (which are called ‘dimension theories’ in [7]) from the set of all open linearly compact $k$-vector subspaces of $U$ to $\mathbb{Z}$ with the property where $Z_1$ and $Z_2$ are arbitrary open linearly compact $k$-vector subspaces of $U$. The group $\mathbb{Z}$ acts on $\operatorname{Dim}(U)$ in the following way:
$$
\begin{equation*}
(m+d)(Z_1)=d(Z_1)+m, \quad \text{where } m \in \mathbb Z.
\end{equation*}
\notag
$$
For any exact sequence of locally linearly compact $k$-vector spaces
$$
\begin{equation}
0 \to U_1 \xrightarrow{\phi_1} U_2 \xrightarrow{\phi_2} U_3 \to 0,
\end{equation}
\tag{2.1}
$$
where $\phi_1$ and $\phi_2$ are continuous maps and $\phi_1$ is a closed embedding, we have a canonical isomorphism:
$$
\begin{equation}
\begin{gathered} \, \operatorname{Dim}(U_1) \otimes_{\mathbb Z}\operatorname{Dim}(U_3) \to\operatorname{Dim}(U_2), \\ d_1 \otimes d_3 \mapsto d_2, \qquad d_2(Z)=d_1(Z \cap U_1)+d_3(\phi_2(U_1)) \notag, \end{gathered}
\end{equation}
\tag{2.2}
$$
where $Z$ is an open linearly compact $k$-vector subspace of $U_2$. For any locally linearly compact $k$-vector space $U$ we define the locally linearly compact $k$-vector space $\check{U}$ as a $k$-vector subspace of the dual $k$-vector space $U^*$ in the following way: where $W$ runs over all open linearly compact $k$-vector subspaces of $U$, the $k$-vector subspace $W^{\perp} \subset U^*$ is the annihilator of $W$ in $U^*$, and $W^{\perp}$, which is the dual vector space of the discrete vector space $U/W$, is an open linearly compact $k$-vector subspace of $\check{U}$. In other words, $\check{U}$ is the continuous dual space, that is, it consists of all continuous linear functionals. We have a canonical isomorphism:
$$
\begin{equation}
\operatorname{Dim}(U) \otimes_{\mathbb Z}\operatorname{Dim}(\check{U}) \simeq \mathbb Z, \qquad d_1 \otimes d_2 \mapsto d_1(Z)+d_2(Z^{\perp}),
\end{equation}
\tag{2.3}
$$
where $Z \subset U$ is a linearly compact $k$-vector subspace, and the result does not depend on the choice of $Z$. Let $D = \sum_{i} a_i C_i$ be a divisor on $X$, where the $C_i$ are irreducible curves on $X$. We denote
$$
\begin{equation*}
\mathbb{A}_{12}(D)=\mathbb{A}_X \cap\prod_{x \in C} t_C^{-\nu_C(D)} \mathcal O_{K_{x,C}},
\end{equation*}
\notag
$$
where the intersection is taken inside $\prod_{x \in C} K_{x,C}$, $t_C=0$ is an equation of an (irreducible) curve $C$ in some open subset of $X$, and $\nu_C(D)$ is equal to $a_i$ when $C = C_i$ and to zero otherwise. (The definition of $\mathbb{A}_{12}(D)$ does not depend on the choice of $t_C$.) We note that
$$
\begin{equation*}
\mathbb{A}_X=\varliminf_{D_2} \varliminf_{D_1 \geqslant D_2} \mathbb{A}_{12}(D_2)/\mathbb{A}_{12}(D_1),
\end{equation*}
\notag
$$
$\mathbb{A}_{12}(D_2) / \mathbb{A}_{12}(D_1)$ is a locally linearly compact $k$-vector space, and for any divisors $D_1 \geqslant D_2 \geqslant D_3$ on $X$ the exact sequence
$$
\begin{equation*}
0 \to \mathbb{A}_{12}(D_2)/\mathbb{A}_{12}(D_1) \to \mathbb{A}_{12}(D_3)/\mathbb{A}_{12}(D_1) \to \mathbb{A}_{12}(D_3)/ \mathbb{A}_{12}(D_2) \to 0
\end{equation*}
\notag
$$
is of type (2.1); see, for example, [9], § 2.2.3, or [11], § 2.3. Let $n \geqslant 1$ be an integer. Definition 1. A $k$-vector subspace $E$ of $\mathbb{A}_X^n$ is called a lattice if and only if there are divisors $D_1$ and $D_2$ on $X$ such that and the image of $E$ in $\mathbb{A}_{12}(D_2)^n/ \mathbb{A}_{12}(D_1)^n$ is a closed $k$-vector subspace. If $E_1 \subset E_2$ are lattices, then $E_2/ E_1$ is a locally linearly compact $k$-vector space with quotient topology and induced topology from the locally linearly compact $k$-vector space $ \mathbb{A}_{12}(D_2)^n/ \mathbb{A}_{12}(D_1)^n $. In this case we define the $\mathbb{Z}$-torsor
$$
\begin{equation*}
\operatorname{Dim}(E_1\mid E_2)=\operatorname{Dim}(E_2/E_1) .
\end{equation*}
\notag
$$
If $E_1 \subset E_2 \subset E_3$ are lattices, then the exact sequence is of type (2.1).Now, for arbitrary lattices $E_1$ and $E_2$ we define the $\mathbb{Z}$-torsor
$$
\begin{equation*}
\operatorname{Dim}(E_1\mid E_2)=\varliminf_E\operatorname{Hom}_{\mathbb Z} \bigl(\operatorname{Dim}(E_1/E),\operatorname{Dim}(E_2/E)\bigr),
\end{equation*}
\notag
$$
where the direct limit is taken over all lattices $E \subset \mathbb{A}_X^n$ such that $E \subset E_i$ for $i=1,2$. Here we use the following isomorphisms of $\mathbb{Z}$-torsors for lattices $E \supset E'$ and $i=1,2$:
$$
\begin{equation*}
\operatorname{Dim}(E_i/E) \otimes_{\mathbb Z}\operatorname{Dim}(E/E') \to\operatorname{Dim}(E_i/E'),
\end{equation*}
\notag
$$
so that the transition maps in this direct limit are given by
$$
\begin{equation*}
f \mapsto f', \qquad f'(a \otimes c)=f'(a) \otimes c.
\end{equation*}
\notag
$$
Obviously, for any lattices $E_1$, $E_2$ and $E_3$ we have a canonical isomorphism of $\mathbb{Z}$-torsors
$$
\begin{equation}
\operatorname{Dim}(E_1 \mid E_2) \otimes_{\mathbb Z}\operatorname{Dim}(E_2 \mid E_3) \to\operatorname{Dim}(E_1 \mid E_3),
\end{equation}
\tag{2.4}
$$
which satisfies the associativity diagram for four lattices. It is not difficult to see that for any $g \in \mathrm{GL}_n(\mathbb{A}_X)$ and any lattice $E$ the $k$-vector subspace $g E$ is also a lattice. For any lattices $E_1$ and $E_2$ we have an obvious isomorphism
$$
\begin{equation*}
\operatorname{Dim}(E_1 \mid E_2) \to\operatorname{Dim}(g E_1 \mid g E_2), \qquad d \mapsto g(d).
\end{equation*}
\notag
$$
All this leads to a construction of the central extension
$$
\begin{equation}
0 \to \mathbb Z \to \widetilde{\mathrm{GL}_n(\mathbb{A}_X)} \xrightarrow{\Theta} \mathrm{GL}_n(\mathbb{A}_X) \to 1,
\end{equation}
\tag{2.5}
$$
where $\widetilde{\mathrm{GL}_n(\mathbb{A}_X)}$ consists of all pairs $(g,d)$, where $g\mkern-2mu \in\mkern-2mu \mathrm{GL}_n(\mathbb{A}_X)$, ${d\mkern-2mu\in\mkern-2mu\operatorname{Dim}(\mathbb{A}_{12}^n\mkern-2mu \mid\mkern-2mu g \mathbb{A}_{12}^n)}$. The group operation and the map $\Theta$ are as follows:
$$
\begin{equation}
(g_1, d_1)(g_2, d_2)=(g_1 g_2, d_1 \otimes g_1(d_2)) \quad\text{and}\quad \Theta((g,d))=g.
\end{equation}
\tag{2.6}
$$
Remark 1. The above construction of the central extension $\widetilde{\mathrm{GL}_n(\mathbb{A}_X)}$ is a special case of a more general construction. Let $B$ be a $C_2$-space over $k$ (or, with a few more restrictions, a $2$-Tate vector space over $k$); see [9]. Then there is a canonical central extension 2.2. The second central extensionFrom (2.5) we obtain another central extension (also see more in [11], § 3.1). We note that
$$
\begin{equation*}
\mathrm{GL}_n(\mathbb{A}_X)=\mathrm{SL}_n(\mathbb{A}_X) \rtimes \mathbb{A}_X^*,
\end{equation*}
\notag
$$
where the group of invertible elements $\mathbb{A}_X^*$ of the ring $\mathbb{A}_X$ acts on $\mathrm{SL}_n(\mathbb{A}_X)$ by conjugations, that is, by inner automorphisms $h \mapsto aha^{-1}$, where $h \in \mathrm{SL}_n(\mathbb{A}_X)$ and we embed $\mathbb{A}_X^*$ into $\mathrm{GL}_n(\mathbb{A}_X)$ as $a \mapsto \operatorname{diag}(a,1,\dots,1)$. Now we define
$$
\begin{equation*}
\widehat{\mathrm{GL}_n(\mathbb{A}_X)}=\Theta^{-1}(\mathrm{SL}_n(\mathbb{A}_X)) \rtimes \mathbb{A}_X^*,
\end{equation*}
\notag
$$
where $\mathbb{A}_X^*$ acts on $\Theta^{-1}(\mathrm{SL}_n(\mathbb{A}_X))$ by inner automorphisms in the group $\widetilde{\mathrm{GL}_n(\mathbb{A}_X)}$ via lifting elements of $\mathbb{A}_X^*$ to $\widetilde{\mathrm{GL}_n(\mathbb{A}_X)}$, that is, $a(g)= a'ga'^{-1}$, where $g \in \Theta^{-1}(\mathrm{SL}_n(\mathbb{A}_X))$ and $a' \in \widetilde{\mathrm{GL}_n(\mathbb{A}_X)}$ is any element such that $\Theta(a')= \operatorname{diag}(a,1,\dots,1)$. Clearly, we obtain a central extension:
$$
\begin{equation}
0 \to \mathbb Z \to \widehat{\mathrm{GL}_n(\mathbb{A}_X)} {\to} \mathrm{GL}_n(\mathbb{A}_X) \to 1.
\end{equation}
\tag{2.7}
$$
Restricted to the subgroup $\mathrm{SL}_n(\mathbb{A}_X) \subset \mathrm{GL}_n(\mathbb{A}_X)$, this central extension coincides with the central extension $\widetilde{\mathrm{GL}_n(\mathbb{A}_X)}$ restricted to this subgroup. In addition, by construction, the central extension (2.7) splits canonically over the subgroup $\mathbb{A}_X^*$. Remark 2. We explain what the transition from (2.5) to (2.7) means in terms of group cohomology. Recall (see [3], § 1.7) that a central extension $\widehat{C}$ of a group $C = G \rtimes H$ by a group $A$ is equivalent to the following data:
Using this description and that $H$ is simultaneously a subgroup and a quotient group of $C$, we see that and that there is an exact sequence
$$
\begin{equation}
0 \to H^1(H,\operatorname{Hom}(G,A)) \to T \xrightarrow{\psi} H^2(G,A)^H \xrightarrow{\phi} H^2(H,\operatorname{Hom}(G,A)),
\end{equation}
\tag{2.10}
$$
which we will now explain. (Here the actions of $H$ on $\operatorname{Hom}(G,A)$ and on $H^2(G,A)$ are obtained in the usual way from the action on $G$.) We use that for any central extension the group $\operatorname{Hom}(G,A)$ is canonically isomorphic to the automorphism group of the central extension (2.11), that is, it is isomorphic to the group of automorphisms of $\widehat{G}$ such that these automorphisms induce the identical action on the subgroup $A$ and the quotient group $G$.Consider any central extension $I$ from $H^2(G,A)^H$. We have the group $L(I)$ which consists of the lifts of the actions of elements of $H$ to actions on the group corresponding to the central extension $I$, with the trivial action on the subgroup $A$. The group $L(I)$ is an extension (in general, noncentral) of $H$ by the automorphism group $\operatorname{Hom}(G,A)$ of the central extension $I$. The isomorphism class of this non-central extension is $\phi(I)$. If $\phi(I) =0 $, then the action of the whole group $H$ can be lifted to an action on the group corresponding to the central extension $I$, with the trivial action on the subgroup $A$. An element $t$ of $T$ is an isomorphism class of the following data: a central extension $\psi(t)$ together with a lift of the action of the group $H$ to an action on the group corresponding to this central extension, with the trivial action on the subgroup $A$. Correspondingly, for any central extension $I$ in $\operatorname{Ker} \phi$ we have that $\psi^{-1}(I)$ is the set of equivalence classes of group sections $s$ of the natural homomorphism $L(I) \to H$. Two group sections $s$ and $s'$ are equivalent if $s = \sigma_a s'$, where $\sigma_a$ is an inner automorphism of the group $L(I)$ given by an element $a$ of the automorphism group $\operatorname{Hom}(G,A)$ of the central extension $I$. Hence $\psi^{-1}(I)$ is an $H^1(H,\operatorname{Hom}(G,A))$-torsor. We note that another, computational construction and proof of exact sequence (2.10) and its continuation to the right were given in [19]. Now the transition from the central extension (2.5) to (2.7) is the projection of $H^2(C, A)$ onto $T$ in (2.9). We note that if a group $G$ is perfect, that is, $[G,G]=G$, then $\operatorname{Hom}(G,A)=0$. Therefore, in this case, from (2.9) and (2.10) we obtain This condition is satisfied, for example, when $C\!=\!\mathrm{GL}_n(F)$ and $H\!=\!F^*$, ${G \!=\! \mathrm{SL}_n(F)}$, where $F$ is an infinite field (for example, $F$ is an $n$-dimensional local field). Then we have
$$
\begin{equation*}
H^2(\mathrm{SL}_n(F), A)^{F^*}=\operatorname{Hom}(H_2(\mathrm{SL}_n(F), \mathbb Z)_{F^*}, A)=\operatorname{Hom}(K_2(F), A)
\end{equation*}
\notag
$$
(cf. [10], § 2.2). § 3. Specially constructed elements of $ {\mathbb{Z}}$-torsorsGiven an irreducible curve $C$ on $X$, let the field $K_C$ be the completion of the field $k(X)$ of rational functions on $X$ with respect to the discrete valuation given by $C$. For a point $x$ of $X$ let $K_x= k(X) \cdot \widehat{\mathcal{O}}_{x,X}$ be a subring of the fraction field $\operatorname{Frac} \widehat{\mathcal{O}}_{x,X}$, where $\widehat{\mathcal{O}}_{x,X}$ is the completion of the local ring of $x$ on $X$. We have the natural diagonal embeddings
$$
\begin{equation}
\prod_{C \subset X} K_C\hookrightarrow\prod_{x \in C} K_{x,C} \quad\text{and}\quad \prod_{x \in X}K_x\hookrightarrow\prod_{x \in C} K_{x,C}.
\end{equation}
\tag{3.1}
$$
The adelic ring $\mathbb{A}_X$ has the following subrings:
$$
\begin{equation*}
\mathbb{A}_{01}=\biggl(\prod_{C \subset X} K_C \biggr)\cap \mathbb{A}_X \quad\text{and}\quad \mathbb{A}_{02}=\biggl(\prod_{x \subset X} K_x\biggr)\cap \mathbb{A}_X,
\end{equation*}
\notag
$$
where the intersection is taken inside the ring $\prod_{x \in C} K_{x,C}$. Let $n \geqslant 1$ be an integer. For lattices $E_1 \subset E_2$ of $\mathbb{A}_X^n$ (see Definition 1) we define a $k$-vector subspace by
$$
\begin{equation*}
\widetilde{\mu}_{E_1, E_2}=(E_2 \cap \mathbb{A}_{02}^n)/ (E_1 \cap \mathbb{A}_{02}^n) \subset E_2/ E_1.
\end{equation*}
\notag
$$
It is easy to see that for any divisors $D_1 \leqslant D_2$ on $X$ the $k$-vector subspace
$$
\begin{equation*}
(A_{12}(D_2) \cap \mathbb{A}_{02}^n)/ (A_{12}(D_1) \cap \mathbb{A}_{02}^n)
\end{equation*}
\notag
$$
is an open linearly compact $k$-vector subspace of $\mathbb{A}_{12}(D_2)/ \mathbb{A}_{12}(D_1)$. Hence $\widetilde{\mu}_{E_1, E_2}$ is an open linearly compact $k$-vector subspace of $E_2/E_1$. Now for arbitrary lattices $E_1$ and $E_2$ of $\mathbb{A}_X^n$ we introduce an element which is uniquely defined by the following two rules:1. If $E_1 \subset E_2$, then $\mu_{E_1, E_2}$ is a ‘dimension theory’ that equals $0$ on the $k$-vector subspace $\widetilde{\mu}_{E_1, E_2} \subset E_2/E_1$. 2. For arbitrary lattices $F_1$, $F_2$ and $F_3$ of $\mathbb{A}_X^n$ we have
$$
\begin{equation}
\mu_{F_1, F_2} \otimes \mu_{F_2, F_3}=\mu_{F_1, F_3}
\end{equation}
\tag{3.2}
$$
with respect to the isomorphism (2.4). Remark 3. To construct an element $\mu_{E_1, E_2} \subset \operatorname{Dim}(E_1 \mid E_2)$ it is not important that $X$ be a projective surface. Analogously, for lattices $E_1 \subset E_2$ of $\mathbb{A}_X^n$ we define a $k$-vector subspace by
$$
\begin{equation*}
\widetilde{\nu}_{E_1, E_2}=(E_2 \cap \mathbb{A}_{01}^n)/ (E_1 \cap \mathbb{A}_{01}^n) \subset E_2/ E_1.
\end{equation*}
\notag
$$
Then $\widetilde{\nu}_{E_1, E_2} \subset E_2/E_1$ is a discrete subspace such that the quotient space of $E_2/E_1$ by it is a linearly compact space. (This fact is first easy to see for $E_1=\mathbb{A}_{12}(D_1)$ and $E_2=\mathbb{A}_{12}(D_2)$, where $D_1$ and $D_2$ are divisors on $X$. In this case the above holds since for any projective curve $C$ on $X$ the field of rational functions $k(C)$ is a discrete subspace of the adelic space of $C$ and the quotient space of this adelic space by $k(C)$ is a linearly compact $k$-vector space, and this follows, for example, from the adelic complex of $C$ and the fact that the cohomology spaces of coherent sheaves on $C$ are finite-dimensional $k$-vector spaces.) Now for arbitrary lattices $E_1$ and $E_2$ of $\mathbb{A}_X^n$ we introduce an element which is uniquely defined by the following two rules.1. If $E_1 \subset E_2$, then $\nu_{E_1, E_2}\in\operatorname{Dim}(E_2/E_1)$ is defined from the exact sequence
$$
\begin{equation*}
0 \to \widetilde{\nu}_{E_1, E_2} \to E_2/ E_1 \to (E_2/E_1)/ \widetilde{\nu}_{E_1, E_2} \to 0,
\end{equation*}
\notag
$$
where the first nonzero term is a discrete space and the last nonzero term is a linearly compact space. By (2.4) there is a canonical isomorphism
$$
\begin{equation*}
\operatorname{Dim}(\widetilde{\nu}_{E_1, E_2}) \otimes_{\mathbb Z}\operatorname{Dim}((E_2/E_1)/ \widetilde{\nu}_{E_1, E_2}) \to\operatorname{Dim}(E_2/ E_1).
\end{equation*}
\notag
$$
Now $\nu_{E_1, E_2} = \nu_1 \otimes \nu_2$, where $\nu_1\in\operatorname{Dim}(\widetilde{\nu}_{E_1,E_2})$ is the ‘dimension theory’ that equals $0$ on the zero-subspace of the discrete space $\widetilde{\nu}_{E_1,E_2}$ and $\nu_2 \in \operatorname{Dim}((E_2/E_1)/\widetilde{\nu}_{E_1,E_2})$ is the ‘dimension theory’ that equals $0$ on the whole linearly compact $k$-vector space $(E_2/E_1)/\widetilde{\nu}_{E_1,E_2}$. 2. For arbitrary lattices $F_1$, $F_2$ and $F_3$ of $\mathbb{A}_X^n$ we have
$$
\begin{equation}
\nu_{F_1, F_2} \otimes \nu_{F_2, F_3}=\nu_{F_1, F_3}
\end{equation}
\tag{3.3}
$$
with respect to the isomorphism (2.4). Remark 4. To construct an element $\nu_{E_1,E_2}\subset\operatorname{Dim}(E_1\mid E_2)$ it is important that $X$ be a projective surface. Note that any rank $n$ locally free subsheaf of $\mathcal{O}_X$-modules $\mathcal{E} \subset k(X)^n$ on $X$ gives the lattice $\mathbb{A}_{12}(\mathcal{E})\,{\subset}\,\mathbb{A}_X^n$ (see [9], § 2.2.3) which generalizes the case of ${\mathbb{A}_{12}(\mathcal{O}_X^n)\,{=}\,\mathbb{A}_{12}^n}$ and the case of $\mathbb{A}_{12}(\mathcal{O}_X(D))=\mathbb{A}_{12}(D)$ for $n=1$, where $D$ is a divisor on $X$. Proposition 1. For any rank $n$ locally free subsheaves of $\mathcal{O}_X$-modules $\mathcal{F}$ and $\mathcal{G}$ of the constant sheaf $k(X)^n$ on $X$ we have where subtraction on the left-hand side of the formula makes sense because it is applied to elements of the $\mathbb{Z}$-torsor $\operatorname{Dim}(\mathbb{A}_{12}(\mathcal{F})\mkern-2mu\mid\mkern-2mu \mathbb{A}_{12}(\mathcal{G}))$, and for any Zariski sheaf $\mathcal{E}$ on $X$ is its Euler characteristic on $X$. Proof. From the properties of the left-hand side (see (3.2) and (3.3)) and the right-hand side of (3.4) we see that it is enough to suppose that $\mathcal{F} \subset \mathcal{G}$. For any quasicoherent sheaf $\mathcal{E}$ of $\mathcal{O}_X$-modules on $X$ we have the adelic complex $\mathcal{A}_X(\mathcal{E})$, which has nonzero terms only in degrees $0$, $1$, $2$, and $H^i(X,\mathcal{E})=H^i(\mathcal{A}_X(\mathcal{E}))$ (see [8], for example).
We have canonical embedding of $\mathcal{A}_X(\mathcal{F})$ into $\mathcal{A}_X(\mathcal{G})$ such that the quotient complex looks as follows:
$$
\begin{equation*}
\begin{gathered} \, \widetilde{\nu}_{\mathbb{A}_{12}(\mathcal F), \mathbb{A}_{12}(\mathcal G)} \oplus \widetilde{\mu}_{\mathbb{A}_{12}(\mathcal F), \mathbb{A}_{12}(\mathcal G)} \to \mathbb{A}_{12}(\mathcal G)/ \mathbb{A}_{12}(\mathcal F), \\ x \oplus y \mapsto x+y. \end{gathered}
\end{equation*}
\notag
$$
Hence we obtain formula (3.4), since the Euler characteristic of the latest complex equals $\chi(\mathcal{A}(\mathcal{G}))-\chi(\mathcal{A}(\mathcal{F}))$. The proposition is proved. § 4. Intersection index of divisorsNow we recall how it is possible to obtain the intersection index of divisors on $X$ by means of the central extension (2.5) when $n=1$:
$$
\begin{equation*}
0 \to \mathbb Z \to \widetilde{\mathrm{GL}_1(\mathbb{A}_X)} \xrightarrow{\Theta} \mathbb{A}_X^* \to 1.
\end{equation*}
\notag
$$
We consider a bimultiplicative and antisymmetric map
$$
\begin{equation*}
\langle \,\cdot\,{,} \,\cdot\, \rangle \colon \mathbb{A}_X^* \times \mathbb{A}_X^* \to \mathbb Z
\end{equation*}
\notag
$$
which is given as follows for any $x, y \in \mathbb{A}_X^*$:
$$
\begin{equation*}
\langle x, y \rangle=[x', y']=x' y' {x'}^{-1} {y'}^{-1},
\end{equation*}
\notag
$$
where $x', y' \in \widetilde{\mathrm{GL}_1(\mathbb{A}_X)}$ are any elements such that $\Theta(x')=x$ and $\Theta(y')=y$. This definition does not depend on the choice of $x'$ and $y'$. We consider a divisor $D$ on $X$. For an irreducible curve $C$ on $X$ let $j_C^D \in K_C^*$ be a local equation of the restriction of $D$ to $\operatorname{Spec} \mathcal{O}_{K_C}$ under the natural morphism $\operatorname{Spec} \mathcal{O}_{K_C} \to X$, where $\mathcal{O}_{K_C}$ is the discrete valuation ring of $K_C$. Note that $j_C^D \cdot v$, where $v \in \mathcal{O}_{K_C}^*$, is also such a local equation. Then under the first diagonal embedding in (3.1) we obtain that the collection $j_{1,D} = \prod_C j_C^D$, where $C$ runs over all irreducible curves $C$ of $X$, belongs to $\mathbb{A}_{01}^*$. For any point $x$ on $X$ let $j_x^D \in K_x^*$ be a local equation of the restriction of $D$ to $\operatorname{Spec} \widehat{\mathcal{O}}_{x,X}$ under the natural morphism $\operatorname{Spec} \widehat{\mathcal{O}}_{x,X} \to X$. Note that $j_x^D \cdot w$, where $w \in \widehat{\mathcal{O}}_{x,X}^*$, is again such a local equation. Then under the second diagonal embedding from (3.1) we obtain that the collection $j_{2,D}= \prod_x j_x^D$, where $x$ runs over all points $x$ of $X$, belongs to $\mathbb{A}_{02}^*$. For any divisors $S$ and $T$ on $X$ we have (see Proposition 2 in [11], which is based on [15], § 2.2), where $(S,T) \in \mathbb{Z}$ is the intersection index of $S$ and $T$ on $X$.Now we give another presentation for the intersection index of divisors which is based on (4.1) and special lifts of elements as in § 3. Proposition 2. For any divisors $S$ and $T$ on $X$ we have Proof. The second equality follows from the first because $(S,T)=(-S,-T)$.
We prove the first equality. We note that for any divisor $D$ on $X$ we have $j_{1,D}\cdot j_{2,D }^{-1} \in \mathbb{A}_{12}^*$ and equality of $k$-vector subspaces of $\mathbb{A}_X$:
$$
\begin{equation*}
j_{1,D} \mathbb{A}_{12}=j_{2, D} \mathbb{A}_{12}=\mathbb{A}_{12}(-D).
\end{equation*}
\notag
$$
Therefore we can take the following special lifts of elements $j_{1,T}$ and $j_{2,S}$ to $\widetilde{\mathrm{GL}_1(\mkern-1mu\mathbb{A}_X\mkern-1mu)}$ to calculate $\langle j_{1,T},j_{2,S}\rangle$:
$$
\begin{equation}
j_{1, T} \mapsto (j_{1, T}, \mu_{\mathbb{A}_{12}, \mathbb{A}_{12}(-T)}), \qquad j_{2,S} \mapsto (j_{2,S}, \nu_{\mathbb{A}_{12}, \mathbb{A}_{12}(-S)}).
\end{equation}
\tag{4.2}
$$
Immediately from the constructions we have
$$
\begin{equation}
j_{1,T} (\nu_{\mathbb{A}_{12}, \mathbb{A}_{12}(-S)})=\nu_{\mathbb{A}_{12}(-T), \mathbb{A}_{12}(-T-S)}
\end{equation}
\tag{4.3}
$$
and
$$
\begin{equation}
j_{2, S} (\mu_{\mathbb{A}_{12}, \mathbb{A}_{12}(-T)})=\mu_{\mathbb{A}_{12}(-S), \mathbb{A}_{12}(-T-S)}.
\end{equation}
\tag{4.4}
$$
Now from formula (4.1) we have the following equality in $\widetilde{\mathrm{GL}_1(\mathbb{A}_X)}$:
$$
\begin{equation*}
\begin{aligned} \, &(j_{1,T}, \mu_{\mathbb{A}_{12}, \mathbb{A}_{12}(-T)})(j_{2,S}, \nu_{\mathbb{A}_{12}, \mathbb{A}_{12}(-S)}) \\ &\qquad =(j_{2, S}, \nu_{\mathbb{A}_{12}, \mathbb{A}_{12}(-S)})(j_{1,T}, \mu_{\mathbb{A}_{12}, \mathbb{A}_{12}(-T)}) (S,T), \end{aligned}
\end{equation*}
\notag
$$
where we consider the intersection index $(S, T )$ as an element of the central subgroup ${\mathbb{Z} \subset \widetilde{\mathrm{GL}_1(\mathbb{A}_X)}}$. Now, using the definition of the group operation in $\widetilde{\mathrm{GL}_1(\mathbb{A}_X)}$ and formulae (4.3) and (4.4), we obtain the statement of the proposition. Remark 5. A lift of elements analogous to (4.2) was considered in [13], § 5, for divisors $S$ and $T = (\omega) - S$, where $\omega \in \Omega^2_{k(X)/k}$. § 5. Canonical splittingsWe construct canonical splittings of the central extensions (2.5) and (2.7) over some subgroups of $\mathrm{GL}_n(\mathbb{A}_X)$ and investigate the properties of these splittings. Proposition 3. Consider the central extension (2.5). 1. The map
$$
\begin{equation}
\mathrm{GL}_n(\mathbb{A}_{12}) \ni g \mapsto (g, 0) \in \widetilde{\mathrm{GL}_n(\mathbb{A}_X)},
\end{equation}
\tag{5.1}
$$
where $0$ is the zero ‘dimension theory’, gives a splitting of this central extension over the subgroup $\mathrm{GL}_n(\mathbb{A}_{12})$. 2. The map
$$
\begin{equation}
\mathrm{GL}_n(\mathbb{A}_{02}) \ni g \mapsto (g, \mu_{\mathbb{A}_{12}^n, g \mathbb{A}_{12}^n}) \in \widetilde{\mathrm{GL}_n(\mathbb{A}_X)}
\end{equation}
\tag{5.2}
$$
gives a splitting of this central extension over the subgroup $\mathrm{GL}_n(\mathbb{A}_{02})$. 3. The map
$$
\begin{equation}
\mathrm{GL}_n(\mathbb{A}_{01}) \ni g \mapsto (g, \nu_{\mathbb{A}_{12}^n, g \mathbb{A}_{12}^n}) \in \widetilde{\mathrm{GL}_n(\mathbb{A}_X)}
\end{equation}
\tag{5.3}
$$
gives a splitting of this central extension over the subgroup $\mathrm{GL}_n(\mathbb{A}_{01})$. 4. The splittings (5.1) and (5.3) coincide over the subgroup $\mathrm{GL}_n(\prod_C \mathcal{O}_{K_C})$. The splittings (5.1) and (5.2) coincide over the subgroup $\mathrm{GL}_n(\prod_x \widehat{\mathcal{O}}_{x,X})$. Splittings (5.3) and (5.2) coincide over the subgroup $\mathrm{GL}_n(k(X))$. (Here we use the diagonal embeddings of subgroups; see (3.1).) Remark 6. For the construction of the splittings (5.1) and (5.2) it is not important that $X$ be projective, but for the construction of the splitting (5.3) it is important. Proof of Proposition 3. Item 1 is evident, since $g \mathbb{A}_{12}^n = \mathbb{A}_{12}^n$ for any $g\in \mathrm{GL}_n(\mathbb{A}_{12})$.
Items 2 and 3 follow from the equalities:
$$
\begin{equation*}
g \widetilde{\mu}_{E_1, E_2}=\widetilde{\mu}_{g E_1, gE_2} \quad\text{and}\quad h \widetilde{\nu}_{E_1, E_2}=\widetilde{\nu}_{hE_1, hE_2},
\end{equation*}
\notag
$$
where $E_1 \subset E_2$ are any lattices in $\mathbb{A}_X^n$, $g\in \mathrm{GL}_n(\mathbb{A}_{02})$ and ${h\in\mathrm{GL}_n(\mathbb{A}_{01})}$.
In item 4 the only nonevident statement is that the splittings (5.3) and (5.2) coincide over the subgroup $\mathrm{GL}_n(\mkern-1mu k(\mkern-1mu X)\mkern-1mu)$. To prove this we note that for any ${g \!\in\! \mathrm{GL}_n\mkern-1mu(\mkern-1mu k(\mkern-1mu X)\mkern-1mu)}$ we have $g\mathbb{A}_{12}^n=\mathbb{A}_{12}(g \mathcal{O}_X^n)$, and by Proposition 1 we have
$$
\begin{equation*}
\nu_{\mathbb{A}_{12}(\mathcal O_X^n), \mathbb{A}_{12}(g \mathcal O_X^n)} - \mu_{\mathbb{A}_{12}(\mathcal O_X^n), \mathbb{A}_{12}(g \mathcal O_X^n)} =\chi(g\mathcal O_X^n)- \chi(\mathcal O_X^n).
\end{equation*}
\notag
$$
We note that the sheaf $g \mathcal{O}_X^n$ is isomorphic to the sheaf $\mathcal{O}_X^n$. Therefore, ${\chi(\mathcal{O}_X^n) = \chi(g\mathcal{O}_X^n)}$. Hence
$$
\begin{equation*}
\nu_{\mathbb{A}_{12}^n, g\mathbb{A}_{12}^n}=\mu_{\mathbb{A}_{12}^n, g\mathbb{A}_{12}^n},
\end{equation*}
\notag
$$
and the two splittings coincide. (Note that we could use Proposition 1 only when $n=1$, since $\mathrm{GL}_n(k(X)) = \mathrm{SL}_n(k(X)) \rtimes k(X)^*$ and the group $\mathrm{SL}_n(k(X))$ is perfect, which implies that any two sections of any central extension of $\mathrm{SL}_n(k(X))$ coincide.)
The proposition is proved. Let $n = n_1 + n_2$, where $n_1$ and $n_2$ are positive integers. We consider the parabolic subgroup
$$
\begin{equation*}
P_{n_1, n_2}=\left\{ \begin{pmatrix} \mathrm{GL}_{n_1}(\mathbb{A}_X) & * \\ 0 & \mathrm{GL}_{n_2}(\mathbb{A}_X) \end{pmatrix}\right\} \subset \mathrm{GL}_n(\mathbb{A}_X).
\end{equation*}
\notag
$$
Let $p_i\colon \mathrm{GL}_n(\mathbb{A}_X) \to P_{n_i}$, where $i=1$ or $i=2$, be natural homomorphisms. Proposition 4. The pullback of the central extension $\widetilde{\mathrm{GL}_n(\mathbb{A}_X)}$ via the embedding ${P_{n_1,n_2} \hookrightarrow \mathrm{GL}_n(\mathbb{A}_X)}$ is isomorphic to the Baer sum of the pullbacks $p_1^*(\widetilde{\mathrm{GL}_{n_1}(\mathbb{A}_X)})$ and $p_2^*(\widetilde{\mathrm{GL}_{n_2}(\mathbb{A}_X)})$. Moreover, the splittings (over special subgroups) in Proposition 3 are compatible with respect to this isomorphism. Proof. The construction of the isomorphism is obtained directly from the exact triple
$$
\begin{equation*}
0 \to \mathbb{A}_X^{n_1} \to \mathbb{A}_X^n \to \mathbb{A}_X^{n_2} \to 0,
\end{equation*}
\notag
$$
the fact that the action of $P_{n_1, n_2}$ preserves this triple, the construction of the group $\widetilde{\mathrm{GL}_n(\mathbb{A}_X)}$ and the isomorphism (2.2).
The compatibility of sections follows from the exact triples
$$
\begin{equation*}
0 \to \mathbb{A}_{01}^{n_1} \to \mathbb{A}_{01}^n \to \mathbb{A}_{01}^{n_2} \to 0 \quad\text{and}\quad 0 \to \mathbb{A}_{02}^{n_1} \to \mathbb{A}_{02}^n \to \mathbb{A}_{02}^{n_2} \to 0
\end{equation*}
\notag
$$
and the fact that the corresponding groups $P_{n_1,n_2} \cap \mathrm{GL}_n(\mathbb{A}_{01})$ or $P_{n_1,n_2} \cap \mathrm{GL}_n(\mathbb{A}_{02})$ act on these triples.
The proposition is proved. Remark 7. A direct analogue of Proposition 3 is valid for the central extension (2.7) (see [11], Proposition 4), where we use that The behaviour of the pullback of central extension (2.7) via the map $P_{n_1,n_2} \hookrightarrow \mathrm{GL}_n(\mathbb{A}_X)$ differs from what is described in Proposition 4 for the central extension (2.5); see [11], Proposition 3. This is because (2.7) is connected with the second Chern number of a rank $n$ locally free sheaf of $\mathcal{O}_X$-modules; see [11], § 3.3, and Remark 9 below. § 6. Trivializations of locally free sheavesWe describe trivializations of locally free sheaves of $\mathcal{O}_X$-modules over the completions of the local rings of scheme points of $X$. We consider a point $x \in X$ as a closed scheme point of $X$, the generic point of an irreducible curve $C$ on $X$ as a non-closed point of $X$, and we also consider the generic point of $X$. Let $\mathcal{E}$ be a locally free sheaf of $\mathcal{O}_X$-modules of rank $n$ on $X$. For a (closed) point $x$ of $X$, the completion of the stalk of $\mathcal{E}$ at $x$ is a free $\widehat{\mathcal{O}}_{x,X}$-module. Let $e_x$ be a basis of this module. We will also call such a basis a basis of $\mathcal{E}$ restricted to $\operatorname{Spec} \widehat{\mathcal{O}}_{x,X}$. For an irreducible curve $C$ on $X$, the completion of the stalk of $\mathcal{E}$ at the generic point of $C$ is a free $\mathcal{O}_{K_C}$-module. Let $e_C$ be a basis of this module. We will also call such a basis a basis of $\mathcal{E}$ restricted to $\operatorname{Spec} \mathcal{O}_{K_C}$. The stalk of $\mathcal{E}$ at the generic point of $X$ is a $k(X)$-vector space. Let $e_0$ be a basis of this vector space. We will call such a basis a basis of $\mathcal{E}$ restricted to $\operatorname{Spec} k(X)$. By embedding the completions of stalks of $\mathcal{E}$ at scheme points of $X$ into the tensor products (over the local rings of the points) of the stalks of $\mathcal{E}$ and the corresponding rings $K_x$, $K_C$ or $K_{x,C}$, we obtain the transition matrices
$$
\begin{equation*}
\alpha_{02, x} \in \mathrm{GL}_n(K_x), \qquad \alpha_{01,C} \in \mathrm{GL}_n(K_C)\quad\text{and} \quad \alpha_{21,x,C} \in \mathrm{GL}_n(\mathcal O_{K_{x,C}})
\end{equation*}
\notag
$$
which are defined by the equalities
$$
\begin{equation*}
e_0=\alpha_{02,x}e_x, \qquad e_0=\alpha_{01,C}e_C\quad\text{and} \quad e_x=\alpha_{21,x,C} e_C.
\end{equation*}
\notag
$$
Denote collections of matrices $\alpha_{02} =\prod_x \alpha_{02,x}$, where $x$ runs over all closed points of $X$, $\alpha_{01} = \prod_C \alpha_{01,C}$, where $C$ runs over all irreducible curves $C$ on $X$ and $\alpha_{21} = \prod_{x \in C} \alpha_{21,x,C} $, where $x \in C$ runs over all pairs $x \in C$ with $x$ a closed point on an irreducible curve $C$ on $X$. Now set $\alpha_{ij} = \alpha_{ji}^{-1}$, where $i$ and $j$, $i \ne j$, are from the set $\{0, 1, 2 \}$. Then for any $i,\, j,\, k$ from $\{0,1,2\}$, $i \ne j \ne k$, we have an equality (cocycle identity) in $\mathrm{GL}_n(\prod_{x \in C} K_{x,C})$, where we use the diagonal embeddings (3.1): Note that if we change the chosen bases, then the matrices change in the following way:
$$
\begin{equation}
\alpha_{02}\mapsto\alpha_{0} \cdot \alpha_{02}\cdot\alpha_{2}^{-1}, \qquad \alpha_{01}\mapsto\alpha_{0} \cdot \alpha_{01}\cdot\alpha_{1}^{-1}, \qquad \alpha_{21}\mapsto\alpha_{2} \cdot \alpha_{21}\cdot\alpha_{1}^{-1},
\end{equation}
\tag{6.2}
$$
where
$$
\begin{equation*}
\alpha_{0} \in \mathrm{GL}_n(k(X)) \subset \mathrm{GL}_n(\mathbb{A}_X), \qquad \alpha_{1} \in \mathrm{GL}_n \biggl(\prod_C \mathcal O_{K_C}\biggr)\subset \mathrm{GL}_n(\mathbb{A}_X)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\alpha_{2} \in \mathrm{GL}_n \biggl(\prod_x \widehat{\mathcal O}_{x,X}\biggr)\subset \mathrm{GL}_n(\mathbb{A}_X).
\end{equation*}
\notag
$$
Using the diagonal embeddings (3.1) and simple reasoning one can show that
$$
\begin{equation}
\alpha_{02} \in \mathrm{GL}_n(\mathbb{A}_{02}), \qquad \alpha_{01} \in \mathrm{GL}_n(\mathbb{A}_{01})\quad\text{and} \quad \alpha_{21} \in \mathrm{GL}_n(\mathbb{A}_{12})
\end{equation}
\tag{6.3}
$$
(see [11], § 3.3). We will use the notation $\alpha_{ij, \mathcal{E}}$ instead of $\alpha_{ij}$ when it is not clear from the context with which this notation is associated. Remark 8. If $n=1$, then $\mathcal{E} = \mathcal{O}_X(D)$ for some divisor $D$ on $X$. We have that Definition 2. Let $\mathcal{E}$ be a locally free sheaf of $\mathcal{O}_X$-modules of rank $n$ on $X$. For any $i$ and $j$ from $\{0,1,2\}$, $i \ne j$, let $\widetilde{\alpha_{ij}}$ be the canonical lift of $\alpha_{ij}$ for $\mathcal{E}$ to $\widetilde{\mathrm{GL}_n(\mathbb{A}_X)}$ by means of the splitting over the corresponding subgroup; see Proposition 3 and formula (6.3). Then we define This definition is consistent because it follows from formula (6.1) that $f_E \in \mathbb{Z}$, and by Proposition 3 and formula (6.2), the integer $f_{\mathcal{E}}$ depends only on $\mathcal{E}$, that is, this integer does not depend on the choice of the bases $e_0$, $\{e_x\}$ and $\{e_C\}$ for $\mathcal{E}$. As we have already mentioned before, the goal of this paper is to calculate $f_E$ and to relate the calculations of this integer in two different ways to the Riemann-Roch theorem for $\mathcal{E}$ on $X$ (without Noether’s formula for $\mathcal{O}_X$). Remark 9. For the central extension (2.7), § 7. The first way to calculate $f_{\mathcal{E}}$Let $\mathcal{E}$ be a locally free sheaf of $\mathcal{O}_X$-modules of rank $n$ on $X$. We calculate the integer $f_{\mathcal{E}}$ (see Definition 2) in the first way. Proposition 5. The following properties are satisfied. 1. Consider an exact triple of finite rank locally free sheaves of $\mathcal{O}_X$-modules:
$$
\begin{equation*}
0 \to {\mathcal E}_1 \to {\mathcal E} \to {\mathcal E_2} \to 0.
\end{equation*}
\notag
$$
Then $f_{\mathcal{E}} = f_{\mathcal{E}_1} + f_{\mathcal{E}_2}$. 2. Let $\pi\colon Y \to X$ be the blow-up of a point on $X$. We have $f_{\mathcal{E}} = f_{\pi^*(\mathcal{E})}$. Proof. 1. The integers $f_{\mathcal{E}_i}$, where $i \in \{1,2,3 \}$, do not depend on the choice of bases of the $\mathcal{E}_i$ over the completions of the local rings of scheme points of $X$. Therefore we can choose first bases for $\mathcal{E}_1$ and then complete them to the bases of $\mathcal{E}$. Hence all the transition matrices $\alpha_{ij}$ belong to the subgroup $P_{n_1,n_2} \subset \mathrm{GL}_n(\mathbb{A}_X)$, where $n_1$ and $n_2$ are the ranks of $\mathcal{E}_1$ and $\mathcal{E}_2$, respectively. Now we apply Proposition 4.
2. (Cf. the proof of Theorem 1 in [11].) Let $\pi$ be the blow-up of a point $x \in X$, and let $\pi^{-1}(x)= R$. Note that
$$
\begin{equation}
\mathbb{A}_Y=\mathbb{A}_X \times \sideset{}{'}\prod_{p \in R}K_{p,R}.
\end{equation}
\tag{7.1}
$$
Since $f_{\mathcal{E}}$ and $f_{\pi^*(\mathcal{E})}$ do not depend on the choice of bases of $\mathcal{E}$ and $\pi^*(\mathcal{E})$, we choose the special bases. Fix a trivialization of $\mathcal{E}$ on an open neighbourhood of $x$ on $X$. This trivialization induces the bases $e_0$, $e_x$, $e_R$ and $e_p$, where $p \in R$. We have a canonical isomorphism
$$
\begin{equation*}
\delta \colon \widetilde{\mathrm{GL}_n(\mathbb{A}_X)} \to \gamma^*(\widetilde{\mathrm{GL}_n(\mathbb{A}_Y)}),
\end{equation*}
\notag
$$
where the embedding $\gamma\colon \mathrm{GL}_n(\mathbb{A}_X) \hookrightarrow \mathrm{GL}_n(\mathbb{A}_Y)$ is induced by the decomposition (7.1). Now for any $i$ and $ j$ from $\{0, 1, 2 \}$, $i \ne j$, we have where we consider the $\widetilde{\alpha_{ij, \pi^*(\mathcal{E})}}$ as elements of the group $\gamma^*(\widetilde{\mathrm{GL}_n(\mathbb{A}_Y)})$. This implies the statement. In the sequel we denote also the intersection index by $\mathcal{E} \cdot \mathcal{F} = (\mathcal{E}, \mathcal{F}) \in \mathbb{Z}$ or $c_1(\mathcal{E})^2= \mathcal{E} \cdot \mathcal{E} = (\mathcal{E}, \mathcal{E}) \in \mathbb{Z}$, where $\mathcal{E}$ and $\mathcal{F}$ are invertible sheaves on $X$. Theorem 1. Let $\mathcal{E}$ be a locally free sheaf of $\mathcal{O}_X$-modules of rank $n$ on a smooth projective surface $X$ over a field $k$. Then Proof. The second equality is just a reformulation of the first equality in new notation. Therefore, we prove the first equality.
The left-hand and right-hand sides of the equality are additive with respect to short exact sequences of locally free sheaves of $\mathcal{O}_X$-modules and are preserved by blow-ups of points. Indeed, for the left-hand side of the equality this follows from Proposition 5, and for $\mathrm{ch}_2$ on the right-hand side this follows from the properties of $\mathrm{ch}_2$, and the difference of Euler characteristics is preserved by the blow-up of a point since this follows from the simple case $\chi(\mathcal{F}) - \chi(\mathcal{E}) = \chi(\mathcal{F} / \mathcal{E})$, where $\mathcal{F}$ is a locally free sheaf of $\mathcal{O}_X$-modules of rank $n$, $\mathcal{E} \subset \mathcal{F}$ and the sheaf $\mathcal{F}$ coincides with the sheaf $\mathcal{E}$ in a neighbourhood of the blown-up point. Therefore, by the splitting principle for locally free sheaves on smooth surfaces (cf. the proof of Theorem 1 in [11]), it is enough to prove the equality for $n=1$. So, suppose $n=1$. The chosen basis $e_0$ of $\mathcal{E}$ at the generic point of $X$ gives an embedding of $\mathcal{E}$ into the constant sheaf $k(X)$ on $X$. (Therefore, $\mathcal{E} = \mathcal{O}_X(D)$ for some divisor $D$ on $X$.) We also fix other bases for $\mathcal{E}$ and hence the transition matrices $\alpha_{ij}$ for $\mathcal{E}$, where $i$ and $j$, $i \ne j$, are from $\{1,2\}$ as in § 6. We have to prove that
$$
\begin{equation}
f_{\mathcal E}=\bigl(\chi(\mathcal E) -n \chi(\mathcal O_X)\bigr) -(\mathcal E, \mathcal E).
\end{equation}
\tag{7.2}
$$
From formula (6.4), which describes the intersection index of invertible sheaves as the commutator of the lifts of the corresponding elements to $\widetilde{\mathrm{GL}_1(\mathbb{A}_X)}$, we have
$$
\begin{equation}
\widetilde{\alpha_{10}} \cdot \widetilde{\alpha_{02}} =(-(\mathcal E, \mathcal E)) \cdot \widetilde{\alpha_{02}} \cdot \widetilde{\alpha_{10}},
\end{equation}
\tag{7.3}
$$
where we consider $-(\mathcal{E}, \mathcal{E})$ as an element of the central subgroup $\mathbb{Z} \subset \widetilde{\mathrm{GL}_1(\mathbb{A}_X)}$.
Furthermore, using conjugation we obtain
$$
\begin{equation*}
f_{\mathcal E}=\widetilde{\alpha_{02}} \cdot \widetilde{\alpha_{21}} \cdot \widetilde{\alpha_{10}}=\widetilde{\alpha_{10}} \cdot \widetilde{\alpha_{02}} \cdot \widetilde{\alpha_{21}} \cdot \widetilde{\alpha_{10}} \cdot \widetilde{\alpha_{10}}^{-1} =\widetilde{\alpha_{10}} \cdot \widetilde{\alpha_{02}} \cdot \widetilde{\alpha_{21}}.
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation}
\widetilde{\alpha_{10}} \cdot \widetilde{\alpha_{02}} =f_{\mathcal E} \cdot \widetilde{\alpha_{12}},
\end{equation}
\tag{7.4}
$$
where we consider $f_{\mathcal{E}}$ as an element of the central subgroup $\mathbb{Z} \subset \widetilde{\mathrm{GL}_1(\mathbb{A}_X)}$.
From (7.3) and (7.4) we see that formula (7.2) will follow from the following lemma. Lemma. Let $\mathcal{E}$ be an invertible sheaf on $X$. Then Note that since $\alpha_{02}$ and $\alpha_{10}$ are from $\mathbb{A}_X^*$, we have that ${\alpha_{02} \cdot \alpha_{10} \,{=}\, \alpha_{10} \cdot \alpha_{02} \,{=}\, \alpha_{12}}$. Proof of the lemma. The second equality in the statement of the lemma follows immediately from the first equality and Proposition 1. Therefore, we prove the first equality.
Let $c \in \mathbb{Z} \subset \widetilde{\mathrm{GL}_1(\mathbb{A}_X)}$ be such that $\widetilde{\alpha_{02}} \cdot \widetilde{\alpha_{10}} = c \cdot \widetilde{\alpha_{12}} $. Then we have
$$
\begin{equation}
\widetilde{\alpha_{10}}=c \cdot \widetilde{\alpha_{20}} \cdot \widetilde{\alpha_{12}}.
\end{equation}
\tag{7.5}
$$
Note that $\alpha_{12} \mathbb{A}_{12} = \mathbb{A}_{12}$. Therefore, $\widetilde{\alpha_{12}} = (\alpha_{12}, 0)$, where ${0 \in \mathbb{Z} = \operatorname{Dim}(\mathbb{A}_{12} \mid \mathbb{A}_{12})}$. Hence taking the product with $\widetilde{\alpha_{12}}$ on the right-hand side of (7.5) does not affect the ‘dimension theory’.
Hence (see also Remark 8, where transition matrices for an invertible sheaf are explicitly given) by direct calculation we obtain This finishes the proof of the lemma and, consequently, the proof of the theorem. Remark 10. From Theorem 1, Remark 9 and formula (6.4) we obtain the following ‘local (adelic) decomposition’ for the difference of Euler characteristics for a rank $n$ locally free sheaf $\mathcal{E}$ of $\mathcal{O}_X$-modules and the sheaf $\mathcal{O}_X^n$ using the central extensions (2.5) and (2.7) and transition matrices for $\mathcal{E}$: § 8. The second way to calculate $f_{\mathcal{E}}$ and the Riemann-Roch theoremThere is another way to calculate $f_{\mathcal{E}}$ for a locally free sheaf $\mathcal{E}$ of $\mathcal{O}_X$-modules of rank $n$ on $X$. We take this way in this section. This other way leads to an answer which also uses other invariants of $\mathcal{E}$ and $X$; see Theorem 2 below. And the comparison of this answer with the answer obtained in Theorem 1 immediately gives us the Riemann-Roch theorem for $\mathcal{E}$ on $X$ (without Noether’s formula); see the corollary below. In the sequel we suppose that the ground field $k$ is perfect (this is important for the theory of two-dimensional residues used below). The idea for the new calculation of $f_{\mathcal{E}}$ is to use the fact that $\mathbb{A}_X$ is self-dual as a $C_2$-space over $k$ (or a $2$-Tate vector space) and to make some calculations as on the ‘dual side’. More exactly, the self-duality of $\mathbb{A}_X$ is given by the following pairing (cf. [13], § 2). We fix $\omega \in \Omega^2_{k(X)/k}$ such that $\omega \ne 0$. Then via $\omega$ we construct a bilinear symmetric nondegenerate pairing by means of residues on two-dimensional local fields (see more about residues on $n$-dimensional local fields in [14] and [21]):
$$
\begin{equation}
\mathbb{A}_X \times \mathbb{A}_X \to k \colon \{f_{x,C}\} \times \{ g_{x,C} \} \mapsto \sum_{x \in C}\operatorname{Tr}_{k(x)/k} \circ \operatorname{res}_{x,C} (f_{x,C} g_{x,C}\omega),
\end{equation}
\tag{8.1}
$$
where $\{f_{x,C}\}$ and $\{g_{x,C}\}$ are from $\sideset{}{'}{\textstyle\prod}_{\!\!x \in C}K_{x,C}=\mathbb{A}_X$, and if $K_{x, C} = \prod_i K_i$, then $\operatorname{res}_{x, C}\colon \Omega^2_{K_{x,C}/k} \to k(x)$ equals $\sum_i \operatorname{Tr}_{k_i/ k(x)} \circ \operatorname{res}_{K_i}$, where
$$
\begin{equation}
\operatorname{res}_{K_i} \colon \Omega^2_{K_i/k(x)} \to \Omega^2_{K_i/k_i} \to \widetilde{\Omega}^2_{K_i/k_i} \to k_i,
\end{equation}
\tag{8.2}
$$
and where $K_i \simeq k_i((u))((t))$ and this isomorphism is a homeomorphism with respect to the natural topologies. The first map in (8.2) is the natural map, the second is the map to ‘continuous differential forms’, that is, to the quotient module by the $K_i$-submodule generated by the elements
$$
\begin{equation*}
f_1 \,df_2 \wedge df_3-f_1\,df_2 \wedge \biggl(\frac{\partial{f_3}}{\partial{u}}\,du+\frac{\partial{f_3}}{\partial{t}}\, dt\biggr),
\end{equation*}
\notag
$$
and the last map is
$$
\begin{equation*}
\sum_{j,l} a_{j,l} u^j t^l \,du \wedge dt \mapsto a_{-1,-1},
\end{equation*}
\notag
$$
where $a_{j,l} \in k_i$. We note also that the sum in (8.1) is finite. Now we present a theorem. Theorem 2. Let $\mathcal{E}$ be a locally free sheaf of $\mathcal{O}_X$-modules of rank $n$ on a smooth projective surface $X$ over a perfect field $k$. Then where $K \simeq \mathcal{O}_X(\omega)$, $\omega \in \Omega^2_{k(X)/k}$ and $\omega \ne 0$. Comparing the formulations of Theorem 1 and Theorem 2, we immediately obtain a corollary. Remark 11. If $\mathcal{E}$ is an invertible sheaf on $X$, then the quantity Proof of Theorem 2. It is enough to prove the first equality in the statement of Theorem 2. Using the same arguments as in the beginning of the proof of Theorem 1 we obtain that it is enough to suppose that $n=1$, which we will do.
For any $k$-vector subspace $V \subset \mathbb{A}_X$ we denote by $V^{\perp}$ the annihilator of $V$ in $\mathbb{A}_X$ with respect to the pairing (8.1). From the reciprocity laws on $X$ for residues of differential $2$-forms on two-dimensional local fields (these reciprocity laws are ‘along an irreducible curve’ and ‘around a point’), for any divisor $D$ on $X$ it is possible to obtain (see [14])
$$
\begin{equation}
\mathbb{A}_{12}(D)^{\perp}=\mathbb{A}_{12}((\omega)-D), \qquad \mathbb{A}_{02}^{\perp}=\mathbb{A}_{02}\quad\text{and}\quad {A}_{01}^{\perp}=\mathbb{A}_{01}.
\end{equation}
\tag{8.3}
$$
There is a canonical isomorphism of groups (see [12], § 5.5.5)
$$
\begin{equation*}
\varphi \colon \widetilde{\mathrm{GL}_1(\mathbb{A}_X)} \to \widetilde{\mathrm{GL}_1(\mathbb{A}_X)}_{\mathbb{A}_{12}^{\perp}},
\end{equation*}
\notag
$$
where the central extension $\widetilde{\mathrm{GL}_1(\mathbb{A}_X)}_{\mathbb{A}_{12}^{\perp}}$ is constructed similarly to the central extension $\widetilde{\mathrm{GL}_1(\mathbb{A}_X)}$, but starting from the lattice $\mathbb{A}_{12}^{\perp}$ instead of $\mathbb{A}_{12}$ (cf. Remark 1). More precisely, $\widetilde{\mathrm{GL}_1(\mathbb{A}_X)}_{\mathbb{A}_{12}^{\perp}}$ consists of all pairs $(g,d)$, where $g \in \mathrm{GL}_1(\mathbb{A}_X)$ and ${d \in \operatorname{Dim}(\mathbb{A}_{12}^{\perp} \mid g \mathbb{A}_{12}^{\perp})}$ with the multiplication law as in formula (2.6). Explicitly, the isomorphism $\varphi$ is given by where we use the canonical isomorphism
$$
\begin{equation*}
\operatorname{Dim}(\mathbb{A}_{12}\mid g \mathbb{A}_{12}) \simeq\operatorname{Dim}(\mathbb{A}_{12}^{\perp}\mid g^{-1} \mathbb{A}_{12}^{\perp}),
\end{equation*}
\notag
$$
which is based on the equality $g^{-1} \mathbb{A}_{12}^{\perp} = (g\mathbb{A}_{12})^{\perp}$ and on the canonical isomorphism $\operatorname{Dim}(E_1 \mid E_2) \simeq \operatorname{Dim} (E_1^{\perp} \mid E_2^{\perp})$, where $E_1$ and $E_2$ are lattices in $\mathbb{A}_X$. The last isomorphism comes from the following chain of isomorphisms, where $E_3$ is a lattice such that $E_3 \subset E_1$ and $E_3\subset E_2$:
$$
\begin{equation*}
\begin{aligned} \, \operatorname{Dim}(E_1\mid E_2) &\simeq\operatorname{Dim}(E_1\mid E_3) \otimes_{\mathbb Z}\operatorname{Dim}(E_3\mid E_2) \simeq \operatorname{Dim}(E_1/E_3)^* \otimes_{\mathbb Z}\operatorname{Dim}(E_2/ E_3) \\ &\simeq \operatorname{Dim}(E_3^{\perp}/E_1^{\perp}) \otimes_{\mathbb Z}\operatorname{Dim}(E_3^{\perp}/ E_2^{\perp})^* \simeq\operatorname{Dim}(E_1^{\perp}\mid E_2^{\perp}); \end{aligned}
\end{equation*}
\notag
$$
where ${}^*$ means the dual $\mathbb{Z}$-torsor and we have used the canonical isomorphisms (2.2) and (2.3).
We note that the isomorphism $\varphi$ restricted to the central subgroup $\mathbb{Z} \subset \widetilde{\mathrm{GL}_1(\mathbb{A}_X)}$ is the identity morphism to the central subgroup $\mathbb{Z} \subset \widetilde{\mathrm{GL}_1(\mathbb{A}_X)} $. The chosen basis $e_0$ of $\mathcal{E}$ at the generic point of $X$ gives the embedding of $\mathcal{E}$ into the constant sheaf $k(X)$ on $X$. Therefore, $\mathcal{E} = \mathcal{O}_X(D)$ for some divisor $D$ on $X$. We also fix the other bases for $\mathcal{E}$ and hence the transition elements $\alpha_{ij}$ for $\mathcal{E}$, where $i$ and $j$, $i \ne j$, are from $\{1,2\}$ as in § 6. Therefore, we have
$$
\begin{equation*}
f_{\mathcal E}=\varphi(f_{\mathcal E})=\varphi(\widetilde{\alpha_{02}} \cdot \widetilde{\alpha_{21}} \cdot \widetilde{\alpha_{10}}) =\varphi(\widetilde{\alpha_{02}}) \cdot \varphi(\widetilde{\alpha_{21}}) \cdot \varphi(\widetilde{\alpha_{10}}).
\end{equation*}
\notag
$$
Hence, the proof of the theorem will follow from the proof of the formula
$$
\begin{equation*}
\varphi(\widetilde{\alpha_{01}}) \cdot \varphi(\widetilde{\alpha_{12}}) \cdot \varphi(\widetilde{\alpha_{20}})=\widetilde{\alpha_{02}} \cdot \widetilde{\alpha_{21}} \cdot \widetilde{\alpha_{10}} +(\mathcal E, \mathcal E)+(\mathcal E, \mathcal O_X(\omega)),
\end{equation*}
\notag
$$
where we have used that
$$
\begin{equation*}
-(\widetilde{\alpha_{02}} \cdot \widetilde{\alpha_{21}} \cdot \widetilde{\alpha_{10}})=\widetilde{\alpha_{01}} \cdot \widetilde{\alpha_{12}} \cdot \widetilde{\alpha_{20}}.
\end{equation*}
\notag
$$
From formula (6.4) for the intersection index of invertible sheaves we obtain
$$
\begin{equation*}
(\mathcal E, \mathcal E)=\langle \widetilde{\alpha_{01}}, \widetilde{\alpha_{02}} \rangle=\widetilde{\alpha_{01}} \cdot \widetilde{\alpha_{02}} \cdot \widetilde{\alpha_{10}} \cdot \widetilde{\alpha_{20}}.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
\begin{aligned} \, &\widetilde{\alpha_{02}} \cdot \widetilde{\alpha_{21}} \cdot \widetilde{\alpha_{10}}+(\mathcal E, \mathcal E)=\widetilde{\alpha_{02}} \cdot \widetilde{\alpha_{21}} \cdot \widetilde{\alpha_{10}} \cdot \widetilde{\alpha_{01}}\cdot\widetilde{\alpha_{02}}\cdot \widetilde{\alpha_{10}} \cdot \widetilde{\alpha_{20}} \\ &\qquad =\widetilde{\alpha_{02}}\cdot (\widetilde{\alpha_{21}} \cdot \widetilde{\alpha_{02}} \cdot \widetilde{\alpha_{10}})\cdot\widetilde{\alpha_{02}}^{-1}=\widetilde{\alpha_{21}} \cdot \widetilde{\alpha_{02}}\cdot\widetilde{\alpha_{10}} \\ &\qquad =\widetilde{\alpha_{12}}\cdot(\widetilde{\alpha_{21}} \cdot \widetilde{\alpha_{02}} \cdot \widetilde{\alpha_{10}})\widetilde{\alpha_{12}}^{-1}=\widetilde{\alpha_{02}} \cdot \widetilde{\alpha_{10}}\cdot\widetilde{\alpha_{21}}, \end{aligned}
\end{equation*}
\notag
$$
where we have used that conjugation does not change the result. Hence, to prove the theorem it is enough to prove the formula
$$
\begin{equation*}
\varphi(\widetilde{\alpha_{01}}) \cdot \varphi(\widetilde{\alpha_{12}}) \cdot \varphi(\widetilde{\alpha_{20}}) -\widetilde{\alpha_{02}} \cdot \widetilde{\alpha_{10}} \cdot \widetilde{\alpha_{21}}=(\mathcal E, \mathcal O_X(\omega)).
\end{equation*}
\notag
$$
Let $c = \widetilde{\alpha_{02}} \cdot \widetilde{\alpha_{10}} \cdot \widetilde{\alpha_{21}}$. Then $c \cdot \widetilde{\alpha_{12}} = \widetilde{\alpha_{02}} \cdot \widetilde{\alpha_{10}}$. Therefore, by the lemma (see § 7) we have
$$
\begin{equation*}
c=\nu_{\mathbb{A}_{12}, \mathbb{A}_{12}(\mathcal E)} -\mu_{\mathbb{A}_{12}, \mathbb{A}_{12}(\mathcal E)}.
\end{equation*}
\notag
$$
Now we calculate $d = \varphi(\widetilde{\alpha_{01}}) \cdot \varphi(\widetilde{\alpha_{12}}) \cdot \varphi(\widetilde{\alpha_{20}}) = \varphi(\widetilde{\alpha_{12}}) \cdot \varphi(\widetilde{\alpha_{20}}) \cdot \varphi(\widetilde{\alpha_{01}})$. We have
$$
\begin{equation}
d \cdot \varphi(\widetilde{\alpha_{21}})=\varphi(\widetilde{\alpha_{20}}) \cdot \varphi(\widetilde{\alpha_{01}}).
\end{equation}
\tag{8.4}
$$
From the construction of $\varphi$ and formulae (8.3) we obtain
$$
\begin{equation*}
\begin{gathered} \, \varphi (\widetilde{\alpha_{01}})=\varphi ((\alpha_{01}, \nu_{\mathbb{A}_{12}, \alpha_{01} \mathbb{A}_{12}}))=(\alpha_{10}, \nu_{\mathbb{A}_{12}^{\perp}, \alpha_{10} \mathbb{A}_{12}^{\perp}})=(\alpha_{10}, \nu_{\mathbb{A}_{12}((\omega)), \mathbb{A}_{12}((\omega)+D)}), \\ \varphi (\widetilde{\alpha_{02}})=\varphi ((\alpha_{02}, \mu_{\mathbb{A}_{12}, \alpha_{02} \mathbb{A}_{12}}))=(\alpha_{20}, \mu_{\mathbb{A}_{12}^{\perp}, \alpha_{20} \mathbb{A}_{12}^{\perp}})=(\alpha_{20}, \mu_{\mathbb{A}_{12}((\omega)), \mathbb{A}_{12}((\omega)+D)}), \\ \varphi(\widetilde{\alpha_{21}})=\varphi((\alpha_{21}, 0))=(\alpha_{12}, 0). \end{gathered}
\end{equation*}
\notag
$$
From these formulae and (8.4), for the same reason as in the calculation of $c$ above (see the proof of the lemma) we obtain
$$
\begin{equation*}
d=\nu_{\mathbb{A}_{12}((\omega)), \mathbb{A}_{12}((\omega)+D)} -\mu_{\mathbb{A}_{12}((\omega)), \mathbb{A}_{12}((\omega)+D)}.
\end{equation*}
\notag
$$
Now we have
$$
\begin{equation*}
\begin{aligned} \, d -c &=(\nu_{\mathbb{A}_{12}((\omega)), \mathbb{A}_{12}((\omega)+D)} - \mu_{\mathbb{A}_{12}((\omega)), \mathbb{A}_{12}((\omega)+D)})-(\nu_{\mathbb{A}_{12}, \mathbb{A}_{12}(D)} -\mu_{\mathbb{A}_{12}, \mathbb{A}_{12}(D)}) \\ &=(\nu_{\mathbb{A}_{12}((\omega)), \mathbb{A}_{12}((\omega)+D)} - \mu_{\mathbb{A}_{12}((\omega)), \mathbb{A}_{12}((\omega)+D)})+(\mu_{\mathbb{A}_{12}, \mathbb{A}_{12}(D)} -\nu_{\mathbb{A}_{12}, \mathbb{A}_{12}(D)} ) \\ &\qquad +(\nu_{\mathbb{A}_{12}(D), \mathbb{A}_{12}(\omega)} -\nu_{\mathbb{A}_{12}(D), \mathbb{A}_{12}(\omega)}) \\ &=(\mu_{\mathbb{A}_{12}, \mathbb{A}_{12}(D)} -\nu_{\mathbb{A}_{12}, \mathbb{A}_{12}(D)} ) +(\nu_{\mathbb{A}_{12}(D), \mathbb{A}_{12}(\omega)} -\nu_{\mathbb{A}_{12}(D), \mathbb{A}_{12}(\omega)}) \\ &\qquad+(\nu_{\mathbb{A}_{12}((\omega)), \mathbb{A}_{12}((\omega)+D)} - \mu_{\mathbb{A}_{12}((\omega)), \mathbb{A}_{12}((\omega)+D)}) \\ &=\mu_{\mathbb{A}_{12}, \mathbb{A}_{12}(D)} \otimes \nu_{\mathbb{A}_{12}(D), \mathbb{A}_{12}(\omega)} \otimes \nu_{\mathbb{A}_{12}((\omega)), \mathbb{A}_{12}((\omega)+D)} \\ &\qquad -\nu_{\mathbb{A}_{12}, \mathbb{A}_{12}(D)} \otimes \nu_{\mathbb{A}_{12}(D), \mathbb{A}_{12}(\omega)} \otimes \mu_{\mathbb{A}_{12}((\omega)), \mathbb{A}_{12}((\omega)+D)} \\ &=\mu_{\mathbb{A}_{12}, \mathbb{A}_{12}(D)} \otimes \nu_{\mathbb{A}_{12}(D), \mathbb{A}_{12}((\omega)+D)} -\nu_{\mathbb{A}_{12}, \mathbb{A}_{12}(\omega)} \otimes \mu_{\mathbb{A}_{12}((\omega)), \mathbb{A}_{12}((\omega)+D)}. \end{aligned}
\end{equation*}
\notag
$$
Hence from Proposition 2 we obtain $d-c=(D,(\omega))$.
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\Bibitem{Osi22} |
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