Russian Mathematical Surveys
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Uspekhi Mat. Nauk:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Russian Mathematical Surveys, 2023, Volume 78, Issue 4, Pages 791–793
DOI: https://doi.org/10.4213/rm10138e
(Mi rm10138)
 

This article is cited in 1 scientific paper (total in 1 paper)

Brief communications

Determinant central extension and $\cup$-products of 1-cocycles

D. V. Osipov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
References:
Funding agency Grant number
Russian Science Foundation 23-11-00033
This work was supported by the Russian Science Foundation under grant no. 23-11-00033, https://rscf.ru/en/project/23-11-00033/.
Received: 05.06.2023
Russian version:
Uspekhi Matematicheskikh Nauk, 2023, Volume 78, Issue 4(472), Pages 207–208
DOI: https://doi.org/10.4213/rm10138
Bibliographic databases:
Document Type: Article
MSC: 14C40, 18G15, 19D45
Language: English
Original paper language: Russian

In this note the class of the determinant central extension of some group functor defined over the commutative $\mathbb{Q}$-algebras is calculated as a product of $2$-cocycles that consist of the Contou-Carrère symbol applied to pairwise $\cup$-products of $1$-cocycles. This is a local Riemann–Roch theorem for invertible sheaves in relative dimension $1$, as written in the second cohomology group of group functors.

By a functor (commutative group functor or group functor) $H$ we mean a covariant functor from the category of commutative rings to the category of sets (Abelian groups or groups, respectively). Let $H_{\mathbb{Q}}$ denote the restriction of $H$ to the commutative $\mathbb{Q}$-algebras. Let $H^{\times n}$ denote the functor that is the $n$th direct product of $H$ with itself. In all groups we use the multiplicative notation for the group laws.

Let $R$ be a commutative ring. Let $G$ and $F$ be a group functor and a commutative group functor, respectively, defined over the commutative $R$-algebras, and assume that $G$ acts on $F$ (so that $F$ is a $G$-module). We denote by $\operatorname{Hom}(G^{\times n},F)$ the Abelian group of all (without regard to group structures) morphisms of functors $G^{\times n } \to F$. Then $H^n(G, F)$ (see [4], § 2.3.1) is the $n$th cohomology group of the complex

$$ \begin{equation} C^0 (G,F) \xrightarrow{\delta_0} C^{1}(G, F) \xrightarrow{\delta_1} \cdots \xrightarrow{\delta_{n-1}} C^{n}(G,F) \xrightarrow{\delta_n} \cdots, \end{equation} \tag{1} $$
where $C^0(G,F)=F(R)$, $C^k(G,F)=\operatorname{Hom}(G^{\times k},F)$ for $k\geqslant 1$, and
$$ \begin{equation*} \begin{aligned} \, \delta_{n}c(g_1,\dots,g_{n+1})&=g_1 c(g_2,\dots,g_{n+1}) \\ &\qquad\times\prod_{i=1}^n c(g_1,\dots,g_i g_{i+1},\dots,g_{n+1})^{(-1)^i}\cdot c(g_1,\dots,g_n)^{(-1)^{n+1}}, \end{aligned} \end{equation*} \notag $$
where $c\in C^n(G,F)$ and $g_j \in G(A)$, $1 \leqslant j \leqslant n+1$, for an arbitrary commutative $R$-algebra $A$. An element of $\operatorname{Ker}\delta_n$ is called an $n$-cocycle on $G$ with coefficients in $F$.

For any $1$-cocycles $\lambda_1$ and $\lambda_2$ on $G$ with coefficients in $G$-modules $F_1$ and $F_2$, respectively, we obtain a $2$-cocycle $\lambda_1 \cup \lambda_2$ on $G$ with coefficients in $F_1 \otimes F_2$ such that $(\lambda_1 \cup \lambda_2) (g_1, g_2) = \lambda_1 (g_1) \otimes g_1 ( \lambda_2(g_2))$, where for any commutative $R$-algebra $A$ we have $(F_1 \otimes F_2)(A) = F_1(A) \otimes_{\mathbb{Z}} F_2(A)$, and where $g_1$ and $ g_2 $ are arbitrary elements of $G(A)$. This induces a $\cup$-product between the first cohomology groups. Any morphism of $G$-modules induces a homomorphism of the corresponding cohomology groups of $G$. Fixing a commutative $R$-algebra $A$ gives us a map from the complex (1) to the bar-complex for the $G(A)$-module $F(A)$ and a homomorphism $H^n(G,F) \to H^n(G(A), F(A))$ compatible with the $\cup$-products.

In what follows $A$ is an arbitrary commutative ring. A central extension of group functors is a short exact sequence of group functors that becomes a central extension of groups after fixing any $A$. Central extensions of $G$ by $F$ that admit a section from $G$ (just as functors) are classified up to isomorphism by elements of $H^2(G,F)$, where $F$ is a trivial $G$-module.

Let ${\mathbb G}_m(A)=A^*$. Let $L{\mathbb G}_m $ be a commutative group functor such that $L {\mathbb G}_m (A)=A((t))^*$, where $A((t))=A[[t]][t^{-1}]$. Then $A((t))$ is a topological ring with the following neighbourhood basis of zero: $U_l=t^l A[[t]]$, $l \in \mathbb{Z}$. Let ${{\mathcal Aut}^{\rm c, alg} ({\mathcal L})}$ be a group functor such that ${{\mathcal Aut}^{\rm c,alg}({\mathcal L})}(A)$ is the group of all $A$-automorphisms of the $A$-algebra $A((t))$ that are homeomorphisms; see [4], § 2.1. Each element $\varphi \in {{\mathcal Aut}^{\rm c,alg}({\mathcal L})}(A)$ is uniquely defined by the element $\widetilde{\varphi}=\varphi(t)\in A((t))^*$ (this gives the structure of a functor).

Since $L {\mathbb G}_m$ is an ${{\mathcal Aut}^{\rm c,alg}({\mathcal L})}$-module, we can define the group functor $\mathcal{G}=L{\mathbb G}_m \rtimes{\mathcal Aut}^{\rm c,alg}({\mathcal L})$. Note that $L{\mathbb G}_m$ is a $\mathcal{G}$-module because of the natural morphism ${\mathcal{G} \to {{\mathcal Aut}^{\rm c,alg}({\mathcal L})}}$. We also have a natural continuous action of ${\mathcal G}(A)$ on $A((t))$ such that $(h,\varphi)(f)=h \cdot \varphi(f)$, where $f \in A((t))$, $h \in L{\mathbb G}_m(A)$, and $\varphi \in {\mathcal Aut}^{\rm c,alg}({\mathcal L})(A)$.

For any $g_1,g_2\in {\mathcal G}(A)$ there exists $l\in {\mathbb Z}$ such that ${t^{l} A[[t]] \subset g_i(A[[t]])}$ and $g_i(A[[t]]) / t^l A[[t]]$ are projective $A$-modules of finite rank for $i=1$ and $i=2$ (see [4], § 3.2). We obtain the definition of the relative determinant (independent of the choice of $l$ up to canonical isomorphism):

$$ \begin{equation*} \begin{aligned} \, &\det(g_1(A[[t]]) \mid g_2 (A[[t]])) \\ &\qquad=\operatorname{Hom}_A(\wedge^{\max}_A (g_1(A[[t]])/ t^l A[[t]]), \wedge^{\max}_A (g_2(A[[t]])/ t^l A[[t]])). \end{aligned} \end{equation*} \notag $$

Proposition. A relative determinant is a free $A$-module of rank $1$.

Let $g_1, g_2, g_3, g \in {\mathcal G}(A)$. Then there are canonical isomorphisms of $A$-modules:

$$ \begin{equation*} \begin{gathered} \, \det(g_1(A[[t]])\,{\mid}\,g_2(A[[t]]))\,{\otimes_A}\,\det(g_2(A[[t]])\,{\mid}\,g_3(A[[t]]))\,{\xrightarrow{\sim}}\, \det(g_1(A[[t]])\,{\mid}\,g_3(A[[t]])), \\ g\colon \det(g_1(A[[t]])\,{\mid}\,g_2(A[[t]])) \xrightarrow{\sim} \det(gg_1(A[[t]])\,{\mid}\,gg_2(A[[t]])). \end{gathered} \end{equation*} \notag $$

Let the group $\widetilde{{\mathcal G}}(A)$ be the set of pairs $(g,s)$ such that $g \in {\mathcal G}(A)$ and $s$ is an element of the $A$-module ${\det(g (A[[t]]) \mid A[[t]])}$ that generates this $A$-module. The group law is as follows: $(g_1,s_1)(g_2,s_2)=(g_1 g_2,g_1(s_2) \otimes s_1)$. The correspondence $A \mapsto \widetilde{{\mathcal G}}(A)$ is a group functor that defines the determinant central extension of ${\mathcal G}$ by ${\mathbb G}_m$, where the homomorphism $\widetilde{{\mathcal G}}(A) \twoheadrightarrow {\mathcal G}(A)$ is defined by $(g,s) \mapsto g$.

The Contou-Carrère symbol $\operatorname{CC}$ is a morphism of group functors $ L {\mathbb G}_m \otimes L {\mathbb G}_m \to {\mathbb G}_m$ which has a lot of interesting properties; see [1] and [5], § 2. In particular, $\operatorname{CC}$ is a morphism of $ {\mathcal G}$-modules, where $ {\mathcal G}$ acts diagonally on $ L {\mathbb G}_m \otimes L {\mathbb G}_m $ and trivially on ${\mathbb G}_m$; see [3].

For any two $1$-cocycles $\lambda_1$ and $\lambda_2$ on ${\mathcal G}$ with coefficients in $L{\mathbb G}_m$ we define the $2$-cocycle $\langle \lambda_1,\lambda_2 \rangle= \operatorname{CC} \mathrel{\circ} (\lambda_1 \cup \lambda_2)$ on ${\mathcal G}$ with coefficients in ${\mathbb G }_m$, where $\circ$ means the composition of morphisms of functors.

We introduce $1$-cocycles $\Lambda$ and $\Omega$ on ${\mathcal G}$ with coefficients in $L {\mathbb G }_m$, where $\Lambda ((h, \varphi)) =h$ and $\Omega((h,\varphi))={\widetilde{\varphi}}^{\,\prime}=d\varphi(t)/dt$ for $(h,\varphi) \in {\mathcal G}(A)$.

Theorem. The determinant central extension of the group functor ${\mathcal G} $ by the group functor ${\mathbb G}_m$ admits a natural section ${\mathcal G} \to \widetilde{\mathcal G}$ as functors, and therefore this central extension yields an element $\mathcal D$ of $H^2({\mathcal G} , {\mathbb G}_m )$. The following equality holds in the group $H^2({\mathcal G}_{\mathbb{Q}},{{\mathbb G}_m}_{\mathbb{Q}})$:

$$ \begin{equation} {\mathcal D}^{12}=\langle \Lambda, \Lambda \rangle^{6} \cdot \langle\Lambda,\Omega\rangle^{-6} \cdot \langle\Omega,\Omega\rangle. \end{equation} \tag{2} $$

Formally, (2) looks like the Deligne–Riemann–Roch isomorphism from [2]. Consider a projective curve $C$ over a field $k$ and a commutative $k$-algebra $A$. Elements of ${\mathcal G}(A)$ reglue the scheme $C_A = C \times_k A$ and the sheaf ${\mathcal O}_{C_A}$ along the punctured formal neighbourhood of a constant section (to a smooth point). Fibres of the determinant central extension over elements of ${\mathcal G}(A)$ are canonically isomorphic to the difference between the determinants of the higher direct images of the reglued sheaves and the sheaf ${\mathcal O}_{C_A}$.


Bibliography

1. C. Contou-Carrère, C. R. Acad. Sci. Paris Sér. I Math., 318:8 (1994), 743–746  mathscinet  zmath
2. P. Deligne, Current trends in arithmetical algebraic geometry (Arcata, CA 1985), Contemp. Math., 67, Amer. Math. Soc., Providence, RI, 1987, 93–177  mathscinet  zmath
3. S. O. Gorchinskiy and D. V. Osipov, Funct. Anal. Appl., 50:4 (2016), 268–280  mathnet  crossref  mathscinet  zmath
4. D. V. Osipov, Proc. Steklov Inst. Math., 320 (2023), 226–257  mathnet  crossref  mathscinet  zmath
5. D. Osipov and Xinwen Zhu, J. Algebraic Geom., 25:4 (2016), 703–774  crossref  mathscinet  zmath

Citation: D. V. Osipov, “Determinant central extension and $\cup$-products of 1-cocycles”, Uspekhi Mat. Nauk, 78:4(472) (2023), 207–208; Russian Math. Surveys, 78:4 (2023), 791–793
Citation in format AMSBIB
\Bibitem{Osi23}
\by D.~V.~Osipov
\paper Determinant central extension and $\cup$-products of 1-cocycles
\jour Uspekhi Mat. Nauk
\yr 2023
\vol 78
\issue 4(472)
\pages 207--208
\mathnet{http://mi.mathnet.ru/rm10138}
\crossref{https://doi.org/10.4213/rm10138}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4687813}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2023RuMaS..78..791O}
\transl
\jour Russian Math. Surveys
\yr 2023
\vol 78
\issue 4
\pages 791--793
\crossref{https://doi.org/10.4213/rm10138e}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=001146060800007}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85169451918}
Linking options:
  • https://www.mathnet.ru/eng/rm10138
  • https://doi.org/10.4213/rm10138e
  • https://www.mathnet.ru/eng/rm/v78/i4/p207
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Успехи математических наук Russian Mathematical Surveys
    Statistics & downloads:
    Abstract page:529
    Russian version PDF:23
    English version PDF:32
    Russian version HTML:197
    English version HTML:131
    References:30
    First page:22
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024