Tannaka duality on quotient stacks
type: Item
instance of: scholarly article
title: Tannaka duality on quotient stacks
zbMATH Open document ID: 1094.14002
author: Valentin Savin
published in: Manuscripta Mathematica
publication date: 2006-03-21T00:00:00Z
review text: The paper under review is concerned with the general problem of constructing dual objects in the category of projective schemes, meaning that a given scheme should be canonically isomorphic to its associated double dual. Prominent examples in algebraic geometry are provided by the dual of an abelian variety, by S. Mukai's duality theory of polarized \(K3\) surfaces, or by certain moduli spaces of stable vector bundles over a smooth projective curve [à la \textit{M. S. Narasimhan} and \textit{S. Ramanan}, Ann. Math. (2) 101, 391--417 (1975; Zbl 0314.14004)]. With these classical examples in mind, the author's underlying philosophy is that certain stacks of vector bundles should be natural categorical candidates for dual objects in algebraic geometry. Indeed, pursuing this strategy, he is able to construct dual objects for quotient stacks of schemes in the following way. Let \({\mathcal X}= [X/G]\) be the quotient stack associated with a scheme \(X\) acted on by an affine group scheme \(G\), and denote by \(\text{VB}({\mathcal X})\) the category of all vector bundles on the stack \({\mathcal X}\). For any affine base scheme \(S\), a \(S\)-valued fiber functor on \(\text{VB}({\mathcal X})\) is defined to be an exact additive tensor functor from \(\text{VB}({\mathcal X})\) to \(\text{VB}(S)\), and these fiber functors form a fibered category denoted by \(\text{FIB}(\text{VB}({\mathcal X}))\). The basic technical framework for this construction is explained in the first sections of the present paper. Then it is shown that there is a natural morphism of fibered categories \(D_{{\mathcal X}}:{\mathcal X}\to\text{FIB} (\text{VB}({\mathcal X}))\), and the author's main theorem establishes the following fact: Assume that the scheme \(X\) is quasi-compact and that there is a line bundle on the quotient stack \({\mathcal X}= [X/G]\) whose underlying line bundle on \(X\) is very ample. Then the associated ``duality functor'' \(D_{{\mathcal X}}:{\mathcal X}\to\text{FIB} ({\mathca…
reviewed by: Werner Kleinert
Mathematics Subject Classification ID: 14A20
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 18D30
Mathematics Subject Classification ID: 18F20
zbMATH DE Number: 5014046
zbMATH Keywords: algebraic stacks
zbMATH Keywords: group actions on schemes
zbMATH Keywords: tensor categories
zbMATH Keywords: Grothendieck topologies
zbMATH Keywords: duality theory
zbMATH Keywords: vector bundles
zbMATH Keywords: scheme
zbMATH Keywords: fibered categories
zbMATH Keywords: Tannakian categories
MaRDI profile type: Publication
full work available at URL: s00229-005-0616-8
OpenAlex ID: W1989785135
cites work: Hodge cycles, motives, and Shimura varieties
cites work: Q5698979
cites work: Q4693151
cites work: Q4263534
cites work: Q4090463
cites work: Deformations of the moduli space of vector bundles over an algebraic curve
DOI: 10.1007/S00229-005-0616-8
description: scientific article; zbMATH DE number 5014046