Derived Azumaya algebras and generators for twisted derived categories
type: Item
instance of: scholarly article
title: Derived Azumaya algebras and generators for twisted derived categories
zbMATH Open document ID: 1275.14017
author: Bertrand Toën
published in: Inventiones Mathematicae
publication date: 2012-09-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1002.2599
review text: The author defines the notion of a derived Azumaya algebra over a scheme \(X\) as a sheaf of \(\mathcal{O}_X\)-dg-algebras \(B\) so that \(B\) forms a perfect complex on \(X\), we have \(B\otimes^{\mathbb{L}}_{\mathcal{O}_X} k(x)\not=0\) for each point \(x\in X\), and the natural morphism of complexes of \(\mathcal{O}_X\)-modules \[ B\otimes^{\mathbb{L}}_{\mathcal{O}_X} B^{\mathrm{op}}\rightarrow\mathbb{R}\underline{\mathrm{Hom}}{}_{\mathcal{O}_X}(B,B) \] is a quasi-isomorphism. He associates a characteristic class \(\phi(B)\in H_{\text{ét}}^2(X,\mathbb{G}_m)\) to every derived Azumaya algebra \(B\), and he proves that this mapping, from derived Azumaya algebras to the étale cohomology group \(H_{\text{ét}}^2(X,\mathbb{G}_m)\), is surjective when \(X\) is a quasi-compact and quasi-separated scheme. He more precisely establishes the existence of an isomorphism \[ \mathrm{dBr}(X)\mathop{\rightarrow}^{\simeq} H_{\text{ét}}^1(X,\mathbb{Z})\times H_{\text{ét}}^2(X,\mathbb{G}_m),\leqno{(*)} \] for any scheme of this form \(X\), where \(\mathrm{dBr}(X)\) denotes the group of Morita equivalence classes of derived Azumaya algebras over \(X\). On one hand, this statement has to be compared with a result of \textit{O. Gabber} [Lect. Notes Math. 844, 129--209 (1981; Zbl 0472.14013)] stating that the group of ordinary Azumaya algebras on a scheme \(X\) surjects onto the subgroup of torsion classes in \(H_{\text{ét}}^2(X,\mathbb{G}_m)\) when \(X\) is a quasi-compact separated scheme equipped with an ample line bundle. On the other hand, some results hint that ordinary Azumaya algebras are not enough to retrieve the entire cohomology group \(H_{\text{ét}}^2(X,\mathbb{G}_m)\) in general (for instance, because the characteristic class of an ordinary Azumaya algebra is always torsion while we have examples of schemes, like the Mumford complex algebraic surface (see [\textit{A. Grothendieck}, Adv. Stud. Pure Math. 3, 46--66 (1968; Zbl 0193.21503)]), so that the group \(H_{\text{ét}}^2…
Mathematics Subject Classification ID: 14F22
Mathematics Subject Classification ID: 14A20
Mathematics Subject Classification ID: 16E45
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 6084026
zbMATH Keywords: derived Azumaya algebras
zbMATH Keywords: derived Brauer group
zbMATH Keywords: twisted derived categories
reviewed by: Benoît Fresse
MaRDI profile type: Publication
OpenAlex ID: W2104988791
arXiv ID: 1002.2599
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DOI: 10.1007/S00222-011-0372-1
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description: scientific article; zbMATH DE number 6084026