Derived Azumaya algebras and generators for twisted derived categories
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 \(H_{\text{ét}}^2(X,\mathbb{G}_m)\) is certainly not entirely torsion). The author more generally considers derived Azumaya algebras over derived stacks. In a first step, he proves that a derived stack \(\mathbb{D}g^{Az}\) which represents the functor \(F\mapsto\mathbb{D}\mathrm{eraz}^{\mathrm{dg}}(F)\) assigning a classifying space of derived Azumaya dg-categories to any derived stack for the fppf topology \(F\) is equivalent to \(K(\mathbb{Z},1)\times K(\mathbb{G}_m,2)\). The set \(\mathrm{dBr}(F) = \pi_0(\mathbb{D}\mathrm{eraz}^{\mathrm{dg}}(F))\) defines a (derived) group of Morita equivalences of derived Azumaya algebras over \(F\), and the statement on the derived stack \(\mathbb{D}g^{Az}\) implies that this group embeds into \(H^1_{\mathrm{fppf}}(F,\mathbb{Z})\times H^2_{\mathrm{fppf}}(F,\mathbb{G}_m)\). This result proves the injectivity of mapping (\(*\)) for any scheme \(X\). In a second step, the author studies a stack of locally presentable dg-categories over schemes. To any locally presentable dg-category \(\alpha\) over a scheme \(X\), he associates a dg-category of global sections \(L_{\alpha}(X)\), which he calls the twisted derived category of \(X\) with coefficients in \(\alpha\). He uses that the dg-categories of quasi-coherent sheaves of dg-modules over a dg-algebra \(B\) are identified with the dg-categories \(\alpha\) such that the twisted dg-category \(L_{\alpha}(X)\) is equipped with an object \(E\) whose restriction to any open affine subscheme \(U = \mathrm{Spec} A\subset X\) defines a compact generator of the category \(L_{\alpha}(U)\). He then deduces the surjectivity of mapping (\(*\)) from a general statement asserting that, in the case of a quasi-compact quasi-separated scheme \(X\), if we have an fppf covering \(X'\rightarrow X\) such that \(L_{\alpha}(X')\) admits a compact generator, then the category \(L_{\alpha}(X)\) admits a compact generator \(E\in L_{\alpha}(X)\) which is also a compact local generator (in the above sense).
scientific article; zbMATH DE number 6084026
scientific article