A Hilbert 90-type theorem for perfect complexes, and the homotopical Skolem--Noether theorem
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This is joint work with Ajneet Dhillon. We prove two results about the infinity-stack of perfect complexes. Perfect complexes have better properties than coherent sheaves. For example, if X is a quasi-compact and quasi-separated scheme, U is an open quasi-compact subscheme, and E is a perfect complex on U, then E is the restriction of a perfect complex on X if and only if the K_0-class of E comes from a class in K_0(X) [Thomason--Trobaugh]. The price to pay is that perfect complexes have higher cocycles. Therefore, they need to be classified by a higher stack. We will introduce the notions in higher stack theory we use in the language of quasi-categories.

Let f: X--->S be a proper morphism of schemes, and E a perfect complex on X. We extend the result in [Toën--Vaquié, 2007] to show that the stack of automorphisms of families Aut_{X/S}(E) is algebraic. Then we study the deformation theory of morphisms of perfect complexes, and show that assuming that the cohomology sheaves of E are locally free, the automorphism infinity-group Aut_X(E) is formally smooth. Finally, we show the following Hilbert 90-type result: Let F be another perfect complex on X. Suppose that F is fppf-locally quasi-isomorphic to E. Then F is Zariski locally quasi-isomorphic to E.

A derived Azumaya algebra is a perfect complex with an algebra structure which is locally quasi-isomorphic to the REnd of a perfect complex. Every cohomological Brauer class is induced by one of these objects [Toën, 2012], and the stack of these compactifies the stack of ordinary Azumaya algebras [Lieblich, 2009]. We prove the homotopical Skolem--Noether theorem: for a perfect complex E on X, we have a fibration sequence BG_m -> BAut E -> BAut REnd E. The derived Skolem--Noether theorem in [Lieblich, 2009] is the 0-truncation of this.