# Algebra/Topology Seminar

**Speaker**: Pal Zsamboki (Alfred Renyi Institute of Mathematics, Budapest)

**Title**: Hilbert 90 and Skolem--Noether for perfect complexes.

**Abstract**: This is joint work with Ajneet Dhillon, see arXiv:1901.06816

[math.AG]. We prove two results about perfect complexes, both of which

are generalizations of classical theorems about vector bundles.

Let X be a scheme. The Hilbert 90 theorem says

H^1_fppf(X,GL_n)=H^1_Zar(X,GL_n). That is, if F is an fppf-form of

O_X^{\oplus n}, ie there exists an fppf cover U --> X and an isomorphism

F|U\cong O_U^{\oplus n}, then there exists such a Zariski cover too. In

other words, F is a vector bundle of rank n. We have proven the

following. Suppose that X is a Noetherian scheme with infinite residue

fields. Let E,F be perfect complexes. Suppose that there exists an fppf

cover U --> X and a quasi-isomorphism E|U\simeq F|U. Then there exists

such a Zariski cover too.

The Skolem--Noether theorem says that 1 --> G_m --> GL_n -Ad-> PGL_n -->

1 is a short exact sequence of group sheaves. Delooping this, we get a

fibre sequence of pointed stacks BG_m --> Vec_n -End-> Az_n. In

particular, the gerbe of trivializations of an Azumaya algebra is a

G_m-gerbe, the class of which in H^2(X,G_m) is the Brauer class

[Giraud]. Not every H^2(X,G_m)-class can be represented by an Azumaya

algebra, but Toën has shown [2012] that we can always find a

representative that is a derived Azumaya algebra: an algebra A on a

perfect complex such that locally it is quasi-isomorphic to REnd E for

some perfect complex E. We have shown that Skolem--Noether holds even in

this context: Let E be a perfect complex, and let i: Supp E --> X denote

the inclusion of its support. Then i_* B G_m --> B Aut E -REnd-> B Aut

REnd E is a fibre sequence of pointed infinity-stacks. We show how this

implies the classical, and the derived [Lieblich, 2009] Skolem--Noether

theorems.