# 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.