Automorphismes, graduations et categories triangulees

TL;DR

This paper explores the automorphism groups of finite dimensional algebras and smooth projective varieties, revealing invariance properties under derived equivalences and proposing methods to transfer gradings and construct homological invariants.

Contribution

It introduces a moduli interpretation of outer automorphism groups, proves their invariance under derived and stable equivalences, and applies these results to gradings and homological constructions.

Findings

Out^0 is invariant under derived and stable equivalences

Gradings can be transferred between algebras using automorphism invariants

Invariance of Pic^0 x Aut^0 under derived equivalence for varieties

Abstract

We give a moduli interpretation of the outer automorphism group Out of a finite dimensional algebra similar to that of the Picard group of a scheme. We deduce that Out^0 is invariant under derived and stable equivalences. This allows us to transfer gradings between algebras and gives rise to conjectural homological constructions of interesting gradings on block of finite groups with abelian defect. We give applications to the lifting of stable equivalences to derived equivalences. We give a counterpart of the invariance result for smooth projective varieties: the product Pic^0xAut^0 is invariant under derived equivalence.