Twisted Equivalences in Spectral Algebraic Geometry

TL;DR

This paper explores twisted derived equivalences in spectral algebraic geometry, establishing conditions under which such equivalences imply isomorphisms of stacks and rings, extending classical results to the spectral setting.

Contribution

It introduces the notion of twisted equivalences for spectral stacks and proves that these induce stack isomorphisms, generalizing derived equivalence results to spectral algebraic geometry.

Findings

Twisted equivalences for spectral stacks can imply isomorphisms of the stacks.

A spectral analogue of Rickard's theorem shows derived equivalences induce isomorphisms of centers.

The work extends classical derived equivalence results to the spectral algebraic geometry context.

Abstract

We study twisted derived equivalences for schemes in the setting of spectral algebraic geometry. To this end, we introduce the notion of a twisted equivalence and show that a twisted equivalence for perfect spectral algebraic stacks admitting a quasi-finite presentation supplies an equivalence between the stacks, which compensate for the failure of twisted derived equivalences for non-affine schemes to provide an isomorphism of the schemes. In the case of (not necessarily connective) commutative ring spectra, we also prove a spectral analogue of Rickard's theorem, which shows that a derived equivalence of associative rings induces an isomorphism between their centers.