Abelian and derived deformations in the presence of Z-generating geometric helices

TL;DR

This paper establishes a correspondence between deformations of Z-algebras and abelian categories, especially when the sequence forms a geometric helix, linking algebraic and categorical deformation theories.

Contribution

It demonstrates that under certain conditions, deformations of Z-algebras correspond to deformations of associated Grothendieck categories, and extends this to geometric helices in derived categories.

Findings

Equivalence between linear deformations of Z-algebras and abelian deformations of categories.

Restriction to a thread of objects yields an equivalence with matrix algebra deformations.

Applicable under suitable cohomological conditions.

Abstract

For a Grothendieck category C which, via a Z-generating sequence (O(n))_{n in Z}, is equivalent to the category of "quasi-coherent modules" over an associated Z-algebra A, we show that under suitable cohomological conditions "taking quasi-coherent modules" defines an equivalence between linear deformations of A and abelian deformations of C. If (O(n))_{n in Z} is at the same time a geometric helix in the derived category, we show that restricting a (deformed) Z-algebra to a "thread" of objects defines a further equivalence with linear deformations of the associated matrix algebra.