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

Abstract

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