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

### Olivier De Deken

Universiteit Antwerpen, Belgium### Wendy Lowen

Universiteit Antwerpen, Belgium

## 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.

## Cite this article

Olivier De Deken, Wendy Lowen, Abelian and derived deformations in the presence of ℤ-generating geometric helices. J. Noncommut. Geom. 5 (2011), no. 4, pp. 477–505

DOI 10.4171/JNCG/83