JournalsjncgVol. 5, No. 4pp. 477–505

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

  • Olivier De Deken

    Universiteit Antwerpen, Belgium
  • Wendy Lowen

    Universiteit Antwerpen, Belgium
Abelian and derived deformations in the presence of ℤ-generating geometric helices cover
Download PDF

Abstract

For a Grothendieck category C\mathcal{C} which, via a Z\mathbb{Z}-generating sequence (O(n))nZ(\mathcal{O}(n))_{n \in \mathbb{Z}}, is equivalent to the category of “quasi-coherent modules” over an associated Z\mathbb{Z}-algebra a\mathfrak{a}, we show that under suitable cohomological conditions “taking quasi-coherent modules” defines an equivalence between linear deformations of a\mathfrak{a} and abelian deformations of C\mathcal{C}. If (O(n))nZ(\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) Z\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