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

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.

💡 Research Summary

The paper establishes a precise correspondence between two seemingly different deformation theories—linear (algebraic) deformations of a Z‑algebra and abelian (categorical) deformations of a Grothendieck category—under the presence of a Z‑generating sequence and, when available, a geometric helix in the derived category.

Setting and notation.
Let ( \mathcal{C} ) be a Grothendieck abelian category equipped with a Z‑generating sequence ( ( \mathcal{O}(n) )_{n\in\mathbb{Z}} ). By definition, every object of ( \mathcal{C} ) can be built from sufficiently many shifts of the ( \mathcal{O}(n) )’s, and the Hom‑spaces satisfy suitable vanishing conditions (e.g. ( \operatorname{Ext}^1(\mathcal{O}(m),\mathcal{O}(n))=0 ) for ( m>n )). From this data one constructs a Z‑algebra ( A ) whose objects are the integers and whose morphism spaces are \