The deformation theory of t-dg-categories - Julie Symons

Talk by Julie Symons (University of Antwerp), at the Antwerp Algebra, Geometry and Number Theory Seminar on November 8, 2024.

It is well-known that the Hochschild cohomology of an algebraic object (such as an algebra or a linear category) should govern its deformations. In the case of a dg-category however, the Hochschild cohomology has been shown to parametrize not the dg-deformations, but the curved A-infinity deformations. This phenomenon is known as the curvature problem. We therefore seek a different deformation-theoretic interpretation of the Hochschild cohomology of a dg-category. Motivated by the example of the bounded derived category of an abelian category — which can be approached via the deformation theory of abelian categories in the sense of Lowen-Van den Bergh — we develop the deformation theory of other dg-categories with similar extra data: pretriangulated dg-categories with a t-structure, abbreviated to t-dg-categories.

In this talk, I will establish the deformation theory of t-dg-categories following the prototypical example of the bounded derived category of an abelian category, thus allowing for novel interpretations of the higher Hochschild cohomology groups also in this case. In particular, we will discuss a deformation equivalence between the bounded t-deformations of a bounded t-dg-category on the one hand, and dg-deformations of the dg-category of derived injective ind-dg-objects on the other hand. Since this latter dg-category is cohomologically concentrated in nonpositive degrees, we do not encounter curvature.

This is joint work with Francesco Genovese, Wendy Lowen and Michel Van den Bergh.