The curvature problem for formal and infinitesimal deformations

We interpret all Maurer-Cartan elements in the formal Hochschild complex of a small dg category which is cohomologically bounded above in terms of torsion Morita deformations. This solves the curvature problem, i.e. the phenomenon that such Maurer-Cartan elements naturally parameterize curved A_infinity deformations. In the infinitesimal setup, we show how (n+1)-th order curved deformations give rise to n-th order uncurved Morita deformations.

Non-commutative deformations and quasi-coherent modules

We identify a class of quasi-compact semi-separated (qcss) twisted presheaves of algebras A for which well-behaved Grothendieck abelian categories of quasi-coherent modules Qch(A) are defined. This class is stable under algebraic deformation, giving rise to a 1-1 correspondence between algebraic deformations of A and abelian deformations of Qch(A). For a qcss presheaf A, we use the Gerstenhaber-Schack (GS) complex to explicitely parameterize the first order deformations. For a twisted presheaf A with central twists, we descibe an alternative category QPr(A) of quasi-coherent presheaves which is equivalent to Qch(A), leading to an alternative, equivalent association of abelian deformations to GS cocycles of qcss presheaves of commutative algebras. Our construction applies to the restriction O of the structure sheaf of a scheme X to a finite semi-separating open affine cover (for which we have an equivalence between Qch(O) and Qch(X)). Under a natural identification of Gerstenhaber-Schack cohomology of O and Hochschild cohomology of X, our construction is shown to be equivalent to Todas construction in the smooth case.

Hochschild cohomology with support

In this paper we investigate the functoriality properties of map-graded Hochschild complexes. We show that the category MAP of map-graded categories is naturally a stack over the category of small categories endowed with a certain Grothendieck topology of 3-covers. For a related topology of infinity-covers on the cartesian morphisms in MAP, we prove that taking map-graded Hochschild complexes defines a sheaf. From the functoriality related to injections between map-graded categories, we obtain Hochschild complexes with support. We revisit Kellers arrow category argument from this perspective, and introduce and investigate a general Grothendieck construction which encompasses both the map-graded categories associated to presheaves of algebras and certain generalized arrow categories, which together constitute a pair of complementary tools for deconstructing Hochschild complexes.

On compact generation of deformed schemes

We obtain a theorem which allows to prove compact generation of derived categories of Grothendieck categories, based upon certain coverings by localizations. This theorem follows from an application of Rouquiers cocovering theorem in the triangulated context, and it implies Neemans result on compact generation of quasi-compact separated schemes. We prove an application of our theorem to non-commutative deformations of such schemes, based upon a change from Koszul complexes to Chevalley-Eilenberg complexes.

On deformations of triangulated models

This paper is the first part of a project aimed at understanding deformations of triangulated categories, and more precisely their dg and A infinity models, and applying the resulting theory to the models occurring in the Homological Mirror Symmetry setup. In this first paper, we focus on models of derived and related categories, based upon the classical construction of twisted objects over a dg or $A_{infty}$-algebra. For a Hochschild 2 cocycle on such a model, we describe a corresponding curvature compensating deformation which can be entirely understood within the framework of twisted objects. We unravel the construction in the specific cases of derived A infinity and abelian categories, homotopy categories, and categories of graded free qdg-modules. We identify a purity condition on our models which ensures that the structure of the model is preserved under deformation. This condition is typically fulfilled for homotopy categories, but not for unbounded derived categories.

Cohomology of exact categories and (non-)additive sheaves

We use (non-)additive sheaves to introduce an (absolute) notion of Hochschild cohomology for exact categories as Exts in a suitable bisheaf category. We compare our approach to various definitions present in the literature.

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.