Categorifying non-commutative deformations

Abstract

We define the functor $\textrm{ncDef}_{(Z_1,\ldots,Z_n)}$ of non-commutative deformations of an $n$-tuple of objects in an arbitrary $k$-linear abelian category $\mathcal{Z}$. In our categorified approach, we view the underlying spaces of infinitesimal flat deformations as Deligne finite categories, i.e. finite length abelian categories admitting projective generators, with $n$ isomorphism classes of simple objects. More generally, we define the functor $\textrm{ncDef}_{\zeta}$ of non-commutative deformations of an exact functor $\zeta \colon \mathcal{A} \to \mathcal{Z}$ of abelian categories. Here the role of an infinitesimal non-commutative thickening of $\mathcal{A}$ is played by an abelian category $\mathcal{B}$ containing $\mathcal{A}$ and such that $\mathcal{A}$ generates $\mathcal{B}$ by extensions. The functor $\textrm{ncDef}_{\zeta}$ assigns to such $\mathcal{B}$ the set of equivalence classes of exact functors $\mathcal{B} \to \mathcal{Z}$ which extend $\zeta$. We prove that an exact functor on an infinitesimal extension is fully faithful if and only if it is fully faithful on the first infinitesimal neighbourhood. We show that if $\zeta$ is fully faithful, then the functor $\textrm{ncDef}_{\zeta}$ is ind-represented by the extension closure of the essential image of $\zeta$. We prove that for a flopping contraction $f\colon X\to Y$ with the fiber over a closed point $C = \bigcup_{i=1}^n C_i$, where $C_i$'s are irreducible curves, $\{\mathcal{O}_{C_i}(-1)\}$ is the set of simple objects in the null-category for $f$. We conclude that the null-category ind-represents the functor $\textrm{ncDef}_{(\mathcal{O}_{C_1}(-1),\ldots,\mathcal{O}_{C_n}(-1))}$.