Categorifying non-commutative deformations

Abstract

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