Kavli Affiliate: Alexey Bondal
| Summary:
We define the functor $textrmncDef_(Z_1,ldots,Z_n)$ of non-commutative deformations of an $n$-tuple of objects in an arbitrary $k$-linear abelian category $mathcalZ$. 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 $textrmncDef_ζ$ of non-commutative deformations of an exact functor $ζcolon mathcalA to mathcalZ$ of abelian categories. Here the role of an infinitesimal non-commutative thickening of $mathcalA$ is played by an abelian category $mathcalB$ containing $mathcalA$ and such that $mathcalA$ generates $mathcalB$ by extensions. The functor $textrmncDef_ζ$ assigns to such $mathcalB$ the set of equivalence classes of exact functors $mathcalB to mathcalZ$ which extend $ζ$. 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 $ζ$ is fully faithful, then the functor $textrmncDef_ζ$ is ind-represented by the extension closure of the essential image of $ζ$.
We prove that for a flopping contraction $fcolon Xto Y$ with the fiber over a closed point $C = bigcup_i=1^n C_i$, where $C_i$’s are irreducible curves, $\mathcalO_C_i(-1)$ is the set of simple objects in the null-category for $f$. We conclude that the null-category ind-represents the functor $textrmncDef_(mathcalO_C_1(-1),ldots,mathcalO_C_n(-1))$.
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