Kavli Affiliate: Alexey Bondal
| First 5 Authors: Agnieszka Bodzenta, Alexey Bondal, , ,
| Summary:
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 $fcolon Xto Y$ 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
$textrm{ncDef}_{(mathcal{O}_{C_1}(-1),ldots,mathcal{O}_{C_n}(-1))}$.
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