Categorifying non-commutative deformations
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866912377858949120 |
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| author | Bodzenta, Agnieszka Bondal, Alexey |
| author_facet | Bodzenta, Agnieszka Bondal, Alexey |
| contents | 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 $\textrm{ncDef}_ζ$ of non-commutative deformations of an exact functor $ζ\colon \mathcal{A} \to \mathcal{Z}$ 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 $\textrm{ncDef}_ζ$ assigns to such $\mathcal{B}$ the set of equivalence classes of exact functors $\mathcal{B} \to \mathcal{Z}$ 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 $\textrm{ncDef}_ζ$ is ind-represented by the extension closure of the essential image of $ζ$.
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 $\textrm{ncDef}_{(\mathcal{O}_{C_1}(-1),\ldots,\mathcal{O}_{C_n}(-1))}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_03084 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Categorifying non-commutative deformations Bodzenta, Agnieszka Bondal, Alexey Algebraic Geometry 14A22, 14D15, 14E05, 18E10 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 $\textrm{ncDef}_ζ$ of non-commutative deformations of an exact functor $ζ\colon \mathcal{A} \to \mathcal{Z}$ 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 $\textrm{ncDef}_ζ$ assigns to such $\mathcal{B}$ the set of equivalence classes of exact functors $\mathcal{B} \to \mathcal{Z}$ 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 $\textrm{ncDef}_ζ$ is ind-represented by the extension closure of the essential image of $ζ$. 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 $\textrm{ncDef}_{(\mathcal{O}_{C_1}(-1),\ldots,\mathcal{O}_{C_n}(-1))}$. |
| title | Categorifying non-commutative deformations |
| topic | Algebraic Geometry 14A22, 14D15, 14E05, 18E10 |
| url | https://arxiv.org/abs/2004.03084 |