Moduli spaces of semiorthogonal decompositions in families
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866908651378180096 |
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| author | Belmans, Pieter Okawa, Shinnosuke Ricolfi, Andrea T. |
| author_facet | Belmans, Pieter Okawa, Shinnosuke Ricolfi, Andrea T. |
| contents | To a smooth and proper morphism $\mathcal{X}\to U$ with quasicompact semiseparated target we associate a sheaf in the étale topology, which takes an affine $U$-scheme $V$ to the set of $V$-linear semiorthogonal decompositions (of fixed length) of the category $\operatorname{Perf}\mathcal{X}_V$. We use Artin's criterion to prove that, when $U$ is excellent, this is in fact an algebraic space which is moreover étale (though in general non-quasicompact and non-separated) over $U$. We moreover generalise the construction of the sheaf to families of geometric noncommutative schemes in the sense of Orlov. We also define a subfunctor classifying nontrivial semiorthogonal decompositions, and conjecture it is an open and closed subspace.
Along the way, we prove that for a smooth and proper family of schemes, a semiorthogonal decomposition of the bounded derived category of coherent sheaves of a fibre uniquely deforms over an étale neighbourhood of the point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2002_03303 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Moduli spaces of semiorthogonal decompositions in families Belmans, Pieter Okawa, Shinnosuke Ricolfi, Andrea T. Algebraic Geometry Category Theory K-Theory and Homology To a smooth and proper morphism $\mathcal{X}\to U$ with quasicompact semiseparated target we associate a sheaf in the étale topology, which takes an affine $U$-scheme $V$ to the set of $V$-linear semiorthogonal decompositions (of fixed length) of the category $\operatorname{Perf}\mathcal{X}_V$. We use Artin's criterion to prove that, when $U$ is excellent, this is in fact an algebraic space which is moreover étale (though in general non-quasicompact and non-separated) over $U$. We moreover generalise the construction of the sheaf to families of geometric noncommutative schemes in the sense of Orlov. We also define a subfunctor classifying nontrivial semiorthogonal decompositions, and conjecture it is an open and closed subspace. Along the way, we prove that for a smooth and proper family of schemes, a semiorthogonal decomposition of the bounded derived category of coherent sheaves of a fibre uniquely deforms over an étale neighbourhood of the point. |
| title | Moduli spaces of semiorthogonal decompositions in families |
| topic | Algebraic Geometry Category Theory K-Theory and Homology |
| url | https://arxiv.org/abs/2002.03303 |