$\mathbb{P}^n$-functors
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2019
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| _version_ | 1866909793416904704 |
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| author | Anno, Rina Logvinenko, Timothy |
| author_facet | Anno, Rina Logvinenko, Timothy |
| contents | We propose a new theory of (non-split) P^n-functors. These are F: A -> B for which the adjunction monad RF is a repeated extension of Id_A by powers of an autoequivalence H and three conditions are satisfied: the monad condition, the adjoints condition, and the highest degree term condition. This unifies and extends the two earlier notions of spherical functors and split P^n-functors. We construct the P-twist of such F and prove it to be an autoequivalence. We then give a criterion for F to be a P^n-functor which is stronger than the definition but much easier to check in practice. It involves only two conditions: the strong monad condition and the weak adjoints condition. For split P^n-functors, we prove Segal's conjecture on their relation to spherical functors. Finally, we give four examples of non-split P^n-functors: spherical functors, extensions by zero, cyclic covers, and family P-twists. For the latter, we show the P-twist to be the derived monodromy of associated Mukai flop, the so-called `flop-flop = twist' formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1905_05740 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | $\mathbb{P}^n$-functors Anno, Rina Logvinenko, Timothy Algebraic Geometry Category Theory 14F05, 18E30, 18D20, 18G99 We propose a new theory of (non-split) P^n-functors. These are F: A -> B for which the adjunction monad RF is a repeated extension of Id_A by powers of an autoequivalence H and three conditions are satisfied: the monad condition, the adjoints condition, and the highest degree term condition. This unifies and extends the two earlier notions of spherical functors and split P^n-functors. We construct the P-twist of such F and prove it to be an autoequivalence. We then give a criterion for F to be a P^n-functor which is stronger than the definition but much easier to check in practice. It involves only two conditions: the strong monad condition and the weak adjoints condition. For split P^n-functors, we prove Segal's conjecture on their relation to spherical functors. Finally, we give four examples of non-split P^n-functors: spherical functors, extensions by zero, cyclic covers, and family P-twists. For the latter, we show the P-twist to be the derived monodromy of associated Mukai flop, the so-called `flop-flop = twist' formula. |
| title | $\mathbb{P}^n$-functors |
| topic | Algebraic Geometry Category Theory 14F05, 18E30, 18D20, 18G99 |
| url | https://arxiv.org/abs/1905.05740 |