Resolutions of symmetric ideals via stratifications of derived categories
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909265260707840 |
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| author | Ganapathy, Karthik |
| author_facet | Ganapathy, Karthik |
| contents | We propose a method to unify various stability results about symmetric ideals in polynomial rings by stratifying related derived categories. We execute this idea for chains of $GL_n$-equivariant modules over an infinite field $k$ of positive characteristic. We prove the Le--Nagel--Nguyen--Römer conjectures for such sequences and obtain stability patterns in their resolutions as corollaries of our main result, which is a semiorthogonal decomposition for the bounded derived category of $GL_{\infty}$-equivariant modules over $S = k[x_1, x_2, \ldots, x_n, \ldots]$. Our method relies on finite generation results for certain local cohomology modules. We also outline approaches (i) to investigate Koszul duality for $S$-modules taking the Frobenius homomorphism (of $GL_{\infty}$) into account, and (ii) to recover and extend Murai's results about free resolutions of symmetric monomial ideals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16071 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Resolutions of symmetric ideals via stratifications of derived categories Ganapathy, Karthik Commutative Algebra Representation Theory 13A50, 13D02, 13D45, 18G80 We propose a method to unify various stability results about symmetric ideals in polynomial rings by stratifying related derived categories. We execute this idea for chains of $GL_n$-equivariant modules over an infinite field $k$ of positive characteristic. We prove the Le--Nagel--Nguyen--Römer conjectures for such sequences and obtain stability patterns in their resolutions as corollaries of our main result, which is a semiorthogonal decomposition for the bounded derived category of $GL_{\infty}$-equivariant modules over $S = k[x_1, x_2, \ldots, x_n, \ldots]$. Our method relies on finite generation results for certain local cohomology modules. We also outline approaches (i) to investigate Koszul duality for $S$-modules taking the Frobenius homomorphism (of $GL_{\infty}$) into account, and (ii) to recover and extend Murai's results about free resolutions of symmetric monomial ideals. |
| title | Resolutions of symmetric ideals via stratifications of derived categories |
| topic | Commutative Algebra Representation Theory 13A50, 13D02, 13D45, 18G80 |
| url | https://arxiv.org/abs/2407.16071 |