Stable sheaf cohomology and Koszul--Ringel duality
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909781141225472 |
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| author | Raicu, Claudiu VandeBogert, Keller |
| author_facet | Raicu, Claudiu VandeBogert, Keller |
| contents | We identify a close relationship between stable sheaf cohomology for polynomial functors applied to the cotangent bundle on projective space, and Koszul--Ringel duality on the category of strict polynomial functors as described in the work of Chałupnik, Krause, and Touzé. Combining this with recent results of Maliakas--Stergiopoulou we confirm a conjectured periodicity statement for stable cohomology. In a different direction, we find a remarkable invariance property for $\Ext$ groups between Schur functors associated to hook partitions, and compute all such extension groups over a field of arbitrary characteristic. We show that this is further equivalent to the calculation of $\Ext$ groups for partitions with $2$ rows (or $2$ columns), and as such it relates to Parker's recursive description of $\Ext$ groups for $\SL_2$-representations. Finally, we give a general sharp bound for the interval of degrees where stable cohomology of a Schur functor can be non-zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08923 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stable sheaf cohomology and Koszul--Ringel duality Raicu, Claudiu VandeBogert, Keller Representation Theory Commutative Algebra Algebraic Geometry We identify a close relationship between stable sheaf cohomology for polynomial functors applied to the cotangent bundle on projective space, and Koszul--Ringel duality on the category of strict polynomial functors as described in the work of Chałupnik, Krause, and Touzé. Combining this with recent results of Maliakas--Stergiopoulou we confirm a conjectured periodicity statement for stable cohomology. In a different direction, we find a remarkable invariance property for $\Ext$ groups between Schur functors associated to hook partitions, and compute all such extension groups over a field of arbitrary characteristic. We show that this is further equivalent to the calculation of $\Ext$ groups for partitions with $2$ rows (or $2$ columns), and as such it relates to Parker's recursive description of $\Ext$ groups for $\SL_2$-representations. Finally, we give a general sharp bound for the interval of degrees where stable cohomology of a Schur functor can be non-zero. |
| title | Stable sheaf cohomology and Koszul--Ringel duality |
| topic | Representation Theory Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2509.08923 |