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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.21596 |
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| _version_ | 1866912569967509504 |
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| author | Borić, Bruno Sakthivadivel, Dalton A R |
| author_facet | Borić, Bruno Sakthivadivel, Dalton A R |
| contents | Spencer cohomology theory studies the cohomology of chain complexes of modules over the ring of differential operators $\mathscr{D}$ of a smooth analytic space. In this paper we give a generalisation of Spencer cohomology suitable for singular schemes of finite type over a field. Our motivation was a conjecture of Vinogradov concerning the homological properties of differential operators on singular affine varieties; namely, that complexes of certain such operators are acyclic if and only if the variety is smooth. We will provide a negative answer to Vinogradov's conjecture as stated. In principle Vinogradov's conjecture can also be posed for the Spencer complex of a general $\mathscr{D}$-module -- however the answer is trivial, since singularities prohibit a definition of Spencer cohomology with any good properties. Our main result will be the construction of a Spencer complex on a large class of singular schemes which is suitable as a cohomology theory for the space. Following this we are able to ask the same question as Vinogradov in this case, where we give a more positive answer. Our main technique draws from Hartshorne's construction of de Rham cohomology by formal completion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_21596 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Completions of complexes of differential modules on singular schemes Borić, Bruno Sakthivadivel, Dalton A R Algebraic Geometry 14B05, 14F10, 58H10 Spencer cohomology theory studies the cohomology of chain complexes of modules over the ring of differential operators $\mathscr{D}$ of a smooth analytic space. In this paper we give a generalisation of Spencer cohomology suitable for singular schemes of finite type over a field. Our motivation was a conjecture of Vinogradov concerning the homological properties of differential operators on singular affine varieties; namely, that complexes of certain such operators are acyclic if and only if the variety is smooth. We will provide a negative answer to Vinogradov's conjecture as stated. In principle Vinogradov's conjecture can also be posed for the Spencer complex of a general $\mathscr{D}$-module -- however the answer is trivial, since singularities prohibit a definition of Spencer cohomology with any good properties. Our main result will be the construction of a Spencer complex on a large class of singular schemes which is suitable as a cohomology theory for the space. Following this we are able to ask the same question as Vinogradov in this case, where we give a more positive answer. Our main technique draws from Hartshorne's construction of de Rham cohomology by formal completion. |
| title | Completions of complexes of differential modules on singular schemes |
| topic | Algebraic Geometry 14B05, 14F10, 58H10 |
| url | https://arxiv.org/abs/2508.21596 |