Tensor weight structures and t-structures on the derived categories of schemes
We give a condition which characterises those weight structures on a derived category which come from a Thomason filtration on the underlying scheme. Weight structures satisfying our condition will be called $\otimes ^c$-weight structures. More precisely, for a Noetherian separated scheme $X$, we gi...
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Académie des sciences
2023-07-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.450/ |
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| author | Dubey, Umesh V. Sahoo, Gopinath |
| author_facet | Dubey, Umesh V. Sahoo, Gopinath |
| author_sort | Dubey, Umesh V. |
| collection | DOAJ |
| description | We give a condition which characterises those weight structures on a derived category which come from a Thomason filtration on the underlying scheme. Weight structures satisfying our condition will be called $\otimes ^c$-weight structures. More precisely, for a Noetherian separated scheme $X$, we give a bijection between the set of compactly generated $\otimes ^c$-weight structures on $\mathbf{D} (\mathrm{Qcoh}\, X)$ and the set of Thomason filtrations of $X$. We achieve this classification in two steps. First, we show that the bijection [12, Theorem 4.10] restricts to give a bijection between the set of compactly generated $\otimes ^c$-weight structures and the set of compactly generated tensor t-structures. We then use our earlier classification of compactly generated tensor t-structures to obtain the desired result. We also study some immediate consequences of these classifications in the particular case of the projective line. We show that in contrast to the case of tensor t-structures, there are no non-trivial tensor weight structures on $\mathbf{D}^b (\mathrm{Coh}\, \mathbb{P}^1_k)$. |
| format | Article |
| id | doaj-art-1870f104d6c64176835e856ff93b772b |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-07-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-1870f104d6c64176835e856ff93b772b2025-02-07T11:08:07ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-07-01361G587788810.5802/crmath.45010.5802/crmath.450Tensor weight structures and t-structures on the derived categories of schemesDubey, Umesh V.0Sahoo, Gopinath1Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute, Chhatnag Road, Jhunsi, Prayagraj 211019, IndiaHarish-Chandra Research Institute, A CI of Homi Bhabha National Institute, Chhatnag Road, Jhunsi, Prayagraj 211019, IndiaWe give a condition which characterises those weight structures on a derived category which come from a Thomason filtration on the underlying scheme. Weight structures satisfying our condition will be called $\otimes ^c$-weight structures. More precisely, for a Noetherian separated scheme $X$, we give a bijection between the set of compactly generated $\otimes ^c$-weight structures on $\mathbf{D} (\mathrm{Qcoh}\, X)$ and the set of Thomason filtrations of $X$. We achieve this classification in two steps. First, we show that the bijection [12, Theorem 4.10] restricts to give a bijection between the set of compactly generated $\otimes ^c$-weight structures and the set of compactly generated tensor t-structures. We then use our earlier classification of compactly generated tensor t-structures to obtain the desired result. We also study some immediate consequences of these classifications in the particular case of the projective line. We show that in contrast to the case of tensor t-structures, there are no non-trivial tensor weight structures on $\mathbf{D}^b (\mathrm{Coh}\, \mathbb{P}^1_k)$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.450/ |
| spellingShingle | Dubey, Umesh V. Sahoo, Gopinath Tensor weight structures and t-structures on the derived categories of schemes Comptes Rendus. Mathématique |
| title | Tensor weight structures and t-structures on the derived categories of schemes |
| title_full | Tensor weight structures and t-structures on the derived categories of schemes |
| title_fullStr | Tensor weight structures and t-structures on the derived categories of schemes |
| title_full_unstemmed | Tensor weight structures and t-structures on the derived categories of schemes |
| title_short | Tensor weight structures and t-structures on the derived categories of schemes |
| title_sort | tensor weight structures and t structures on the derived categories of schemes |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.450/ |
| work_keys_str_mv | AT dubeyumeshv tensorweightstructuresandtstructuresonthederivedcategoriesofschemes AT sahoogopinath tensorweightstructuresandtstructuresonthederivedcategoriesofschemes |