On t-structures adjacent and orthogonal to weight structures
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908709462999040 |
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| author | Bondarko, Mikhail V. |
| author_facet | Bondarko, Mikhail V. |
| contents | We study $t$-structures (on triangulated categories) that are closely related to weight structures.
A $t$-structure couple $t=(C_{t\le 0},C_{t\ge 0})$ is said to be adjacent to a weight structure $w=(C_{w\le 0}, C_{w\ge 0})$ if $C_{t\ge 0}=C_{w\ge 0}$.
For a category $C$ that satisfies the Brown representability property we prove that $t$ that is adjacent to $w$ exists if and only if $w$ is smashing (that is, "respects C-coproducts"). The heart $Ht$ of this $t$ is the category of those functors $Hw^{op}\to Ab$ that respect products (here $Hw$ is the heart of $w$); the result has important applications.
We prove several more statements on constructing $t$-structures starting from weight structures; we look for a strictly orthogonal $t$-structure $t$ on some $C'$ (where $C,C'$ are triangulated subcategories of a common $D$) such that $C'_{t\le 0}$ (resp. $C'_{t\ge 0}$) is characterized by the vanishing of morphisms from $C_{w\ge 1}$ (resp. $C_{w\le -1}$). Some of these results generalize properties of semi-orthogonal decompositions proved in the previous paper, and can be applied to various derived categories of (quasi)coherent sheaves on a scheme $X$ that is projective over an affine noetherian one. We also study hearts of orthogonal $t$-structures and their restrictions, and prove some statements on "reconstructing" weight structures from orthogonal $t$-structures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07855 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On t-structures adjacent and orthogonal to weight structures Bondarko, Mikhail V. K-Theory and Homology Algebraic Geometry Category Theory Representation Theory 18G80, 14F08 (Primary) 18F20, 18G05, 18E10, 14A15, 14G40 (Secondary) We study $t$-structures (on triangulated categories) that are closely related to weight structures. A $t$-structure couple $t=(C_{t\le 0},C_{t\ge 0})$ is said to be adjacent to a weight structure $w=(C_{w\le 0}, C_{w\ge 0})$ if $C_{t\ge 0}=C_{w\ge 0}$. For a category $C$ that satisfies the Brown representability property we prove that $t$ that is adjacent to $w$ exists if and only if $w$ is smashing (that is, "respects C-coproducts"). The heart $Ht$ of this $t$ is the category of those functors $Hw^{op}\to Ab$ that respect products (here $Hw$ is the heart of $w$); the result has important applications. We prove several more statements on constructing $t$-structures starting from weight structures; we look for a strictly orthogonal $t$-structure $t$ on some $C'$ (where $C,C'$ are triangulated subcategories of a common $D$) such that $C'_{t\le 0}$ (resp. $C'_{t\ge 0}$) is characterized by the vanishing of morphisms from $C_{w\ge 1}$ (resp. $C_{w\le -1}$). Some of these results generalize properties of semi-orthogonal decompositions proved in the previous paper, and can be applied to various derived categories of (quasi)coherent sheaves on a scheme $X$ that is projective over an affine noetherian one. We also study hearts of orthogonal $t$-structures and their restrictions, and prove some statements on "reconstructing" weight structures from orthogonal $t$-structures. |
| title | On t-structures adjacent and orthogonal to weight structures |
| topic | K-Theory and Homology Algebraic Geometry Category Theory Representation Theory 18G80, 14F08 (Primary) 18F20, 18G05, 18E10, 14A15, 14G40 (Secondary) |
| url | https://arxiv.org/abs/2403.07855 |