Künneth and extension theorems for Morita invariants of non-commutative schemes
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915060944732160 |
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| author | Nicolle, Alexandre |
| author_facet | Nicolle, Alexandre |
| contents | In this article, we apply the derived Morita theory of dg-categories to show how to extend the domain of validity of many identities relating Morita invariants from associative dg-algebras toward non-commutative scheme. Doing so, we obtain that the dg-category of associative algebras can be used to test the exactness of any sequence and the commutativity of any diagram involving Morita invariants depending multilinearly on their arguments, under a mild condition of stability by cofibrant replacement. This gives a simple picture of how the underlying algebra of a non-commutative scheme captures its Morita invariant properties. As an application, we use a Künneth formula on non-commutative schemes to factorise the Hochschild cohomology of a product of quasi-phantom categories such as built by Orlov and Gorshinsky in [GO13]. This is an occasion to explore how the derived Morita theory interacts with the Čech dg-enhancement of a projective scheme and to shed an unifying light upon the landscape of dg-modules over dg-categories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_08892 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Künneth and extension theorems for Morita invariants of non-commutative schemes Nicolle, Alexandre Algebraic Geometry In this article, we apply the derived Morita theory of dg-categories to show how to extend the domain of validity of many identities relating Morita invariants from associative dg-algebras toward non-commutative scheme. Doing so, we obtain that the dg-category of associative algebras can be used to test the exactness of any sequence and the commutativity of any diagram involving Morita invariants depending multilinearly on their arguments, under a mild condition of stability by cofibrant replacement. This gives a simple picture of how the underlying algebra of a non-commutative scheme captures its Morita invariant properties. As an application, we use a Künneth formula on non-commutative schemes to factorise the Hochschild cohomology of a product of quasi-phantom categories such as built by Orlov and Gorshinsky in [GO13]. This is an occasion to explore how the derived Morita theory interacts with the Čech dg-enhancement of a projective scheme and to shed an unifying light upon the landscape of dg-modules over dg-categories. |
| title | Künneth and extension theorems for Morita invariants of non-commutative schemes |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2412.08892 |