A new approach to formal moduli problems
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866917678659141632 |
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| author | Grignou, Brice Le Lucio, Victor Roca i |
| author_facet | Grignou, Brice Le Lucio, Victor Roca i |
| contents | The main goal of this paper is to introduce a framework for infinitesimal deformation problems, using new methods coming from operadic calculus. We construct an adjunction between infinitesimal deformation problems over some type of algebras and their Koszul dual algebras, in any characteristic. This adjunction is an equivalence if and only if some algebras are equivalent to their completions. We give a concrete homological criterion for it. It gives us a new proof of the celebrated Lurie--Pridham theorem, as well as of many other generalizations of it. Our methods are effective, meaning they directly produce point-set models for the algebras that encode infinitesimal deformation problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_07227 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A new approach to formal moduli problems Grignou, Brice Le Lucio, Victor Roca i Algebraic Topology Algebraic Geometry Category Theory 18M70, 18N40, 14D15, 14D23 The main goal of this paper is to introduce a framework for infinitesimal deformation problems, using new methods coming from operadic calculus. We construct an adjunction between infinitesimal deformation problems over some type of algebras and their Koszul dual algebras, in any characteristic. This adjunction is an equivalence if and only if some algebras are equivalent to their completions. We give a concrete homological criterion for it. It gives us a new proof of the celebrated Lurie--Pridham theorem, as well as of many other generalizations of it. Our methods are effective, meaning they directly produce point-set models for the algebras that encode infinitesimal deformation problems. |
| title | A new approach to formal moduli problems |
| topic | Algebraic Topology Algebraic Geometry Category Theory 18M70, 18N40, 14D15, 14D23 |
| url | https://arxiv.org/abs/2306.07227 |