Sequence-regular commutative DG-rings
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866929273676234752 |
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| author | Shaul, Liran |
| author_facet | Shaul, Liran |
| contents | We introduce a new class of commutative noetherian DG-rings which generalizes the class of regular local rings. These are defined to be local DG-rings $(A,\bar{\mathfrak{m}})$ such that the maximal ideal $\bar{\mathfrak{m}} \subseteq \mathrm{H}^0(A)$ can be generated by an $A$-regular sequence. We call these DG-rings sequence-regular DG-rings, and make a detailed study of them. Using methods of Cohen-Macaulay differential graded algebra, we prove that the Auslander-Buchsbaum-Serre theorem about localization generalizes to this setting. This allows us to define global sequence-regular DG-rings, and to introduce this regularity condition to derived algebraic geometry. It is shown that these DG-rings share many properties of classical regular local rings, and in particular we are able to construct canonical residue DG-fields in this context. Finally, we show that sequence-regular DG-rings are ubiquitous, and in particular, any eventually coconnective derived algebraic variety over a perfect field is generically sequence-regular. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_08109 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Sequence-regular commutative DG-rings Shaul, Liran Commutative Algebra Algebraic Geometry Algebraic Topology 16E45, 13D09, 13H05, 13H10 We introduce a new class of commutative noetherian DG-rings which generalizes the class of regular local rings. These are defined to be local DG-rings $(A,\bar{\mathfrak{m}})$ such that the maximal ideal $\bar{\mathfrak{m}} \subseteq \mathrm{H}^0(A)$ can be generated by an $A$-regular sequence. We call these DG-rings sequence-regular DG-rings, and make a detailed study of them. Using methods of Cohen-Macaulay differential graded algebra, we prove that the Auslander-Buchsbaum-Serre theorem about localization generalizes to this setting. This allows us to define global sequence-regular DG-rings, and to introduce this regularity condition to derived algebraic geometry. It is shown that these DG-rings share many properties of classical regular local rings, and in particular we are able to construct canonical residue DG-fields in this context. Finally, we show that sequence-regular DG-rings are ubiquitous, and in particular, any eventually coconnective derived algebraic variety over a perfect field is generically sequence-regular. |
| title | Sequence-regular commutative DG-rings |
| topic | Commutative Algebra Algebraic Geometry Algebraic Topology 16E45, 13D09, 13H05, 13H10 |
| url | https://arxiv.org/abs/2106.08109 |