Asymptotic Universal Koszulity in Galois Cohomology
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918464745111552 |
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| author | Palaisti, Marina |
| author_facet | Palaisti, Marina |
| contents | We introduce the notion of asymptotic universal Koszulity for graded-commutative algebras generated in degree~$1$, capturing the idea that an infinite-dimensional algebra can be approximated by a filtered system of finite-type universally Koszul quadratic subalgebras. We establish basic structural properties of this class, including stability under filtered colimits, direct products, and base change, as well as a local finite-type criterion expressed in terms of finite-dimensional subspaces of the degree-one component. In the context of Galois cohomology, we prove a colimit theorem for pro-$p$ groups under mild assumptions, showing that cohomology rings arise as filtered colimits of finite quotients. This yields a general criterion under which the cohomology algebra of a profinite group is asymptotically universally Koszul. We further analyze finitely generated quotients via a finite-type capture result, identifying their cohomology with canonical quadratic subalgebras of the ambient algebra. Finally, we formulate conditional local--global and patching principles that isolate the mechanisms by which asymptotic universal Koszulity may arise in arithmetic settings. These results provide a flexible structural framework linking homological algebra, quadratic algebras, and Galois cohomology, and suggest several directions for further investigation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_27032 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic Universal Koszulity in Galois Cohomology Palaisti, Marina Number Theory Rings and Algebras 12G05, 12F10, 16S37, 20E18 We introduce the notion of asymptotic universal Koszulity for graded-commutative algebras generated in degree~$1$, capturing the idea that an infinite-dimensional algebra can be approximated by a filtered system of finite-type universally Koszul quadratic subalgebras. We establish basic structural properties of this class, including stability under filtered colimits, direct products, and base change, as well as a local finite-type criterion expressed in terms of finite-dimensional subspaces of the degree-one component. In the context of Galois cohomology, we prove a colimit theorem for pro-$p$ groups under mild assumptions, showing that cohomology rings arise as filtered colimits of finite quotients. This yields a general criterion under which the cohomology algebra of a profinite group is asymptotically universally Koszul. We further analyze finitely generated quotients via a finite-type capture result, identifying their cohomology with canonical quadratic subalgebras of the ambient algebra. Finally, we formulate conditional local--global and patching principles that isolate the mechanisms by which asymptotic universal Koszulity may arise in arithmetic settings. These results provide a flexible structural framework linking homological algebra, quadratic algebras, and Galois cohomology, and suggest several directions for further investigation. |
| title | Asymptotic Universal Koszulity in Galois Cohomology |
| topic | Number Theory Rings and Algebras 12G05, 12F10, 16S37, 20E18 |
| url | https://arxiv.org/abs/2603.27032 |