Towards the classification of DGAs with polynomial homology
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912590729314304 |
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| author | Bayındır, Haldun Özgür Land, Markus |
| author_facet | Bayındır, Haldun Özgür Land, Markus |
| contents | We study the classification of $\mathbb{Z}$-DGAs with polynomial homology $\mathbb{F}_p[x]$ with $\lvert x \rvert >0$, motivated by computations in algebraic $K$-theory. This classification problem was left open in work of Dwyer, Greenlees, and Iyengar. We prove that there are infinitely many such DGAs for even $\lvert x \rvert$ and that for $\lvert x \rvert \geq 2p-2$ any such DGA is formal as a ring spectrum. Through this, we obtain examples of triangulated categories with infinitely many DG-enhancements and a classification of prime DG-division rings.
Combining our results with earlier work of the second author and Tamme, we obtain new (relative) algebraic $K$-theory computations for rings such as the mixed characteristic coordinate axes $\mathbb{Z}[x]/px$ and the group ring $\mathbb{Z}[C_{p^n}]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14015 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Towards the classification of DGAs with polynomial homology Bayındır, Haldun Özgür Land, Markus Algebraic Topology K-Theory and Homology Rings and Algebras We study the classification of $\mathbb{Z}$-DGAs with polynomial homology $\mathbb{F}_p[x]$ with $\lvert x \rvert >0$, motivated by computations in algebraic $K$-theory. This classification problem was left open in work of Dwyer, Greenlees, and Iyengar. We prove that there are infinitely many such DGAs for even $\lvert x \rvert$ and that for $\lvert x \rvert \geq 2p-2$ any such DGA is formal as a ring spectrum. Through this, we obtain examples of triangulated categories with infinitely many DG-enhancements and a classification of prime DG-division rings. Combining our results with earlier work of the second author and Tamme, we obtain new (relative) algebraic $K$-theory computations for rings such as the mixed characteristic coordinate axes $\mathbb{Z}[x]/px$ and the group ring $\mathbb{Z}[C_{p^n}]$. |
| title | Towards the classification of DGAs with polynomial homology |
| topic | Algebraic Topology K-Theory and Homology Rings and Algebras |
| url | https://arxiv.org/abs/2509.14015 |