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Towards the classification of DGAs with polynomial homology
- Part of
-
- Journal:
- Compositio Mathematica / Volume 162 / Issue 10 / October 2026
- Published online by Cambridge University Press:
- 07 October 2026, pp. 2327-2361
- Print publication:
- October 2026
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- Article
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We study the classification of
$\mathbb{Z}$-DGAs (differential graded algebras) with polynomial homology 
$\mathbb{F}_p[x]$ with 
$\lvert x \rvert \gt0$, 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 
$\mathbb{Z}[C_{p^n}]$.
