Flat dimension for power series over valuation rings
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866908572150923264 |
|---|---|
| author | Jones, Adam |
| author_facet | Jones, Adam |
| contents | We examine the power series ring $R[[X]]$ over a valuation ring $R$ of rank 1, with proper, dense value group. We give a counterexample to Hilbert's syzygy theorem for $R[[X]]$, i.e. an $R[[X]]$-module $C$ that is flat over $R$ and has flat dimension at least 2 over $R[[X]]$, contradicting a previously published result. The key ingredient in our construction is an exploration of the valuation theory of $R[[X]]$. We also use this theory to give a new proof that $R[[X]]$ is not a coherent ring, a fact which is essential in our construction of the module $C$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_03483 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Flat dimension for power series over valuation rings Jones, Adam Commutative Algebra Rings and Algebras 16E10, 13F25, 13F30, 13C11, 16P70, 13A18 We examine the power series ring $R[[X]]$ over a valuation ring $R$ of rank 1, with proper, dense value group. We give a counterexample to Hilbert's syzygy theorem for $R[[X]]$, i.e. an $R[[X]]$-module $C$ that is flat over $R$ and has flat dimension at least 2 over $R[[X]]$, contradicting a previously published result. The key ingredient in our construction is an exploration of the valuation theory of $R[[X]]$. We also use this theory to give a new proof that $R[[X]]$ is not a coherent ring, a fact which is essential in our construction of the module $C$. |
| title | Flat dimension for power series over valuation rings |
| topic | Commutative Algebra Rings and Algebras 16E10, 13F25, 13F30, 13C11, 16P70, 13A18 |
| url | https://arxiv.org/abs/2305.03483 |