Two theorems about maximal Cohen--Macaulay modules
| dc.creator | Huneke, Craig | |
| dc.creator | Leuschke, Graham J. | |
| dc.date | 2004-04-09 | |
| dc.date.accessioned | 2026-07-07T05:07:20Z | |
| dc.date.available | 2026-07-07T05:07:20Z | |
| dc.description | This paper contains two theorems concerning the theory of maximal Cohen--Macaulay modules. The first theorem proves that certain Ext groups between maximal Cohen--Macaulay modules $M$ and $N$ must have finite length, provided only finitely many isomorphism classes of maximal Cohen--Macaulay modules exist having ranks up to the sum of the ranks of $M$ and $N$. This has several corollaries. In particular it proves that a Cohen--Macaulay local ring of finite Cohen--Macaulay type has an isolated singularity. A well-known theorem of Auslander gives the same conclusion but requires that the ring be Henselian. Other corollaries of our result include statements concerning when a ring is Gorenstein or a complete intersection on the punctured spectrum, and the recent theorem of Leuschke and Wiegand that the completion of an excellent Cohen--Macaulay local ring of finite Cohen--Macaulay type is again of finite Cohen--Macaulay type. The second theorem proves that a complete local Gorenstein domain of positive characteristic $p$ and dimension $d$ is $F$-rational if and only if the number of copies of $R$ splitting out of $R^{1/p^e}$ divided by $p^{de}$ has a positive limit. This result generalizes work of Smith and Van den Bergh. We call this limit the $F$-signature of the ring and give some of its properties. | |
| dc.description | 14 pages | |
| dc.identifier | https://arxiv.org/abs/math/0404204 | |
| dc.identifier | http://arxiv.org/abs/math/0404204 | |
| dc.identifier | Mathematische Annalen, 324 (2002) 391-404 | |
| dc.identifier | doi:10.1007/s00208-002-0343-3 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/70818 | |
| dc.subject | Commutative Algebra | |
| dc.subject | 13C14; 13A35 | |
| dc.title | Two theorems about maximal Cohen--Macaulay modules | |
| dc.type | text |