#### Document Type

Article

#### Date

4-9-2004

#### Disciplines

Mathematics

#### Description/Abstract

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.

#### Recommended Citation

Huneke, Craig and Leuschke, Graham J., "Two Theorems about Maximal Cohen--Macaulay Modules" (2004). *Mathematics Faculty Scholarship*. 31.

https://surface.syr.edu/mat/31

#### Source

Harvested from arXiv.org

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 3.0 License.

## Additional Information

This manuscript is from arXiv.org, for more information see http://arxiv.org/abs/math/0404204