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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.Throughout this paper we will work with a Noetherian local ring (R, m). Recall that a finitely generated R-module M is maximal Cohen-Macaulay (MCM) if depth M = dim R. We say that R has finite Cohen-Macaulay type (or finite CM type) provided there are only finitely many isomorphism classes of indecomposable MCM R-modules. There is a large body of work devoted to the classification of Cohen-Macaulay local rings of finite CM type, for example see the book [22]. One of the first and most famous results is that of M. Auslander [1]: if R is complete Cohen-Macaulay and has finite Cohen-Macaulay type, then R has an isolated singularity, i.e. for all primes p = m, R p is a regular local ring. (Yoshino [22] points out that R need only be Henselian and have a canonical module.) A key point in Auslander's proof is to prove that the modules Ext 1 R (M, N ) are of finite length for arbitrary MCM modules M and N , and he accomplishes this by using the theory of almost split sequences. The first result in this paper gives a proof of the finite length

We show that the F -signature of a local ring of characteristic p, defined by Huneke and Leuschke, is positive if and only if the ring is strongly F -regular.

We show that determinantal varieties defined by maximal minors of a generic matrix have a non-commutative desingularization, in that we construct a maximal Cohen-Macaulay module over such a variety whose endomorphism ring is Cohen-Macaulay and has finite global dimension. In the case of the determinant of a square matrix, this gives a non-commutative crepant resolution.

Abstract. For a commutative local ring R, consider (noncommutative) R-algebras Λ of the form Λ = End R (M) where M is a reflexive R-module with nonzero free direct summand. Such algebras Λ of finite global dimension can be viewed as potential substitutes for, or analogues of, a resolution of singularities of Spec R. For example, Van den Bergh has shown that a three-dimensional Gorenstein normal C-algebra with isolated terminal singularities has a crepant resolution of singularities if and only if it has such an algebra Λ with finite global dimension and which is maximal Cohen-Macaulay over R (a "noncommutative crepant resolution of singularities"). We produce algebras Λ = End R (M) having finite global dimension in two contexts: when R is a reduced one-dimensional complete local ring, or when R is a Cohen-Macaulay local ring of finite Cohen-Macaulay type. If in the latter case R is Gorenstein, then the construction gives a noncommutative crepant resolution of singularities in the sense of Van den Bergh.