PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 11, Pages 3177–3187 S 0002-9939(04)07525-2 Article electronically published on May 12, 2004 CHARACTERIZING COHEN-MACAULAY LOCAL RINGS BY FROBENIUS MAPS RYO TAKAHASHI AND YUJI YOSHINO (Communicated by Bernd Ulrich) Abstract. Let R be a commutative noetherian local ring of prime charac- teristic. Denote by e R the ring R regarded as an R-algebra through e-times composition of the Frobenius map. Suppose that R is F-finite, i.e., 1 R is a finitely generated R-module. We prove that R is Cohen-Macaulay if and only if the R-modules e R have finite Cohen-Macaulay dimensions for infinitely many integers e. 1. Introduction Throughout the present paper, we assume that all rings are commutative and noetherian. Let R be a local ring of characteristic p, and let f : R → R be the Frobenius map, given by a → a p . For an integer e, we denote by f e : R → R the e-th power of f ; that is, it is given by f e (a)= a p e . We denote by e R the R-algebra R whose R-algebra structure is given via f e . The ring R is said to be F-finite if 1 R, hence every e R, is a finitely generated R-module. E. Kunz [18] proved that R is regular if and only if 1 R is R-flat. A. G. Rodicio [21] gave a generalization of this result as follows. Theorem 1.1 (Rodicio). A local ring R of prime characteristic is regular if and only if the R-module 1 R has finite flat dimension. A similar result concerning the complete intersection property and complete intersection dimension (abbr. CI-dimension) was proved by Blanco and Majadas [9]. They actually proved that a local ring R of prime characteristic is a complete intersection if and only if there exists a Cohen factorization R → S →  1 R such that the CI-dimension of  1 R over S is finite, where  1 R denotes the completion of the local ring 1 R. (For the definition of a Cohen factorization, see [5].) Therefore, we have in particular the following. Theorem 1.2 (Blanco and Majadas). An F-finite local ring R is a complete in- tersection if and only if the R-module 1 R has finite CI-dimension. Received by the editors May 15, 2002 and, in revised form, April 9, 2003 and August 7, 2003. 2000 Mathematics Subject Classification. Primary 13A35, 13D05, 13H10. Key words and phrases. Frobenius map, CM-dimension, G-dimension, flat dimension, injective dimension. c 2004 American Mathematical Society 3177 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use