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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A problem on Noetherian local rings of characteristic $p$
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Proc. Amer. Math. Soc. 64 (1977), 199-205 Request permission

Abstract:

Let (A, m, k) be a one-dimensional Noetherian local ring of characteristic p ($p > 0$, a prime number) and assume that the Frobenius endomorphism F of A is finite. Further assume that the field k is algebraically closed and that it is contained in A. Let B denote A when it is regarded as an A-algebra by F. Then, if $\operatorname {Hom}_A(B,A) \cong B$ as B-modules, A is a Macaulay local ring and $r(A) \equiv \dim _k\operatorname {Ext}_A^1(k,A) \leqslant \max \{ \sharp {\text {Ass}}\hat A - 1,1\}$ where  denotes the m-adic completion of A. Thus, in case $\sharp {\text {Ass}}{\mkern 1mu} \hat A \leqslant 2,A$ is a Gorenstein local ring if and only if $\operatorname {Hom}_A(B,A) \cong B$ as B-modules. If $\sharp {\text {Ass}}\hat A \geqslant 3$ this assertion is not true and the counterexamples are given.
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 64 (1977), 199-205
  • MSC: Primary 13E05; Secondary 13H10
  • DOI: https://doi.org/10.1090/S0002-9939-1977-0447212-6
  • MathSciNet review: 0447212