A problem on Noetherian local rings of characteristic 𝑝

Goto, Shiro

doi:10.1090/S0002-9939-1977-0447212-6

A problem on Noetherian local rings of characteristic $p$
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by Shiro Goto

Proc. Amer. Math. Soc. 64 (1977), 199-205

DOI: https://doi.org/10.1090/S0002-9939-1977-0447212-6

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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 Â 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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