## A problem on Noetherian local rings of characteristic $p$

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**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

*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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## 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