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A characterization of Gorenstein rings in characteristic \(\lowercase{p} (> 0)\). (English) Zbl 0890.13002

In this note we study how the Gorenstein property for a local ring \((R,m,k= R/m)\) of characteristic \(p\) \((>0)\), dimension \(d\), is connected with the behaviour of \(\lim_{n \to \infty} \ell(F^n(M))/p^{nd}\) and \(\lim_{n\to\infty} \ell(F^n(M^\nu))/p^{nd}\), when \(M\) is a module of finite length and finite projective dimension over \(R\), \(M^\nu= \operatorname{Hom}_R (M,E)\), \(E\) is the injective hull of \(k\) over \(R\) and \(F^n(M)\) is obtained from \(M\) by applying the Frobenius map \(n\) times. It is well known that for a module \(M\) of finite length over any local ring \(R\) of \(\dim d\) and \(\text{char} p\;(>0)\), \(\lim_{n\to\infty} \ell(F^n(M))/p^{nd}\) is positive, and if \(R\) is Cohen-Macaulay and \(M=R/(x_1, \dots, x_d)\), where \(x_1, \dots, x_d\) is a system of parameters in \(R\), the above limit is \(\ell(R/(x_1, \dots, x_d))\). Recall that when \(R\) is Gorenstein \(M^\nu\simeq \text{Ext}^d_R (M,R)\) for any module \(M\) of finite length. Moreover when \(\text{pd}_R M<\infty\), we have \((F^n(M))^\nu \simeq \text{Ext}^d (F^n(M),R) \simeq F^n (\text{Ext}^d (M,R)) \simeq F^n (M^\nu)\). Thus \(\ell(F^n(M)) =\ell(F^n (M^\nu))\), and obviously the above limits are same. This leads us to raise the same question for Cohen-Macaulay rings which are not Gorenstein. Our main theorem shows that the answer, in this case, is in the negative even for cyclic modules of the form \(R/(x_1, \dots, x_d)\), where \(x_1, \dots, x_d\) is a system of parameters of \(R\). In fact our main theorem is the following:
Theorem. A Cohen-Macaulay local ring \(R\) of dimension \(d\) and \(\text{char} p\) \((>0)\) is Gorenstein if and only if there exists a system of parameters \(x_1, \dots, x_d\) such that \[ \lim_{n\to\infty} \ell (F^n((R/ \underline x)^\nu))/p^{nd} =\ell (R/ \underline x). \]

MSC:

13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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References:

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