Frobenius criteria of freeness and Gorensteinness (Q438611)

Let \((R,\mathfrak{m})\) be a noetherian commutative local ring of characteristic \(p>0\). Following the notation in the paper under review, let \(f:R\to R\) denote the Frobenius endomorphism and let \(F^n(M)\) denote the base change \(M\otimes_R{^{f^n}R}\) along \(f^n\). Each iteration \(f^n\) defines a new \(R\)-module structure on \(R\), denoted by \({^{f^n}R}\), via restriction of scalars. A finitely generated \(R\)-module is called \textit{rigid against Frobenius} if either \(M\) has finite projective dimension or \(\mathrm{Tor}^R_i(M,{^{f^n}}R)\neq 0\) for all \(i\) and all \(n\gg 0\). \textit{L. L. Avramov} and \textit{C. Miller} [Math. Res. Lett. 8, No. 1--2, 225--232 (2001; Zbl 1034.13006)] showed that, when \(R\) is a complete intersection, every finitely generated \(R\)-module \(M\) is rigid against Frobenius. It is natural to ask if one can extend Avramov-Miller's rigidity result to Gorenstein rings. \textit{H. Dao, J. Li} and \textit{C. Miller} [Algebra Number Theory 4, No. 8, 1039--1053 (2010; Zbl 1221.13004)] constructed a finitely generated module over a Gorenstein local ring which is not rigid. One of the motivations behind the paper under review is to identify modules over a Cohen-Macaulay local ring that are rigid against Frobenius. One of the main results is: Theorem 1. Let \((R,\mathfrak{m})\) be a Cohen-Macaulay local ring of dimension \(d>0\) in characteristic \(p>0\). Let \(M\) be a finitely generated \(R\)-module that has a rank. Suppose \(\mathrm{Tor}^R_i(M,{^{f^n}R})=0\) for one integer \(n\geq \kappa(R)\) and all \(1\leq i\leq d-\mathrm{depth}(F^n(M))\). Then \(M\) has finite projective dimension. Here \(\kappa(R)\) is defined as follows: \[ \kappa(R):=\inf \{t\mid\text{there is a system of parameters }x_1,\dots,x_d \text{ such that }\mathfrak{m}^{[p^t]}\subset (x_1,\dots,x_d)\}. \] Criteria of Gorensteinness are also obtained along with the previous result: Theorem 2. Let \((R,\mathfrak{m})\) be a Cohen-Macaulay local ring in characteristic \(p>0\). If \(R\) satisfies one the following (a) \(R\) admits a canonical module \(\omega\) that has a rank and \(F^n(\omega)\) is maximal Cohen-Macaulay for an integer \(n\geq \kappa(R)\); or (b) there is an integer \(n\geq \kappa(R)\) such that \({^{f^n}R}\) has a rank and such that \[ \mathrm{Ext}^i_R({^{f^n}R},R)=0 \] for all \(1\leq i\leq d\); or (c). \(R\) admits a canonical module \(\omega\) that has a rank and there is a system of parameters \(x_1,\dots,x_d\) such that \[ \mathrm{Tor}_i(\omega/(x_1,\dots,x_d)\omega,{^{f^n}R})=0 \] for an integer \(i>0\) and an integer \(n\geq \kappa(R)\), then \(R\) is Gorenstein. To prove the aforementioned results, criteria of freeness of a module over a Cohen-Macaulay local ring are also discussed.