Tensoring with the Frobenius endomorphism (Q1707292)

Hereafter, \(R\) will denote a commutative Noetherian ring of prime characteristic \(p,\) and \(F:\;R\rightarrow R\) will denote the Frobenius map raising \(r\in R\) to its \(p\)-th power \(r^p;\) given an \(R\)-module \(M\) and an integer \(n\geq 0,\) \(F_*^n M\) will denote the \(R\)-module obtained by restriction of scalars along \(F^n,\) hence the left action of \(R\) on \(F_*^n M\) is given by \(r\cdot m:=r^{p^n}m\) for \((r,m)\in R\times M.\) Moreover, in this case, one agrees that the left \(R\)-module structure on \(F^{*n}:=M\otimes F_*^n R\) comes from the usual right action of \(R\) on \(F_*^n R\) (that is \(r\cdot (m\otimes s):=m\otimes (sr)).\) In the paper under review, the authors prove the following result (see Theorem 2.1): let \((R,\mathfrak{m})\) be a commutative Noetherian Cohen-Macaulay local ring of prime characteristic \(p\) and positive dimension, and let \(M\) be a finitely generated \(R\)-module. Moreover, assume that there is an integer \(n\geq 0\) such that \(p^n\geq e(R)\) (where \(e(R)\) denotes the multiplicity of \(R\)) and \(F^{*n}M\) is a maximal Cohen-Macaulay module. If, for any minimal prime ideal \(\mathfrak{p}\) of \(R,\) \(M_{\mathfrak{p}}\) is free over \(R_{\mathfrak{p}},\) then \(M\) is free. This recovers and extends a result by \textit{J. Li} [Arch. Math. 98, No. 6, 499--506 (2012; Zbl 1248.13011)].