A Frobenius characterization of finite projective dimension over complete intersections
Let \(R\) be a Noetherian local ring of dimension \(d\) which is a complete intersection and contains a field of characteristic \(p\). Let \(F(-)\) denote the Frobenius functor and \(\ell\) the length function. Let \(M\) be an \(R\)-module of finite length. In this paper the author shows that \(M\) has finite projective dimension if and only if \(\ell(F^n(M))=\ell(M)p^{nd}\) for all \(n\geq 0\). This may be viewed as the converse to a theorem of \textit{S. Dutta} [J. Algebra 85, 424-448 (1983; Zbl 0527.13014)]. As a corollary the author gets that, in a complete intersection \((R,\mathfrak m)\), an \(\mathfrak m\)-primary ideal \(I\) has finite projective dimension if and only if its Hilbert-Kunz multiplicity equals the length of \(R/I\).
scientific article; zbMATH DE number 1406451