The existence of the \(F\)-signature for rings with large \(\mathbb Q\)-Gorenstein locus
Let \((R, m,k)\) be a local ring of positive prime characteristic \(p\) and dimension \(d\). Assume that \(R\) is \(F\)-\textit{finite}, i.e., the Frobenius homomorphism makes \(R\) into a finite \(R\)-algebra. Let \(\alpha=\log_p [k^{1/p}: k]\) and let \(a_q\) denote the number of \(R\)-free direct summands of \(R^{1/q}\) (\(q=p^e\)). The \(F\)-\textit{signature} of \(R\) is \(\lim_{q \to \infty} \frac{a_q}{q^{d+\alpha}}\), provided the limit exists. The author considers the question whether the \(F\)-\textit{signature} exists in the case of strongly \(F\)-regular local rings. Previously established results gave an affirmative answer in the particular cases when \(R\) is Gorenstein, Gorenstein on the punctured spectrum, \(\mathbb{Q}\)-Gorenstein on the punctured spectrum, an affine semigroup ring, and an \(\mathbb{N}\)-graded algebra over a field. All of these are cases when it is know that weak \(F\)-regularity implies strong \(F\)-regularity, and essentially the same techniques can be used to show the existence of the \(F\)-signature. In the paper under review, under a certain boundness condition on local cohomology, the author shows the existence of the \(F\)-\textit{signature} of a strongly \(F\)-regular local ring in the case when the non-\(\mathbb{Q}\)-Gorenstein locus has dimension 1. This is a situation where the boundness assumption is not sufficient to show that weak \(F\)-regularity implies strong \(F\)-regularity.
scientific article; zbMATH DE number 5277624
scientific article