Applications of pseudocanonical covers to tight closure problems
For a local Cohen-Macaulay ring \(R\) which admits a canonical ideal \(I\) the author introduces a pseudocanonical double cover as the \(\mathbb Z/2\mathbb Z\)-graded ring \(R \oplus I\), where for a fixed element \(f\) of \(R\), for any \(r,r' \in R\) and \(i,i' \in I\), \((r,i)\cdot(r',i') = (rr'+ii'f, ri'+r'i)\). The pseudocanonical cover depends on the element \(f\), but under some assumptions on the ring several properties of the cover do not depend on it. For rings \(R\) with further assumptions there exist the notions of canonical and anticanonical covers. Then the ring \(R\) is F-regular if and only if its canonical cover is F-regular [see \textit{K. E. Smith}, Commun. Algebra 28, No.~12, 5915-5929 (2000; Zbl 0979.13007) and \textit{K. Watanabe}, J. Pure Appl. Algebra 71, No.~2/3, 341-350 (1991; Zbl 0738.13005)]\ and under the assumption that the anticanonical cover is Noetherian, \(R\) is F-regular if and only if its anticanonical cover is [see \textit{K. Watanabe} in: Commutative algebra: Syzygies, multiplicities, and birational algebra: AMS-IMS-SIAM Summer research conference, Contemp. Math. 159, 423-432 (1994; Zbl 0822.13012)]). Without the strong assumptions on the ring needed for the existence of canonical and anticanonical covers, in this paper the author uses the notion of pseudocanonical cover to obtain sufficient conditions for strong F-regularity and F-purity of the ring.
scientific article; zbMATH DE number 1878474
scientific article