On the Frobenius closure of parameter ideals in local rings F-injective on the punctured spectrum

Abstract

Let $(R,\mathfrak{m})$ be an excellent generalized Cohen-Macaulay local ring of dimension d that is F-injective on the punctured spectrum. Let $\mathfrak{q}$ be a standard parameter ideal of R. The aim of the paper is to prove that $${\mathcal{l}}_R({\mathfrak{q}}^F/\mathfrak{q})\le \sum _{i=0}^d\left(\begin{array}{c}d\\ i\end{array}\right){\mathcal{l}}_R(0_{H_{\mathfrak{m}}^i(R)}^F).$$

Moreover, if $\mathfrak{q}$ is contained in a large enough power of $\mathfrak{m}$, we have $${\mathfrak{q}}^F/\mathfrak{q}\cong {\oplus }_{i=0}^d{(0_{H_{\mathfrak{m}}^i(R)}^F)}^{\left(\begin{array}{c}d\\ i\end{array}\right)}.$$

Keywords:

• F-injective
• Frobenius closure
• Frobenius test exponent
• generalized Cohen-Macaulay

2020 Mathematics Subject Classification:

• 13A35
• 13H15
• 13D45

Acknowledgments

This paper was developed while the authors visited the Vietnam Institute for Advanced Study in Mathematics (VIASM). The authors are grateful to the referee for careful reading of the paper.

Notes

1 In Question 3 of [14], the second author missed the word ``standard'' in the statement.

Additional information

Funding

The authors are grateful to the VIASM for the financial support and hospitality.