Nakayama closures, interior operations, and core-hull duality

TL;DR

This paper explores dualities between algebraic closures and interiors in Noetherian local rings, providing new characterizations, computational methods, and unified proofs of key properties in tight closure theory.

Contribution

It establishes a duality framework for Nakayama closures and interior operations, and applies it to unify and extend results in tight closure theory.

Findings

Duality between cl-reductions and i-expansions

Unified proof of F-rationality and F-regularity equivalence

New characterization of finitistic tight closure test ideal

Abstract

Exploiting the interior-closure duality developed by Epstein and R.G., we show that for the class of Matlis dualizable modules $\mathcal{M}$ over a Noetherian local ring, when cl is a Nakayama closure and i its dual interior, there is a duality between cl-reductions and i-expansions that leads to a duality between the cl-core of modules in $\mathcal{M}$ and the i-hull of modules in $\mathcal{M}^\vee$. We further show that many algebra and module closures and interiors are Nakayama and describe a method to compute the interior of ideals using closures and colons. We use our methods to give a unified proof of the equivalence of F-rationality with F-regularity, and of F-injectivity with F-purity, in the complete Gorenstein local case. Additionally, we give a new characterization of the finitistic tight closure test ideal in terms of maps from $R^{1/p^e}$. Moreover, we show that the liftable integral spread of a module exists.