2020/07/23 by Neil Epstein, Epstein, Neil, Rebecca R.G. +3
Mathematics · #13B22 #13C60 #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Primary: 13J10 #Rings, Modules, and Algebras #Secondary: 13A35
paper · pdf · doi:10.48550/arxiv.2007.12209
openalex publication_date 2020/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Exploiting the interior-closure duality developed by Epstein and R.G., we show that for the class of Matlis dualizable modules 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 M and the i-hull of modules in 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 R1/pe. Moreover, we show that the liftable integral spread of a module exists.