TL;DR
This paper develops an axiomatic framework for various closure operations in commutative algebra, enabling unified analysis of reductions, independence, and special parts across different closure types.
Contribution
It introduces a general axiomatic approach to closure operations, unifying concepts like reduction and special parts for tight, Frobenius, and integral closures.
Findings
Unified framework for closure operations
Applications to evolutions and Briançon-Skoda theorems
Enhanced understanding of reductions and special parts
Abstract
We provide an axiomatic framework for working with a wide variety of closure operations on ideals and submodules in commutative algebra, including notions of reduction, independence, spread, and special parts of closures. This framework is applied to tight, Frobenius, and integral closures. Applications are given to evolutions and special Brian\c{c}on-Skoda theorems.
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