Closure Operations that Induce Big Cohen-Macaulay Modules and Algebras, and Classification of Singularities
[electronic resource].
Description
- Language(s)
-
English
- Published
-
2016.
- Summary
-
regular closure are not Dietz closures, and that all Dietz closures are contained in liftable integral closure. We give an additional axiom for a closure operation such that the existence of a Dietz closure satisfying this axiom is equivalent to the existence of a big Cohen-Macaulay algebra. We prove that many closure operations satisfy the Algebra Axiom, whether or not they are Dietz closures. We discuss the smallest big Cohen-Macaulay algebra closure on a given ring, and show that every Dietz closure satisfying the Algebra Axiom is contained in a big Cohen-Macaulay algebra closure. This leads to proofs that in rings of characteristic p > 0, every Dietz closure satisfying the Algebra Axiom is contained in tight closure, and there exist Dietz closures that do not satisfy the Algebra Axiom.
Geoffrey Dietz introduced a set of axioms for a closure operation on a complete local domain such that the existence of a closure operation satisfying these axioms is equivalent to the existence of a big Cohen-Macaulay module. These are called Dietz closures. In characteristic p > 0, tight closure and plus closure satisfy the axioms. In order to study these closures, we define module closures and discuss their properties. For many of these properties, there is a smallest closure operation satisfying the property. We discuss properties of big Cohen-Macaulay module closures, and prove that every Dietz closure is contained in a big Cohen-Macaulay module closure. Using this result, we show that under mild conditions, a ring R is regular if and only if all Dietz closures on R are trivial. We also show that solid closure in equal characteristic 0, integral closure, and
- Locate a Print Version
-
Find in a library service is not available from this catalog. Search Worldcat
Viewability