Closure Operations that Induce Big Cohen-Macaulay Modules and Algebras, and Classification of Singularities

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100 1 ⊔ ‡aRebhuhn-Glanz, Rebecca.
245 1 0 ‡aClosure Operations that Induce Big Cohen-Macaulay Modules and Algebras, and Classification of Singularities ‡h[electronic resource].
260 ⊔ ⊔ ‡c2016.
502 ⊔ ⊔ ‡aDissertation (Ph.D.)--University of Michigan. PhD
504 ⊔ ⊔ ‡aIncludes bibliographical references.
520 3 ⊔ ‡aregular 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.
520 3 ⊔ ‡aGeoffrey 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
538 ⊔ ⊔ ‡aMode of access: Internet.
650 ⊔ 4 ‡aCommutative algebra.
650 ⊔ 4 ‡aCohen-macaulay module.
650 ⊔ 4 ‡aClosure operation.
650 ⊔ 4 ‡aTight closure.
690 ⊔ 4 ‡aMathematics.
710 2 ⊔ ‡aUniversity of Michigan. ‡bLibrary. ‡bDeep Blue.
899 ⊔ ⊔ ‡a39015089730066
CID ⊔ ⊔ ‡a102153809
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DAT 1 ⊔ ‡a20231112060855.0 ‡b2023-11-12T14:46:37Z
DAT 2 ⊔ ‡a2017-08-22T18:00:03Z
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