A Guide to Closure Operations in Commutative Algebra

Abstract. We present a non-standard proof of the fact that the existence of a local (i.e. restricted to a point) characteristic-zero, semi-parametric lifting for a variety defined by the zero locus of polynomial equations over the integers is equivalent to the existence of a collection of local semiparametric (positive-characteristic) reductions of such variety for almost all primes (i.e. outside a finite set), and such that there exists a global complexity bounding all the corresponding structures involved. Results of this kind are a fundamental tool for transferring theorems in commutative algebra from a characteristic-zero setting to a positive-characteristic one.Mathematics Subject Classification. 03C20, 03C98, 13B99, 11D72, 11R99.

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“…, y r ) a tuple of elements in S such that Rad(x) = m, and F α (x, y) = 0, for every α. [6] and [2]). …”

Section: Remarkmentioning

confidence: 98%

On Positive-Characteristic Semi-parametric Local Uniform Reductions of Varieties over Finitely Generated $$\mathbb {Q}$$ Q -Algebras

2017

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“…We refer to the intersection of two closure operations cl and cl ′ , as defined in [Eps12]. Let N ⊆ M be finitely-generated R-modules.…”

Section: Properties Of Closure Operationsmentioning

confidence: 99%

“…Let D denote the sum of all Dietz closures. To see that it is a Dietz closure (it will be a closure operation, by [Eps12]), we use the fact [Eps12] that since R is Noetherian, for any particular N ⊆ M finitely-generated R-modules, there is some Dietz closure cl such that N …”

Section: Proof Consider the Ringmentioning

confidence: 99%

Closure operations that induce big Cohen–Macaulay modules and classification of singularities

Rebecca

2016

Journal of Algebra

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Geoffrey Dietz introduced a set of axioms for a closure operation on a complete local domain R so that the existence of such a closure operation is equivalent to the existence of a big Cohen-Macaulay module. These closure operations are called Dietz closures. In complete rings of characteristic p > 0, tight closure and plus closure satisfy the axioms.We define module closures and discuss their properties. For many of these properties, there is a smallest closure operation satisfying the property. In particular, we discuss properties of big CohenMacaulay 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. Finally, we show that solid closure in equal characteristic 0, integral closure, and regular closure are not Dietz closures, and that all Dietz closures are contained in liftable integral closure.

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“…In view of this results, it seems very natural to study in commutative algebra the question of finding a closure operation with "good" properties (see [8]), in terms of finding suitables algebraic-geometrical as well as topological or homological properties of the forcing morphism. This approach goes closer to the philosophy of Grothendieck's EGA of defining and studying the objects in a relative context (see [11] and [10]).…”

Section: ])mentioning

confidence: 99%

Normality and Related Properties of Forcing Algebras

Gomez-Ramirez

Brenner

2016

Communications in Algebra

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Abstract. We present a sufficient condition for irreducibility of forcing algebras and study the (non)-reducedness phenomenon. Furthermore, we prove a criterion for normality for forcing algebras over a polynomial base ring with coefficients in a perfect field. This gives a geometrical normality criterion for algebraic (forcing) varieties over algebraically closed fields. Besides, we examine in detail an specific (enlightening) example with several forcing equations. Finally, we compute explicitly the normalization of a particular forcing algebra by means of finding explicitly the generators of the ideal defining it as an affine ring.Mathematical Subject Classification (2010): 13B22, 14A15, 14R25, 54D05

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A Guide to Closure Operations in Commutative Algebra

Cited by 23 publications

References 45 publications

On Positive-Characteristic Semi-parametric Local Uniform Reductions of Varieties over Finitely Generated $$\mathbb {Q}$$ Q -Algebras

On Positive-Characteristic Semi-parametric Local Uniform Reductions of Varieties over Finitely Generated $$\mathbb {Q}$$ Q -Algebras

Closure operations that induce big Cohen–Macaulay modules and classification of singularities

Normality and Related Properties of Forcing Algebras

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