Flat Covers and Flat Cotorsion Modules

Enochs, Edgar E.

doi:10.2307/2045180

Cited by 49 publications

(81 citation statements)

References 4 publications

(4 reference statements)

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“…Theorem 1.11 is deduced from Main Lemma 1.4, and Theorem 1.12 is deduced from Main Lemma 1. 6. This is done in Section 10.…”

Section: Flat Lemmamentioning

confidence: 99%

“…Proposition 2. 6. Let R be a Noetherian commutative ring, S be a finitely generated commutative R-algebra, and F be a finitely generated S-module.…”

Section: Main Lemma Implies Main Theoremmentioning

confidence: 99%

“…where the maps r n+1 A −→ r n A are provided by the operator of multiplication with r. Sublemma 4. 6. Let R be a commutative ring and r ∈ R be an element.…”

Section: Toy Examples Of a Main Lemmamentioning

confidence: 99%

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Flat morphisms of finite presentation are very flat

Positselski

Slavik

2019

Annali di Matematica

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Principal affine open subsets in affine schemes are an important tool in the foundations of algebraic geometry. Given a commutative ring R, R-modules built from the rings of functions on principal affine open subschemes in Spec R using ordinal-indexed filtrations and direct summands are called very flat. The related class of very flat quasi-coherent sheaves over a scheme is intermediate between the classes of locally free and flat sheaves, and has serious technical advantages over both. In this paper we show that very flat modules and sheaves are ubiquitous in algebraic geometry: if S is a finitely presented commutative R-algebra which is flat as an R-module, then S is a very flat R-module. This proves a conjecture formulated in the February 2014 version of the first author's long preprint on contraherent cosheaves [20]. We also show that the (finite) very flatness property of a flat module satisfies descent with respect to commutative ring homomorphisms of finite presentation inducing surjective maps of the spectra.

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“…Theorem 1.11 is deduced from Main Lemma 1.4, and Theorem 1.12 is deduced from Main Lemma 1. 6. This is done in Section 10.…”

Section: Flat Lemmamentioning

confidence: 99%

“…Proposition 2. 6. Let R be a Noetherian commutative ring, S be a finitely generated commutative R-algebra, and F be a finitely generated S-module.…”

Section: Main Lemma Implies Main Theoremmentioning

confidence: 99%

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Flat morphisms of finite presentation are very flat

Positselski

Slavik

2019

Annali di Matematica

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“…Our constructions of these cotorsion theories will require making some assumptions on the ring R and/or the multiplicative subset S. 7.1. Flat cotorsion theory in S-contramodule R-modules for a Noetherian commutative ring R. We refer to Section 1.1 of the introduction and the paper [9] for the definition of an (Enochs) cotorsion R-module. In this section, as well as below, we will be particularly interested in S-contramodule R-modules that are flat, cotorsion, etc.…”

Section: Contramodule Approximation Sequencesmentioning

confidence: 99%

On strongly flat and weakly cotorsion modules

Positselski

Slavik

2018

Math. Z.

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The aim of this paper is to describe the classes of strongly flat and weakly cotorsion modules with respect to a multiplicative subset or a finite collection of multiplicative subsets in a commutative ring. The strongly flat modules are characterized by a set of conditions, while the weakly cotorsion modules are produced by a generation procedure. Passing to the collection of all countable multiplicative subsets, we define quite flat and almost cotorsion modules, and show that, over a Noetherian ring with countable spectrum, all flat modules are quite flat and all almost cotorsion modules are cotorsion.

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“…Proof. See [6,Theorem] Let S be a multiplicatively closed subset of R and a be an ideal of R. For a cotorsion flat R-module F , we have RHom R (S −1 R, F ) ∼ = Hom R (S −1 R, F ) and LΛ V (a) F ∼ = Λ V (a) F . Moreover, by Proposition 5.1, we may regard F as an Rmodule of the form p∈Spec R T p .…”

Section: Cotorsion Flat Modules and Cosupportmentioning

confidence: 99%

Localization functors and cosupport in derived categories of commutative Noetherian rings

Nakamura

Yoshino

2018

Pacific J. Math.

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Let R be a commutative Noetherian ring. We introduce the notion of localization functors λ W with cosupports in arbitrary subsets W of Spec R; it is a common generalization of localizations with respect to multiplicatively closed subsets and left derived functors of ideal-adic completion functors. We prove several results about the localization functors λ W , including an explicit way to calculate λ W by the notion ofČech complexes. As an application, we can give a simpler proof of a classical theorem by Gruson and Raynaud, which states that the projective dimension of a flat R-module is at most the Krull dimension of R. As another application, it is possible to give a functorial way to replace complexes of flat R-modules or complexes of finitely generated R-modules by complexes of pure-injective R-modules.2010 Mathematics Subject Classification. 13D09, 13D45, 55P60.

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Flat Covers and Flat Cotorsion Modules

Cited by 49 publications

References 4 publications

Flat morphisms of finite presentation are very flat

Flat morphisms of finite presentation are very flat

On strongly flat and weakly cotorsion modules

Localization functors and cosupport in derived categories of commutative Noetherian rings

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