Crit�res de platitude et de projectivit�

Raynaud, Michel; Gruson, Laurent

doi:10.1007/bf01390094

Cited by 560 publications

(185 citation statements)

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“…After Raynaud a Gruson [33], it is well known that over an ℵ 0 -noetherian ring any module of projective dimension at most n is filtered by countably generated (presented) modules of projective dimension at most n. Specializing to the case of projective dimension at most one, we observe in Proposition 5.5 that for right orders in ℵ 0 -noetherian rings (P 1 , P ⊥ 1 ) is of countable type.…”

Section: Introductionmentioning

confidence: 81%

“…Hence, by Auslander-Buchsbaum equality [9], f.dim R = 1 and by [33] F.dim R = 2. Now the conclusion follows from Lemma 9.1.…”

Section: Examplesmentioning

confidence: 88%

“…A combination of a result by Bass [11] and one by Raynaud Gruson [33] shows that, for a commutative noetherian ring, the big finitistic dimension equals the Krull dimension. Moreover, a commutative noetherian ring is artinian if and only if its Krull dimension is zero.…”

Section: Lemma 83mentioning

confidence: 89%

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Cotorsion pairs generated by modules of bounded projective dimension

Bazzoni

Herbera

2009

Isr. J. Math.

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We apply the theory of cotorsion pairs to study closure properties of classes of modules with finite projective dimension with respect to direct limit operations and to filtrations.We also prove that if the ring is an order in an ℵ0-noetherian ring Q of small finitistic dimension 0, then the cotorsion pair generated by the modules of projective dimension at most one is of finite type if and only if Q has big finitistic dimension 0. This applies, for example, to semiprime Goldie rings and Cohen Macaulay noetherian commutative rings.

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Section: Introductionmentioning

confidence: 81%

“…Hence, by Auslander-Buchsbaum equality [9], f.dim R = 1 and by [33] F.dim R = 2. Now the conclusion follows from Lemma 9.1.…”

Section: Examplesmentioning

confidence: 88%

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Cotorsion pairs generated by modules of bounded projective dimension

Bazzoni

Herbera

2009

Isr. J. Math.

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“…Let the Krull dimension of R be n . If M is an R-module, let 0 -M -» E° ->-► E"'1 -C -> 0 be a partial injective resolution of M. Note that for any flat module F, proj.dim^F < n (Raynaud and Gruson [9,Corollary 3.2.7]). Then it is easy to see that C is cotorsion.…”

Section: ^R^l-+7j>^0mentioning

confidence: 99%

The Existence of Flat Covers Over Noetherian Rings of Finite Krull Dimension

Xu¹

1995

Proceedings of the American Mathematical Society

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Abstract. Bass characterized the rings R with the property that every left .R-module has a projective cover. These are the left perfect rings. A ring is left perfect if and only if the class of projective Ä-modules coincides with the class of flat Ä-modules, so the projective covers over these rings are flat covers. This prompts the conjecture that over any ring R, every left Ä-module has a flat cover. Known classes of rings for which the conjecture holds include Von Neumann regular rings (trivially), the left perfect rings (Bass), Prüfer domains (Enochs), and then more generally, all right coherent rings of finite weak global dimension (Belshoff, Enochs, Xu).In this paper we show that the conjecture holds for all commutative Noetherian rings of finite Krull dimension and so for all local rings and all coordinate rings of affine algebraic varieties. PreliminariesWe will use the terminology of Enochs [4]. We recall that if y is a class of left flat 7?-modules, a linear map tp: F -> M with F e y is called an yprecover of M if Hom^C?, F) -> HomÄ(G, M) -* 0 is exact for all G e y. If furthermore any linear map f:F-*F such that ftp = tp is an automorphism of F, then tp: F -> M is called an y-cover. If an y-cover exists, it is unique up to isomorphism. If, for example, y is the class of flat modules, then an y-precover (or cover) is called a flat precover (or cover). Note that since projective modules are flat, flat precovers are necessarily surjective. y-pre-envelopes and envelopes are defined dually.We note that in Auslander and Reiten's terminology [1], an y-precover and an y-cover are called a right y-approximation and a minimal right yapproximation respectively. In their language, our main result says that the class of flat modules is contravariantly finite in the category of left R-modules with R commutative, Noetherian, and of finite Krull dimension.

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“…Thus, we get a direct system of exact sequences, whose direct limit is the exact sequence: (3) 0^B^>Bb^>indlimB/(b")B^>0.…”

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confidence: 99%