## The existence of flat covers over Noetherian rings of finite Krull dimension

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- by Jin Zhong Xu PDF
- Proc. Amer. Math. Soc.
**123**(1995), 27-32 Request permission

## 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

*R*-modules coincides with the class of flat

*R*-modules, so the projective covers over these rings are flat covers. This prompts the conjecture that over any ring

*R*, every left

*R*-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), Prufer 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.

## References

- Maurice Auslander and Idun Reiten,
*Applications of contravariantly finite subcategories*, Adv. Math.**86**(1991), no. 1, 111–152. MR**1097029**, DOI 10.1016/0001-8708(91)90037-8 - Hyman Bass,
*Finitistic dimension and a homological generalization of semi-primary rings*, Trans. Amer. Math. Soc.**95**(1960), 466–488. MR**157984**, DOI 10.1090/S0002-9947-1960-0157984-8 - Richard Belshoff, Edgar E. Enochs, and Jin Zhong Xu,
*The existence of flat covers*, Proc. Amer. Math. Soc.**122**(1994), no. 4, 985–991. MR**1209416**, DOI 10.1090/S0002-9939-1994-1209416-4 - Edgar E. Enochs,
*Injective and flat covers, envelopes and resolvents*, Israel J. Math.**39**(1981), no. 3, 189–209. MR**636889**, DOI 10.1007/BF02760849 - Edgar Enochs,
*Torsion free covering modules*, Proc. Amer. Math. Soc.**14**(1963), 884–889. MR**168617**, DOI 10.1090/S0002-9939-1963-0168617-7 - Edgar Enochs,
*Flat covers and flat cotorsion modules*, Proc. Amer. Math. Soc.**92**(1984), no. 2, 179–184. MR**754698**, DOI 10.1090/S0002-9939-1984-0754698-X - Edgar E. Enochs,
*Covers by flat modules and submodules of flat modules*, J. Pure Appl. Algebra**57**(1989), no. 1, 33–38. MR**984044**, DOI 10.1016/0022-4049(89)90025-X - L. Gruson and C. U. Jensen,
*Dimensions cohomologiques reliées aux foncteurs $\underleftarrow {\mmlToken {mi}{lim}}^{(i)}$*, Paul Dubreil and Marie-Paule Malliavin Algebra Seminar, 33rd Year (Paris, 1980) Lecture Notes in Math., vol. 867, Springer, Berlin-New York, 1981, pp. 234–294 (French). MR**633523** - Michel Raynaud and Laurent Gruson,
*Critères de platitude et de projectivité. Techniques de “platification” d’un module*, Invent. Math.**13**(1971), 1–89 (French). MR**308104**, DOI 10.1007/BF01390094 - Bo Stenström,
*Rings of quotients*, Die Grundlehren der mathematischen Wissenschaften, Band 217, Springer-Verlag, New York-Heidelberg, 1975. An introduction to methods of ring theory. MR**0389953**, DOI 10.1007/978-3-642-66066-5

## Additional Information

- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**123**(1995), 27-32 - MSC: Primary 16D40; Secondary 13C11
- DOI: https://doi.org/10.1090/S0002-9939-1995-1242111-5
- MathSciNet review: 1242111