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Minimal pure injective resolutions of flat modules. (English) Zbl 0614.13005

This paper examines the pure injective minimal resolution of a flat module F over a noetherian ring R. Starting from the fact that the pure injective envelope \(PE(F)\) of a flat module F over a noetherian ring \(R\) can be represented as: \(PE(F)=\oplus_{{\mathfrak p}\in\text{Spec} k}T_{{\mathfrak p}}\), where \(T_{{\mathfrak p}}\) is the completion of a free \(R_{{\mathfrak p}}\)-module, the author proves the following theorem on a minimal pure injective resolution of \(F: F\to PE(F) \to PE^ 1(F) \to ....\)
Let \(\pi_ n({\mathfrak q},F)\) be the cardinality of a base of a free \(R_{{\mathfrak q}}\) module whose completion is \(T_{{\mathfrak q}}\), where \(PE^ n(F) = \prod T_{{\mathfrak q}}\), then:
\(\pi_ n({\mathfrak q},F)=0\), \(\forall {\mathfrak q}\supsetneqq {\mathfrak p} \Rightarrow \pi_{n+1}({\mathfrak q},F)=0\), \(\forall {\mathfrak q}>{\mathfrak p}.\)
As a consequence: \(\dim(R)<\infty\), \(n>\dim(R) \Rightarrow PE^ n(F)=0.\)
This theorem is used to give new proofs of results about the projective dimension of flat modules and at the same time to sharpen these results. A characterization of the modules \(\prod T_{{\mathfrak p}},\) such that \(F\to \prod T_{{\mathfrak p}}\) is a pure injective envelope, enables to give a change of ring theorem and, as a corollary, the result that
\(\dim(R_ 1)=\dim(R_ 2) \Rightarrow\) pure \(inj.\dim._{R_ 1}(R_ 1) =\) pure \(inj.\dim_{R_ 2}(R_ 2),\)
for \(R_ 1, R_ 2\) coordinate rings of affine algebraic varieties over a field \(k\).
Reviewer: C.Massaza

MSC:

13C11 Injective and flat modules and ideals in commutative rings
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
13D05 Homological dimension and commutative rings
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References:

[1] Warfield, R. B., Purity and algebraic compactness for modules, Pacific J. Math., 28, 699-719 (1969) · Zbl 0172.04801
[2] Enochs, E. E., Injective and flat covers, envelopes and resolvents, Israel J. Math., 39, 189-209 (1981) · Zbl 0464.16019
[3] Matlis, E., Injective modules over noetherian rings, Pacific J. Math., 8, 511-528 (1958) · Zbl 0084.26601
[4] Enochs, E. E., Flat covers and flat cotorsion modules, (Proc. Amer. Math. Soc., 92 (1984)), 179-184 · Zbl 0522.13008
[5] Ishikawa, T., On injective modules and flat modules, J. Math. Soc. Japan, 17, 291-296 (1965) · Zbl 0199.07802
[6] Raynaud, M.; Gruson, L., Critères de platitude et de projectivité, Invent. Math., 13, 1-89 (1971) · Zbl 0227.14010
[7] Bass, H., On the ubiquity of Gorenstein rings, Math. Z., 82, 8-28 (1963) · Zbl 0112.26604
[8] Enochs, E.; Jenda, O. M.G, Balanced functors applied to modules, J. Algebra, 92, 303-310 (1985) · Zbl 0554.18006
[9] Jensen, C. U., Les foncteurs dérivés de \(lim\) et leurs appliations en théorie des modules, (Lecture Notes in Mathematics, Vol. 254 (1972), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0238.18007
[10] Nagata, M., Local Rings (1962), Interscience: Interscience New York · Zbl 0123.03402
[11] Gruson, L.; Jensen, C. U., Dimensions cohomologiques reliées aux foncteurs \(lim^{(i)}\), (Lecture Notes in Mathematics, Vol. 867 (1981), Springer-Verlag: Springer-Verlag New York/Berlin), 234-294 · Zbl 0505.18005
[12] Eakin, P., The converse to a well-known theorem on Noetherian rings, Math. Ann., 177, 278-282 (1968) · Zbl 0155.07903
[13] Fuchs, L., Algebraically compact modules over noetherian rings, Indian J. Math., 9, no. 2, 357-374 (1967) · Zbl 0186.35001
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