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Artin-Nagata properties and Cohen-Macaulay associated graded rings. (English) Zbl 0866.13001
This paper is a welcome addition to the existing literature on blow-up algebras. For sometime to come it will contain among the most general results on the subject of the Cohen-Macaulayness of the Rees algebra \({\mathcal R}={\mathcal R}(I)\) and the associated graded ring \(G=\text{gr}_I(R)\), where \(I\subset R\) is an ideal of the local ring \(R\). There has been quite some intense activity on this subject in recent years – one can have an idea of its size by looking at the very references of the present paper. In the case in which the ring \(R\) is regular (pseudo-rational singularity would be enough), J. Lipman [Math. Res. Lett. 1, No. 2, 149-157 (1994; Zbl 0844.13006)] proved that if \(G\) is Cohen-Macaulay then so is \({\mathcal R}\). This settled completely, in the regular case, one of the main problems in the subject. However, the more general situation where \(R\) is Cohen-Macaulay – or even Gorenstein for that matter – has as yet a lot to go.
The authors base their main results on assumptions regarding the Cohen-Macaulayness of certain residual intersection loci of the ideal \(I\). The precise notion entering these assumptions – dubbed by the authors the AN (for Artin-Nagata) property – is perhaps a little too technical to be reproduced here. What they show, in a nonprecise wording, is that the conjunction of the following three properties imply the Cohen-Macaulayness of \(G\) (and of \({\mathcal R}\) in codimension at least 2):
(1) the AN property (sufficiently qualified),
(2) a reasonable growth estimate for the local number of generators of \(I\), and
(3) sufficient “sliding-depth” for the residue rings \(R/I^j\).
Alas! It is not so easy to recognize when one is in presence of all such properties. This is perhaps felt by the very authors who, therefore, took pains in showing that most of the technical features of the hypotheses fall off under some further natural restrictions. They do this in the form of several interesting corollaries.
The proof of the main theorem (theorem 3.1) is quite involved as expected from a result that contains significant improvement on all the previous ones by many authors, including the present second author himself. The proof itself takes most of section 3. The subsequent sections (4 and 5) are respectively concerned with the Castelnuovo regularity and bounds for the reduction number, and the Gorensteinness of \(G\) and the form of the canonical module. The results obtained in these sections also generalize to quite some extent the previously known theorems.
Reviewer: A.Simis (Salvador)

MSC:
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13C14 Cohen-Macaulay modules
13E05 Commutative Noetherian rings and modules
13D45 Local cohomology and commutative rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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References:
[1] I.M. Aberbach , Local reduction numbers and Cohen-Macaulayness of associated graded rings , preprint, 1994. · Zbl 0918.13001 · doi:10.1006/jabr.1995.1380
[2] I.M. Aberbach and S. Huckaba , Reduction number bounds on analytic deviation two ideals and Cohen-Macaulayness of associated graded rings , Comm. Algebra 23 (1995), 2003-2026. · Zbl 0831.13002 · doi:10.1080/00927879508825325
[3] I.M. Aberbach , S. Huckaba and C. Huneke , Reduction numbers, Rees algebras, and Pfaffian ideals , J. Pure and Applied Algebra, to appear. · Zbl 0838.13003 · doi:10.1016/0022-4049(94)00069-U
[4] I.M. Aberbach and C. Huneke , An improved Briançon-Skoda theorem with applications to the Cohen-Macaulayness of Rees algebras , Math. Ann. 297 (1993), 343-369. · Zbl 0788.13001 · doi:10.1007/BF01459507 · eudml:165137
[5] I.M. Aberbach , C. Huneke and N.V. Trung , Reduction numbers, Briançon-Skoda theorems and depth of Rees algebras , preprint, 1993. · Zbl 0854.13003
[6] M. Artin and M. Nagata , Residual intersections in Cohen-Macaulay rings , J. Math. Kyoto Univ. 12 (1972), 307-323. · Zbl 0263.14019 · doi:10.1215/kjm/1250523522
[7] L. Avramov and J. Herzog , The Koszul algebra of a codimension 2 embedding , Math. Z. 175 (1980), 249-280. · Zbl 0461.14014 · doi:10.1007/BF01163026 · eudml:173036
[8] D. Eisenbudand S. Goto , Linear free resolutions and minimal multiplicities , J. Algebra 88 (1984), 89-133. · Zbl 0531.13015 · doi:10.1016/0021-8693(84)90092-9
[9] D. Eisenbud and C. Huneke , Cohen-Macaulay Rees algebras and their specializations , J. Algebra 81 (1983), 202-224. · Zbl 0528.13024 · doi:10.1016/0021-8693(83)90216-8
[10] S. Goto and S. Huckaba , On graded rings associated to analytic deviation one ideals , Amer. J. Math. 116 (1994), 905-919. · Zbl 0803.13002 · doi:10.2307/2375005
[11] S. Goto and Y. Nakamura , On the Gorensteinness of graded rings associated to ideals of analytic deviation one , Contemporary Mathematics 159 (1994), 51-72. · Zbl 0816.13004
[12] S. Goto and Y. Nakamura , Cohen-Macaulay Rees algebras of ideals having analytic deviation two , Tohoku Math. J. 46 (1994), 573-586. · Zbl 0812.13003 · doi:10.2748/tmj/1178225681
[13] S. Goto and Y. Nakamura , Gorenstein graded rings associated to ideals of analytic deviation two , preprint, 1993. · Zbl 0848.13005 · doi:10.1006/jabr.1995.1215
[14] S. Goto , Y. Nakamura , and K. Nishida , Cohen-Macaulayness in graded rings associated to ideals , preprint, 1994. · Zbl 0878.13003 · doi:10.1215/kjm/1250518548
[15] M. Herrmann , S. Ikeda and U. Orbanz , Equimultiplicity and Blowing-up , Springer Verlag, Berlin, 1988. · Zbl 0649.13011
[16] M. Herrmann , C. Huneke and J. Ribbe , On reduction exponents of ideals with Gorenstein formrings , preprint, 1993. · Zbl 0842.13001 · doi:10.1017/S0013091500019258
[17] J. Herzog , A. Simis and W.V. Vasconcelos , Koszul homology and blowing-up rings , in Commutative Algebra, Proceedings, Trento 1981 (S. Greco and G. Valla, Eds.), Lecture Notes in Pure and Applied Math. 84, Marcel Dekker, New York, 1983, 79-169. · Zbl 0499.13002
[18] J. Herzog , A. Simis and W.V. Vasconcelos , On the canonical module of the Rees algebra and the associated graded ring of an ideal , J. Algebra 105 (1987), 285-302. · Zbl 0613.13007 · doi:10.1016/0021-8693(87)90194-3
[19] J. Herzog , W.V. Vasconcelos and R. Villarreal , Ideals with sliding depth , Nagoya Math. J. 99 (1985), 159-172. · Zbl 0561.13014 · doi:10.1017/S0027763000021553
[20] S. Huckaba and C. Huneke , Powers of ideals having small analytic deviation , American J. Math. 114 (1992), 367-403. · Zbl 0758.13001 · doi:10.2307/2374708
[21] S. Huckaba and C. Huneke , Rees algebras of ideals having small analytic deviation , Trans. Amer. Math. Soc. 339 (1993), 373-402. · Zbl 0813.13009 · doi:10.2307/2154225
[22] C. Huneke , Symbolic powers of prime ideals and special graded algebras , Comm. Algebra 9 (1981), 339-366. · Zbl 0454.13003 · doi:10.1080/00927878108822586
[23] C. Huneke , The theory of d-sequences and powers of ideals , Adv. Math. 46 (1982), 249-279. · Zbl 0505.13004 · doi:10.1016/0001-8708(82)90045-7
[24] C. Huneke , On the associated graded ring of an ideal , Illinois J. Math. 26 (1982), 121-137. · Zbl 0479.13008
[25] C. Huneke , Linkage and Koszul homology of ideals , American J. Math. 104 (1982), 1043-1062. · Zbl 0505.13003 · doi:10.2307/2374083
[26] C. Huneke , Strongly Cohen-Macaulay schemes and residual intersections , Trans. Amer. Math. Soc. 277 (1983), 739-763. · Zbl 0514.13011 · doi:10.2307/1999234
[27] C. Huneke , Numerical invariants of liaison classes , Invent. Math. 75 (1984), 301-325. · Zbl 0536.13005 · doi:10.1007/BF01388567 · eudml:143099
[28] C. Huneke and B. Ulrich , Residual intersections , J. reine angew. Math. 390 (1988), 1-20. · Zbl 0732.13004 · doi:10.1515/crll.1988.390.1 · crelle:GDZPPN002205815 · eudml:153055
[29] B. Johnston and D. Katz , Castelnuovo regularity and graded rings associated to an ideal , Proc. Amer. Math. Soc. 123 (1995), 727-734. · Zbl 0826.13014 · doi:10.2307/2160792
[30] J. Lipman , Cohen-Macaulayness in graded algebras , Math. Research Letters 1 (1994), 149-157. · Zbl 0844.13006 · doi:10.4310/MRL.1994.v1.n2.a2
[31] D.G. Northcott and D. Rees , Reductions of ideals in local rings , Math. Proc. Camb. Phil. Soc. 50 (1954), 145-158. · Zbl 0057.02601
[32] A. Ooishi , Castelnuovo’s regularity of graded rings and modules , Hiroshima Math. J. 12 (1982), 627-644. · Zbl 0557.13007
[33] A. Simis , B. Ulrich and W.V. Vasconcelos , Cohen-Macaulay Rees algebras and degrees of polynomial relations , Math. Ann. 301 (1995), 421-444. · Zbl 0819.13003 · doi:10.1007/BF01446637 · eudml:165299
[34] H. Srinivasan , A grade five cyclic Gorenstein module with no minimal algebra resolutions , preprint. · Zbl 0856.13011 · doi:10.1006/jabr.1996.0016
[35] Z. Tang , Rees rings and associated graded rings of ideals having higher analytic deviation , Comm. Algebra 22 (1994), 4855-4898. · Zbl 0803.13003 · doi:10.1080/00927879408825109
[36] N.V. Trung , Reduction exponent and degree bound for the defining equations of graded rings , Proc. Amer. Math. Soc. 101 (1987), 229-236. · Zbl 0641.13016 · doi:10.2307/2045987
[37] N.V. Trung and S. Ikeda , When is the Rees algebra Cohen-Macaulay ?, Comm. Algebra 17 (1989), 2893-2922. · Zbl 0696.13015 · doi:10.1080/00927878908823885
[38] B. Ulrich , Artin-Nagata properties and reductions of ideals , Contemporary Mathematics 159 (1994), 373-400. · Zbl 0821.13008
[39] B. Ulrich and W.V. Vasconcelos , The equations of Rees algebras of ideals with linear presentation , Math. Z. 214 (1993), 79-92. · Zbl 0789.13002 · doi:10.1007/BF02572392 · eudml:174556
[40] P. Valabrega and G. Valla , Form rings and regular sequences , Nagoya Math. J. 72 (1978), 93-101. · Zbl 0362.13007 · doi:10.1017/S0027763000018225
[41] W.V. Vasconcelos , Hilbert functions, analytic spread and Koszul homology , Contemporary Mathematics 159 (1994), 401-422. · Zbl 0803.13012
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