×

zbMATH — the first resource for mathematics

Reduction numbers, Briançon-Skoda theorems and the depth of Rees rings. (English) Zbl 0854.13003
During the last two decades there was – for several reasons – a strong interest in the blow-up rings, i.e. the Rees algebra \(R[ It]=\bigoplus_{n\geq 0} I^n t^n\) and the form ring \(G(I)=\bigoplus_{n\geq 0} I^n/I^{n+1}\). Here \(I\) denotes an ideal of a local ring \((R, {\mathfrak m})\). One of the main questions has been a characterization of their Cohen-Macaulay property. In a certain sense the authors present a complete solution for \(R\) a Cohen-Macaulay ring.
In order to describe their results let \(J\) denote a minimal reduction of \(I\), i.e. an ideal minimal among those satisfying \(I^{n+1}=JI^n\) for a certain integer \(n\in \mathbb{N}\). The smallest possible integer \(n\in \mathbb{N}\) with this property is called the reduction number \(r(I)\) of \(I\). The analytic spread \(l(I)\) of \(I\) is defined as the dimension of the fibre ring \(G(I)/{\mathfrak m} G(I)\).
Suppose that \(R\) is a Cohen-Macaulay ring. Then the following two results are proved:
(1) For \(\text{height }I \geq 1\) the Rees algebra \(R[ It]\) is a Cohen-Macaulay ring if and only if \(G(I)\) is a Cohen-Macaulay ring and \(r(I_{\mathfrak p})\leq \text{height } {\mathfrak p}-1\) for every prime ideal \({\mathfrak p} \supseteq I\) with \(l (I_{\mathfrak p})=\text{height } {\mathfrak p}\).
(2) For \(\text{height }I\geq 2\) the Rees algebra \(R[ It]\) is a Gorenstein ring if and only if \(G(I)\) is a Gorenstein ring, \(r(I_{\mathfrak p})=\text{height } {\mathfrak p}-2\) for every prime ideal \({germ p}\supseteq I\) with height \({\mathfrak p}/I=0\), and \(r(I_{\mathfrak p})\leq \text{height } {\mathfrak p}-2\) for every prime ideal \({\mathfrak p} \supseteq I\) with \(l(I_{\mathfrak p})=\text{height } {\mathfrak p}\leq l(I)\).
This generalizes several results previously known in the literature. In particular for an \({\mathfrak m}\)-primary ideal \(I\) it turns out that \(R[ It]\) is a Cohen-Macaulay ring if and only if \(G(I)\) is a Cohen-Macaulay ring and \(r(I)< \dim R\), see S. Goto and Y. Shimoda [in Commutative algebra: analytical methods, Conf. Fairfax 1979, Lect. Notes Pure Appl. Math. 68, 213-231 (1982; Zbl 0482.13011)], which was one of the first results in this direction and a motivation for the characterization of the Cohen-Macaulayness of the blow-up rings.
The method of the authors’ proofs are based on filter-regular sequences, minimal reduction, and Castelnuovo-Mumford regularity. There are several applications of the authors’ fundamental characterization concerning Serre’s condition \((S_k)\) for \(R[ It]\) and \(G(I)\), reduction numbers of ideals in regular local rings, and the Briançon-Skoda theorem for equicharacteristic local rings [see also the first two authors in Proc. Am. Math. Soc. 124, No. 3, 707-713 (1996; see the preceding review)]. – For related results about the Cohen-Macaulayness of Rees and form rings on pseudo-rational local rings [see also J. Lipman in Math. Res. Lett. 1, No. 2, 149-157 (1994; Zbl 0844.13006)].
Reviewer: P.Schenzel (Halle)

MSC:
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13C40 Linkage, complete intersections and determinantal ideals
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] Aberbach I. and Huneke, C. : An improved Briançon-Skoda theorem with applications to the Cohen-Macaulayness of Rees rings , Math. Ann. 297 (1993) 343-369. · Zbl 0788.13001 · doi:10.1007/BF01459507 · eudml:165137
[2] Aberbach, I.M. , Huckaba, S. and Huneke, C. : Reduction numbers, Rees rings, and Pfaffian ideals , J. Pure Applied Alg. (to appear). · Zbl 0838.13003 · doi:10.1016/0022-4049(94)00069-U
[3] Artin, M. : Algebraic approximation of structures over complete local rings , Publ. Math. I.H.E.S. 36 (1969) 23-56. · Zbl 0181.48802 · doi:10.1007/BF02684596 · numdam:PMIHES_1969__36__23_0 · eudml:103894
[4] Artin, M. and Nagata, M. : Residual intersections in Cohen-Macaulay rings , J. Math. Kyoto Univ. 12 (1972) 307-323. · Zbl 0263.14019 · doi:10.1215/kjm/1250523522
[5] Bayer, D. and Stillman, M. : A criterion for detecting m-regularity , Invent. Math. 87 (1987) 1-11. · Zbl 0625.13003 · doi:10.1007/BF01389151 · eudml:143411
[6] Briançon, J. and Skoda, H. : Sur la clôture intégrale d’un idéal de germes de fonctions holomorphes en un point de Cn , C. R. Acad. Sci. Paris Sér A 278 (1974) 949-951. · Zbl 0307.32007
[7] Burch, L. : On ideals of finite homological dimension in local rings , Proc. of Camb. Phil. Soc. 64 (1968) 941-948. · Zbl 0172.32302
[8] Cowsik, R.C. and Nori, M.V. : On the fibres of blowing up , J. of Indian Math. Soc. 40 (1976) 217-222. · Zbl 0437.14028
[9] Cuong, N.T. , Schenzel, P. , and Trung, N.V. : Verallgemeinerte Cohen-Macaulay-Moduln , Math. Nachr. 85 (1978) 57-73. · Zbl 0398.13014 · doi:10.1002/mana.19780850106
[10] Eisenbud, D. and Goto, S. : Linear free resolutions and minimal multiplicities , J. Algebra 88 (1984) 89-133. · Zbl 0531.13015 · doi:10.1016/0021-8693(84)90092-9
[11] Goto, S. and Huckaba, S. : On graded rings associated with analytic deviation one ideals , Amer. J. Math. (to appear). · Zbl 0803.13002 · doi:10.2307/2375005
[12] Goto, S. and Nakamura, Y. : Cohen-Macaulay Rees algebras of ideals having analytic deviation two (preprint). · Zbl 0812.13003 · doi:10.2748/tmj/1178225681
[13] Goto, S. and Nakamura, Y. : Gorenstein graded rings associated to ideals of analytic deviation two (preprint). · Zbl 0848.13005 · doi:10.1006/jabr.1995.1215
[14] Goto, S. and Nishida, K. : Filtrations and the Gorenstein property of the associated Rees algebras (preprint). · Zbl 0812.13016
[15] Goto, S. and Shimoda, Y. : On the Rees algebras of Cohen-Macaulay local rings , Lect. Notes in Pure and Applied Mathematics, vol. 68, Marcel-Dekker, New York, 1979, pp. 201-231. · Zbl 0482.13011
[16] Goto, S. and Watanabe, K. : On graded rings I , J. Math. Soc. Japan 30 (1978) 179-213. · Zbl 0371.13017 · doi:10.2969/jmsj/03020179
[17] Grothe, A. , Herrmann, M. , and Orbanz, U. : Graded rings associated to equimultiple ideals , Math. Z. 186 (1984) 531-566. · Zbl 0541.13010 · doi:10.1007/BF01162779 · eudml:173463
[18] Herrmann, M. and Ikeda, S. : On the Gorenstein property of Rees algebras , Manu. Math. 58 (1987) 471-490. · Zbl 0641.13013 · doi:10.1007/BF01170849 · eudml:155262
[19] Herrmann, M. , Huneke, C. , and Ribbe, J. : On reduction exponents of ideals with Gorenstein form ring , Proc. Edinburgh Math. Soc. (to appear). · Zbl 0842.13001 · doi:10.1017/S0013091500019258
[20] Herrmann, M. , Ribbe, J. , and Schenzel, P. : On the Gorenstein property of formrings (preprint). · Zbl 0797.13001 · doi:10.1007/BF03025723 · eudml:174531
[21] Herzog, J. , Simis, A. , and Vasconcelos, W.V. : On the arithmetic and homology of algebras of linear type , Trans. Amer. Math. Soc. 283 (1984) 661-683. · Zbl 0541.13005 · doi:10.2307/1999153
[22] Herzog, J. , Simis, A. , and Vasconcelos, W.V. : On the canonical module of the Rees algebra and the associated graded ring of an ideal , J. of Alg. 105 (1987) 285-302. · Zbl 0613.13007 · doi:10.1016/0021-8693(87)90194-3
[23] Hochster M. and Huneke, C. : tight closure, invariant theory, and the Briançon-Skoda theorem , J. Amer. Math. Soc. 3 (1990) 31-116. · Zbl 0701.13002 · doi:10.2307/1990984
[24] Hochster, M. and Ratliff, L.J., Jr. : Five theorems on Macaulay rings , Pac. J. of Math. 44 (1973) 147-172. · Zbl 0239.13016 · doi:10.2140/pjm.1973.44.147
[25] Huckaba, S. and Huneke, C. : Powers of ideals having small analytic deviation , Amer. J. Math. 114 (1992) 367-403. · Zbl 0758.13001 · doi:10.2307/2374708
[26] Huckaba, S. and Huneke, C. : Rees algebras of ideals having small analytic deviation , Trans. Amer. Math. Soc. 339 (1993) 373-402. · Zbl 0813.13009 · doi:10.2307/2154225
[27] Huckaba, S. and Marley, T. : Depth properties of Rees algebras and associated graded rings , J. Algebra 156 (1993) 259-271. · Zbl 0813.13010 · doi:10.1006/jabr.1993.1075
[28] Huneke, C. : On the associated graded ring of an ideal , Illinois J. Math. 26 (1982) 121-137. · Zbl 0479.13008
[29] Huneke, C. : The theory of d-sequences and powers of ideals , Adv. in Math. 46 (1982) 249-279. · Zbl 0505.13004 · doi:10.1016/0001-8708(82)90045-7
[30] Ikeda, S. : On the Gorenstein property of Rees algebras over local rings , Nagoya Math. J. 92 (1986) 135-154. · Zbl 0585.13014 · doi:10.1017/S0027763000000489
[31] Johnston, B. and Katz, D. : Castelnuovo regularity and graded rings associated to an ideal , Proc. Amer. Math. Soc. (to appear). · Zbl 0826.13014 · doi:10.2307/2160792
[32] Lipman, J. : Cohen-Macaulayness in graded algebras , Math. Res. Letters 1 (1994) 149-157. · Zbl 0844.13006 · doi:10.4310/MRL.1994.v1.n2.a2
[33] Lipman, J. and Sathaye, A. : Jacobian ideals and a theorem of Briançon-Skoda , Michigan Math. J. 28 (1981) 199-222. · Zbl 0438.13019 · doi:10.1307/mmj/1029002510
[34] Lipman, J. and Teissier, B. : Pseudo-rational local rings and a theorem of Briançon-Skoda about integral closures of ideals , Michigan Math. J. 28 (1981) 97-116. · Zbl 0464.13005 · doi:10.1307/mmj/1029002461
[35] Matijevic, J. and Roberts, P. : A conjecture of Nagata on graded Cohen-Macaulay rings , J. Math. Kyoto Univ. 14 (1974) 125-138. · Zbl 0278.13013 · doi:10.1215/kjm/1250523283
[36] Matsumura, H. : Commutative Ring Theory , Cambridge University Press, Cambridge, 1986. · Zbl 0603.13001
[37] Mumford, D. : Lectures on curves on an algebraic surface , Princeton University Press, Princeton, New Jersey, 1966. · Zbl 0187.42701 · doi:10.1515/9781400882069
[38] Mcadam, S. : Asymptotic prime divisors , Lect. Notes in Math. 1023, Springer, 1983. · Zbl 0529.13001 · doi:10.1007/BFb0071575
[39] Northcott, S. and Rees, D. : Reductions of ideals in local rings , Math. Proc. Camb. Phil. Soc. 50 (1954) 145-158. · Zbl 0057.02601
[40] Noh, S. and Vasconcelos, W. : The S2-closure of a Rees algebra , Results in Math. 23 (1993) 149-162. · Zbl 0777.13006 · doi:10.1007/BF03323133
[41] Oiishi, A. : Castelnuovo’s regularity of graded rings and modules , Hiroshima Math. J. 12 (1982) 627-644. · Zbl 0557.13007
[42] Sally, J. : On the associated graded ring of a local Cohen-Macaulay ring , J. Math. Kyoto Univ. 17 (1977) 19-21. · Zbl 0353.13017 · doi:10.1215/kjm/1250522807
[43] Sancho De Salas, J.B. : Géométrie Algébrique et applications I, La Rabida, Travaux en Cours , vol. 22, Hermann, Paris, 1987, pp. 201-209. · Zbl 0625.14025
[44] Simis, A. , Ulrich, B. , and Vasconcelos, W. : Cohen-Macaulay Rees algebras and degrees of polynomial relations , Math. Annalen (to appear). · Zbl 0819.13003 · doi:10.1007/BF01446637 · eudml:165299
[45] Simis, A. and Vasconcelos, W.V. : The syzygies of the conormal module , Amer. J. of Math. 103 (1981) 203-224. · Zbl 0467.13009 · doi:10.2307/2374214
[46] Tang, Z. : Rees rings and associated graded rings of ideals having higher analytic deviation (preprint). · Zbl 0803.13003 · doi:10.1080/00927879408825109
[47] Trung, N.V. : Reduction exponents and degree bound for the defining equations of graded rings , Proc. Amer. Soc. 101 (1987) 229-234. · Zbl 0641.13016 · doi:10.2307/2045987
[48] Trung, N.V. : Filter-regular sequences and multiplicity of blow-up rings of ideals of the principal class , J. Math. Kyoto Univ. (to appear). · Zbl 0816.13020 · doi:10.1215/kjm/1250519184
[49] Trung, N.V. : Reduction number, \alpha -invariant, and Rees algebras of ideals having small analytic deviation , Proceedings of the Workshop on Commutative Algebra, Trieste 1992 (to appear). · Zbl 0927.13004
[50] Trung, N.V. and Ikeda, S. : When is the Rees algebra Cohen-Macaulay ?, Comm. Algebra 17(12) (1989) 2893-2922. · Zbl 0696.13015 · doi:10.1080/00927878908823885
[51] Ulrich, B. : Artin-Nagata properties and reductions of ideals , Cont. Math. 159 (1994) 373-400. · Zbl 0821.13008
[52] Ulrich, B. and Vasconcelos, W.V. : 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
[53] Valabrega, P. and Valla, G. : Form rings and regular sequences , Nagoya Math. J. 72 (1978) 91-101. · Zbl 0362.13007 · doi:10.1017/S0027763000018225
[54] Vasconcelos, W.V. : On the equations of Rees algebras , J. reine angew. Math. 418 (1991) 189-218. · Zbl 0727.13002 · doi:10.1515/crll.1991.418.189 · crelle:GDZPPN002208679 · eudml:153338
[55] Vasconcelos, W. : Hilbert functions, analytic spread, and Koszul homology , Cont. Math. 159 (1994) 401-422. · Zbl 0803.13012
[56] Vasconcelos, W. : Arithmetic of Blowup Algebras (manuscript). · Zbl 0813.13008
[57] Viêt, D.Q. : A note on the Cohen-Macaulayness of Rees algebras of filtrations , Comm. in Alg. 21 (1993) 221-229. · Zbl 0772.13002 · doi:10.1080/00927879208824556
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.