×

zbMATH — the first resource for mathematics

An improved Briançon-Skoda theorem with applications to the Cohen- Macaulayness of Rees algebras. (English) Zbl 0788.13001
We give an improved version of the Briançon-Skoda theorem for regular rings containing a field. If \((R,m)\) is such a ring, \(I\) is an ideal of analytic spread \(\ell\) and bigheight \(h\), and \(J\) is a reduction of \(I\) then we show that \(I^ \ell \subseteq J(I^{\ell-h})^{\text{un}}\) (where \(( )^{\text{un}}\) means: take the intersection of the minimal primary components). The proof uses tight closure in characteristic \(p\) and then standard techniques of reduction to characteristic \(p\) for local rings containing fields of characteristic 0.
We then apply this result to study the relationship between the Cohen- Macaulayness of \(R[It]\) and \(\text{Gr}_ I(R)\). In particular we show that if \(R\) is a regular local ring containing a field and \(I\) is an unmixed curve then \(R[It]\) is Cohen-Macaulay if and only if \(\text{Gr}_ I(R)\) is Cohen-Macaulay.
In a final section we continue the investigation of 4-generated unmixed ideals of height 2 in a 3-dimensional regular local ring begun by Vasconcelos.

MSC:
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
13C14 Cohen-Macaulay modules
13H05 Regular local rings
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Aberbach, I.M., Huckaba, S., Huneke, C.: Reduction numbers, Rees algebras and Pfaffian ideals (in preparation) · Zbl 0838.13003
[2] Artin, M.: Algebraic approximation of structures over complete local rings. Publ. Math., Inst. Hautes ?tud. Sci.36, 23-56 (1969) · Zbl 0181.48802 · doi:10.1007/BF02684596
[3] Artin, M., Nagata, M.: Residual intersections in Cohen-Macaulay rings. J. Math. Kyoto Univ.12, 307-323 (1972) · Zbl 0263.14019
[4] Bayer, D., Stillman, M.: A criterion for detectingm-regularity. Invent. Math.87, 1-11 (1987) · Zbl 0625.13003 · doi:10.1007/BF01389151
[5] Brian?on, J., Skoda, H.: Sur la cl?ture int?grale d’un id?al de germes de fonctions holomorphes en un point deC n C.R. Acad. Sci., Paris, S?r. A278, 949-951 (1974) · Zbl 0307.32007
[6] Burch, L.: On ideals of finite homological dimension in local rings. Proc. Camb. Philos. Soc.64, 941-948 (1968) · Zbl 0172.32302 · doi:10.1017/S0305004100043620
[7] Cowsik, R.C., Nori, M.V.: On the fibres of blowing up. J. Indian Math. Soc.40, 217-222 (1976) · Zbl 0437.14028
[8] Goto, S., Huckaba, S.: On graded rings associated to analytic deviation one ideals. Amer. J. Math (to appear) · Zbl 0803.13002
[9] Goto, S., Shimoda, Y.: On the Rees algebras of Cohen-Macaulay rings. Lect. Notes Pure Appl. Math.68, 201-231 (1979) · Zbl 0482.13011
[10] Grothendieck, A. (notes by R. Hartshorne): Local cohomology. (Lect. Notes Math., vol. 41) Berlin Heidelberg New York: Springer 1967 · Zbl 0185.49202
[11] Herzog, J.: Ein Cohen-Macaulay-Kriterium mit Anwendungen auf den Konormalenmodul und den Differentialmodul. Math. Z.163, 149-162 (1978) · Zbl 0382.13008 · doi:10.1007/BF01214062
[12] Herzog, J., Simis, A., Vasconcelos, W.V.: On the arithmetic and homology of algebras of linear type. Trans. Am. Math. Soc.283, 661-683 (1984) · Zbl 0541.13005 · doi:10.1090/S0002-9947-1984-0737891-6
[13] Herzog, J., Simis, A., Vasconcelos, W.V.: On the canonical module of the Rees algebra and the associated graded ring of an ideal. J. Algebra105, 285-302 (1987) · Zbl 0613.13007 · doi:10.1016/0021-8693(87)90194-3
[14] Hochster, M.: Unpublished notes
[15] Hochster, M., Huneke, C.: Tight closure, invariant theory, and the Brian?on-Skoda theorem. J. Am. Math. Soc.3, 31-116 (1990) · Zbl 0701.13002
[16] Hochster, M., Huneke, C.: Applications of the existence of big Cohen-Macaulay algebras. (Preprint) · Zbl 0834.13013
[17] Hochster, M., Ratliff, Jr., L.J.: Five theorems on Macaulay rings. Pac. J. Math.44, 147-172 (1973) · Zbl 0239.13016
[18] Huckaba, S.: On completed-sequences and the defining ideals of Rees algebras. Math. Proc. Camb. Philos. Soc.106, 445-458 (1989) · Zbl 0693.13013 · doi:10.1017/S0305004100068183
[19] Huckaba, S., Huneke, C.: Powers of ideals having small analytic deviation. Am. J. Math.114, 367-403 (1992) · Zbl 0758.13001 · doi:10.2307/2374708
[20] Huckaba, S., Huneke, C.: Rees algebras of ideals having small analytic deviation. Trans. Am. Math. Soc. (to appear) · Zbl 0813.13009
[21] Huckaba, S., Marley, T.: Depth formulas for certain graded rings associated to an ideal. (Preprint) · Zbl 0796.13004
[22] Huckaba, S., Marley, T.: Depth properties of Rees algebras and associated graded rings. J. Algebra156, 259-271 (1993) · Zbl 0813.13010 · doi:10.1006/jabr.1993.1075
[23] Huneke, C.: On the associated graded ring of an ideal. III. J. Math.26, 121-137 (1982) · Zbl 0479.13008
[24] Huneke, C.: The theory ofd-sequences and powers of ideals. Adv. Math.46, 249-279 (1982) · Zbl 0505.13004 · doi:10.1016/0001-8708(82)90045-7
[25] Huneke, C., Ulrich, B.: Residual intersections. J. Reine Angew. Math.390, 1-20 (1988) · Zbl 0732.13004 · doi:10.1515/crll.1988.390.1
[26] Huneke, C., Ulrich, B.: Powers of licci ideals. In: Hochster, M. et al. (eds.) Commutative Algebra, pp. 339-346. Berlin Heidelberg New York: Springer 1989 · Zbl 0731.13008
[27] Lipman, J.: Relative Lipshitz saturation. Am. J. Math.97, 791-813 (1973) · Zbl 0314.13003 · doi:10.2307/2373777
[28] Lipman, J., Sathaye, A.: Jacobian ideals and a theorem of Brian?on-Skoda. Mich. Math. J.28, 199-222 (1981) · Zbl 0461.13010 · doi:10.1307/mmj/1029002510
[29] Lipman, J., Teissier, B.: Pseudo-rational local rings and a theorem of Brian?on-Skoda about integral closures of ideals. Mich. Math. J.28, 97-116 (1981) · Zbl 0464.13005 · doi:10.1307/mmj/1029002461
[30] Matijevic, J., Roberts, P.: A conjecture of Nagata on graded Cohen-Macaulay rings. J. Math. Kyoto Univ.14, 125-128 (1974) · Zbl 0278.13013
[31] Matsumura, H.: Commutative ring theory. Cambridge: Cambridge University Press 1986 · Zbl 0603.13001
[32] Northcott, D.G., Rees, D.: Reductions of ideals in local rings. Proc. Camb. Philos. Soc.50, 145-158 (1954) · Zbl 0057.02601 · doi:10.1017/S0305004100029194
[33] Simis, A., Vasconcelos, W.V.: The syzygies of the conormal module. Am. J. Math.103, 203-224 (1981) · Zbl 0467.13009 · doi:10.2307/2374214
[34] Swanson, I.: Joint reductions, tight closure, and the Brian?on-Skoda theorem. J. Algebra147, 128-136 (1992) · Zbl 0755.13003 · doi:10.1016/0021-8693(92)90256-L
[35] Trung, N.V.: Reduction exponent and degree bound for the defining equations of graded rings. Proc. Am. Math. Soc.101, 229-236 (1987) · Zbl 0641.13016 · doi:10.1090/S0002-9939-1987-0902533-1
[36] Trung, N.V., Ikeda, S.: When is the Rees algebra Cohen-Macaulay? Commun. Algebra17, 2893-2922 (1989) · Zbl 0696.13015 · doi:10.1080/00927878908823885
[37] Ulrich, B., Vasconcelos, W.V.: The equations of Rees algebras of ideals with linear presentation. (Preprint) · Zbl 0789.13002
[38] Valabrega, P., Valla, G.: Form rings and regular sequences. Nagoya Math. J.72, 91-101 (1978) · Zbl 0362.13007
[39] Vasconcelos, W.V.: On the equations of Rees algebras. J. Reine Angew. Math.418, 189-218 (1991) · Zbl 0727.13002 · doi:10.1515/crll.1991.418.189
[40] Vi?t, D.Q.: A note on the Cohen-Macaulayness of Rees algebras of filtrations. (Preprint) · Zbl 0772.13002
[41] Watanabe, J.: A note on Gorenstein rings of embedding codimension three. Nagoya Math. J.50, 227-232 (1973) · Zbl 0242.13019
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.