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When is the Rees algebra Cohen-Macaulay? (English) Zbl 0696.13015
The authors discuss the Cohen-Macaulay property for the Rees algebra $$R=R({\mathfrak a})=\oplus^{\infty}_{i=0}{\mathfrak a}^ i$$ of an ideal $${\mathfrak a}$$ in a noetherian local ring (A,$${\mathfrak m})$$. The main result is a criterion based on the homogeneous components of the local cohomology of the associated graded ring $$G=G({\mathfrak a})=\oplus^{\infty}_{i=0}{\mathfrak a}^ i/{\mathfrak a}^{i+1}$$ with respect to its maximal homogeneous ideal $${\mathfrak M}:$$ Suppose that $${\mathfrak a}$$ is not contained in the nilradical of A; then R($${\mathfrak a})$$ is Cohen-Macaulay if and only if $$[H^ i_{{\mathfrak M}}(G)]_ n=0$$ for $$n\neq -1, i=0,...,d-1$$ and $$n\geq 0, i=d$$ $$(d=\dim (A))$$. Many special results previously obtained by Goto, Ikeda, Schenzel, Shimoda and others can be derived from this result. As a technical tool the authors introduce the notion of a generalized Cohen-Macaulay ring with respect to an ideal. The proof of the main criterion is then based on a careful discussion of the relationship between the local cohomology modules of A, R, and G induced by standard exact sequences.
The criterion is applied to special cases: $${\mathfrak m}$$-primary ideals $${\mathfrak a}$$, the case in which the analytic spread of $${\mathfrak a}$$ equals the height of $${\mathfrak a}$$, and ideals with small reduction number. We illustrate the results obtained by the following example: Let $$\dim (A)=2$$, $${\mathfrak a}$$ be $${\mathfrak m}$$-primary and $$x_ 1, x_ 2$$ a system of parameters generating a minimal reduction of $${\mathfrak a}$$; then R is Cohen-Macaulay if and only if the following conditions are satisfied: $$depth(A)>0$$, the natural homomorphism from the Koszul cohomology of $$(x_ 1,x_ 2)$$ to the local cohomology of A is surjective, $${\mathfrak a}^ 2=(x_ 1,x_ 2){\mathfrak a}$$, $$(x^ 2_ 1,x^ 2_ 2)\cap {\mathfrak a}^ 3=(x^ 2_ 1,x^ 2_ 2){\mathfrak a}$$, and $$((x_ 1):x_ 2)\cap {\mathfrak a}=(x_ 1).$$
Finally the (natural) case in which A is Cohen-Macaulay itself is treated. The condition of the criterion reduces to $$[H^ d_{{\mathfrak M}}(G)]_ n=0$$ for $$n\geq 0$$, and similarly several other theorems have much simpler formulations than in general circumstances.
Reviewer: W.Bruns

##### MSC:
 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 13A15 Ideals and multiplicative ideal theory in commutative rings
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##### References:
 [1] Archilles R., Seminar D. Eisenbud/B. Singh/W. Vogel 2 pp 5– (1982) [2] DOI: 10.2307/1970159 · Zbl 0092.03902 · doi:10.2307/1970159 [3] DOI: 10.1016/0021-8693(73)90076-8 · Zbl 0256.13017 · doi:10.1016/0021-8693(73)90076-8 [4] DOI: 10.1016/0021-8693(83)90207-7 · Zbl 0475.14001 · doi:10.1016/0021-8693(83)90207-7 [5] DOI: 10.1016/0021-8693(84)90043-7 · Zbl 0543.14003 · doi:10.1016/0021-8693(84)90043-7 [6] DOI: 10.1017/S0305004100047198 · doi:10.1017/S0305004100047198 [7] DOI: 10.1002/mana.19780850106 · Zbl 0398.13014 · doi:10.1002/mana.19780850106 [8] DOI: 10.1016/0021-8693(83)90216-8 · Zbl 0528.13024 · doi:10.1016/0021-8693(83)90216-8 [9] DOI: 10.1016/0021-8693(82)90035-7 · Zbl 0479.13007 · doi:10.1016/0021-8693(82)90035-7 [10] Goto S., J. Math. Kyoto Univ 20 pp 691– (1980) [11] Goto S., Lecture Notes in Pure and Applied Mathematics 68 pp 201– (1982) [12] Goto S., J. Math. Soc 30 pp 179– (1978) · Zbl 0371.13017 · doi:10.2969/jmsj/03020179 [13] Grothendieck A., Publ. Math. I.H.E.S 11 (1961) [14] Grothendieck A., Lect. Notes in Math 41 (1967) [15] Herrmann M., Teubner-Texte zur Mathematik (1977) [16] Hochster M., Pacific J. Math 44 pp 147– (1973) · Zbl 0239.13016 · doi:10.2140/pjm.1973.44.147 [17] DOI: 10.1016/0021-8693(80)90179-9 · Zbl 0439.13001 · doi:10.1016/0021-8693(80)90179-9 [18] Huneke C., Illinois J. Math 26 pp 121– (1982) [19] Ikeda S., Nagoya Math. J 89 pp 47– (1983) · Zbl 0518.13015 · doi:10.1017/S0027763000020237 [20] DOI: 10.1017/S0305004100029194 · doi:10.1017/S0305004100029194 [21] Rees D.., A note on asyptotically unmixed ideals · Zbl 0578.13006 · doi:10.1017/S0305004100063210 [22] DOI: 10.1016/0021-8693(79)90331-4 · Zbl 0401.13016 · doi:10.1016/0021-8693(79)90331-4 [23] Sally J., Composition Math 40 pp 167– (1980) [24] DOI: 10.1002/mana.19750690121 · Zbl 0327.13016 · doi:10.1002/mana.19750690121 [25] DOI: 10.1007/BF01168455 · Zbl 0483.13008 · doi:10.1007/BF01168455 [26] DOI: 10.1007/BF01263267 · Zbl 0483.13009 · doi:10.1007/BF01263267 [27] SHIMODA Y., J. Math. Kyoto Univ 19 pp 327– (1979) [28] DOI: 10.2307/2373908 · Zbl 0429.14001 · doi:10.2307/2373908 [29] Takeuchi Y., Proceedings of the 4th symposium on Commutative Algebra in Japan pp 152– (1982) [30] DOI: 10.1017/S0305004100060308 · Zbl 0509.13024 · doi:10.1017/S0305004100060308 [31] DOI: 10.1002/mana.19841180104 · Zbl 0584.14034 · doi:10.1002/mana.19841180104 [32] Trung N.V., Nagoya Math. J 102 pp 1– (1986) · doi:10.1017/S0027763000000416 [33] DOI: 10.1016/0021-8693(76)90112-5 · Zbl 0338.13013 · doi:10.1016/0021-8693(76)90112-5 [34] Vallabrega P., Nagoya Math. J 72 pp 93– (1978) · Zbl 0362.13007 · doi:10.1017/S0027763000018225 [35] DOI: 10.1007/BF01162779 · Zbl 0541.13010 · doi:10.1007/BF01162779 [36] Herrman M., Nagoya Math. J 92 pp 121– (1983) · Zbl 0502.13012 · doi:10.1017/S0027763000020602
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