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The equations of Rees algebras of ideals with linear presentation. (English) Zbl 0789.13002
Let $$(R,m)$$ be a local Gorenstein ring (or a polynomial ring over a field) of dimension $$d$$, and let $$I$$ be a (homogeneous) $$R$$-ideal of height $$g \geq 1$$. The objective of this paper is to investigate properties of the Rees algebra of $$I$$, $${\mathcal R} (I) =R[It]$$ (where $$t$$ is a variable), and to study the canonical epimorphism $$\alpha:S(I) \twoheadrightarrow {\mathcal R} (I)$$ from the symmetric algebra onto the Rees algebra. As a standing hypothesis, suppose that $$I$$ is strongly Cohen- Macaulay, i.e., that all Koszul homology modules of a (homogeneous) generating set of $$I$$ are Cohen-Macaulay. By a well known result from J. Herzog, A. Simis, and W. V. Vasconcelos [cf. J. Algebra 74, 466-493 (1982; Zbl 0484.13006)], $$\alpha$$ is an isomorphism (one says $$I$$ is of linear type) and $${\mathcal R} (I)$$ is Cohen-Macaulay, if one assumes that $$I$$ satisfies $${\mathcal F}_ 1$$, meaning, if the minimal number of generators $$v(I_ p)$$ of $$I_ p$$ is at most $$\dim R_ p$$ for every prime ideal $$p$$ containing $$I$$.
In the paper under review, the condition $${\mathcal F} _ 1$$ is replaced by the next weaker assumption that $$I$$ satisfies $${\mathcal F}_ 1$$ locally on the punctured spectrum of $$R$$ and that $$v(I)=d+1$$. In this case, $$I$$ cannot be of linear type. It is shown that the analytic spread of $$I$$, $$\ell(I)=\dim {\mathcal R} (I) \bigoplus_ R R/m$$, has the maximal possible value, namely $$d$$, and that the initial degree of $$\ker \alpha$$ is $$d-g+ 2$$ if $$g \geq 2$$. Furthermore, the graded component of $$\ker \alpha$$ in this minimal degree is identified to be $$\text{Ext}^ d_ R (R/I_ 1(\varphi),R)$$, where $$I_ 1(\varphi)$$ denotes the ideal generated by the entries of a minimal presenting matrix $$\varphi$$ of $$I$$. – One can say more if $$R$$ is a polynomial ring over a field and $$I$$ is presented by a matrix with linear entries. In this case, $$I$$ has reduction number $$d- g+1$$ (and hence $$I^{d-g+2} =JI^{d-g+1}$$ for an ideal $$J \subset I$$ generated by a homogeneous system of parameters), $$\ker\alpha$$ is cyclic, and $$S(I)$$ is reduced. Furthermore, $$I$$ is generated by forms of degree $$\delta \leq {d \over g-1}$$, where equality holds if and only if $${\mathcal R} (I)$$ is Cohen-Macaulay, which in turn is equivalent to the Cohen- Macaulayness of the associated graded ring of $$I$$, $$gr_ I(R)={\mathcal R} (I) \bigoplus_ RR/I$$. These equivalent conditions are always satisfied if $$g \leq 4$$.
Finally, consider a strongly Cohen-Macaulay prime ideal $$I$$ in a regular local ring $$R$$, without any assumptions on the local number of generators or the grading of $$I$$. It is shown that if $$\text{Spec(gr}_ I (R))$$ is irreducible of if gr$$_ I(R)$$ is reduced, then $$I$$ is of linear type (in which case $${\mathcal R} (I)$$ is Cohen-Macaulay and gr$$_ I(R)$$ is a Gorenstein domain).

##### MSC:
 13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 13C14 Cohen-Macaulay modules 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 13H05 Regular local rings 13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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##### References:
 [1] Abhyankar, S.: Concepts of order and rank on a complex space, and a condition for normality. Math. Ann.141, 171–192 (1960) · Zbl 0107.15001 · doi:10.1007/BF01360171 [2] Aoyama, Y.: A remark on almost complete intersections. Manuscr. Math.22, 225–228 (1977) · Zbl 0367.13005 · doi:10.1007/BF01172664 [3] Avramov, L., Herzog, J.: The Koszul algebra of a codimension 2 embedding. Math. Z.175, 249–280 (1980) · Zbl 0461.14014 · doi:10.1007/BF01163026 [4] Hartshorne, R.: Complete intersections and connectedness. Am. J. Math.84, 497–508 (1962) · Zbl 0108.16602 · doi:10.2307/2372986 [5] Herzog, J., Simis, A., Vasconcelos, W.V.: Approximation complexes of blowing-up rings. J. Algebra74, 466–493 (1982) · Zbl 0484.13006 · doi:10.1016/0021-8693(82)90034-5 [6] Herzog, J., Simis, A., Vasconcelos, W.V.: Approximation complexes of blowing-up rings. II. J. Algebra82, 53–83 (1983) · Zbl 0515.13018 · doi:10.1016/0021-8693(83)90173-4 [7] 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 [8] Herzog, J., Vasconcelos, W.V., Villarreal, R.: Ideals with sliding depth. Nagoya Math. J.99, 159–172 (1985) · Zbl 0561.13014 [9] 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 [10] Huneke, C.: On the associated graded ring of an ideal. Ill. J. Math.26, 121–137 (1982) · Zbl 0479.13008 [11] Huneke, C.: Strongly Cohen-Macaulay schemes and residual intersections. Trans. Am. Math. Soc.277, 739–763 (1983) · Zbl 0514.13011 · doi:10.1090/S0002-9947-1983-0694386-5 [12] Huneke, C., Simis, A., Vasconcelos, W.V.: Reduced normal cones are domains. Contemp. Math.88, 95–101 (1989) · Zbl 0676.13011 [13] 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 [14] Simis, A., Ulrich, B., Vasconcelos, W.V.: Jacobian dual fibrations. Am. J. Math.115, 47–75 (1993) · Zbl 0791.13007 · doi:10.2307/2374722 [15] 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
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