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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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