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Cohen-Macaulay Rees algebras and degrees of polynomial relations. (English) Zbl 0819.13003
The Rees algebra of an ideal $$I$$ of a Noetherian ring $$R$$ is the subalgebra of the ring of polynomials $$R[t]$$, $$R[It] = \sum_{i \geq 0} I^ it^ i$$. It realizes algebraically the notion of blowing up a variety along a subvariety, so that it is of interest to uncover its arithmetical properties, such as normality or Cohen-Macaulayness. Here we are concerned with how the latter occurs in two distinct ways.
In a manner of speaking, $$R[It]$$ is made up of two parts, $$R$$ itself and the associated graded ring $$\text{gr}_ I (R) = \sum_{i \geq 0} I^ i/I^{i+1}$$. One basic set of issues is how the Cohen-Macaulayness of $$R[It]$$ depends on that of $$R$$ and $$\text{gr}_ I(R)$$. On account of an argument of Huneke, the issue can be narrowed down to expressing the Cohen-Macaulayness of $$R[It]$$ vis-a-vis $$G = \text{gr}_ I (R)$$ and $$a(G)$$, the so-called $$a$$-invariant of $$G$$. After clarifying several cases where $$G$$ is Gorenstein, we give our first main result, a calculation of $$a(G)$$, in terms of the analytic spread $$\ell (I)$$ and reduction number $$r(I)$$ of $$I$$, under the hypothesis that $$I$$ satisfies a mild condition on the local number of generators in codimension below $$\ell (I)$$.
Our second main result bypasses the use of $$\text{gr}_ I (R)$$, focusing instead on the degrees of the generators of modules of syzygies and the polynomial relations amongst the elements of a generating set of the ideal. We explore how the Cohen-Macaulayness of $$R[It]$$ is described in terms of these sets of degrees and other integers that arise from a specification of the ideal through a presentation matrix. Some of the results are explicit formulas for the degrees of polynomial relations in terms of homological properties of the ideal, therefore linking aspects of elimination theory with homology.