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Gröbner basis and depth of Rees algebras. (English) Zbl 1033.13003
From the introduction: Let $$B=K[X_1, \dots, X_n]$$ be a polynomial ring over a field $$K$$ and $$A=B/J$$ a quotient ring of $$B$$ by a homogeneous ideal $$J$$. Let $$m$$ denote the maximal graded ideal of $$A$$. Then the Rees algebra $$R=A[mt]$$ may be considered a standard graded $$K$$-algebra and has a presentation $$B[Y_1, \dots, Y_n]/I_J.$$
The purpose of a paper by J. Herzog, D. Popescu and N. V. Trung [ see J. Lond. Math. Soc., II. Ser. 65, No. 2, 320-338 (2002; Zbl 1094.13503)] is to compare the homological properties of $$A$$ and $$R$$. In particular, the Castelnuovo-Mumford regularity of $$R$$, $$\text{reg }R$$, is $$\leq \text{reg }A+1$$ [see also V. Ene, Math. Rep., Bucur. 3(53), No. 2, 163-168 (2001; Zbl 1023.13004)]. Unfortunately, $$\text{depth }R$$ could be $$>\text{depth } A+1$$, as shows an example of S. Goto [J. Algebra 85, 490-534 (1983; Zbl 0529.13010)], but if $$A$$ is a polynomial algebra in one variable over a standard graded $$K$$-algebra then it holds $$\text{depth }R\leq \text{depth }A+1$$ (see the first cited paper above).
The proof of this paper uses a description of the local cohomology of $$R$$ in terms of the local cohomology of $$A$$. Our section 2 contains a direct proof of the above inequality, which does not use the local cohomology.
##### MSC:
 13C15 Dimension theory, depth, related commutative rings (catenary, etc.) 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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