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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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