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Regularity of Rees algebras. (English) Zbl 1094.13503

Summary: Let \(B = k[x_1, \ldots, x_n]\) be a polynomial ring over a field \(k\) , and let \(A\) be a quotient ring of \(B\) by a homogeneous ideal \(J\) . Let \(\mathfrak{m}\) denote the maximal graded ideal of \(A\) . Then the Rees algebra \(R = A[{\mathfrak{m}} t]\) also has a presentation as a quotient ring of the polynomial ring \(k[x_1, \ldots, x_n, y_1, \ldots, y_n]\) by a homogeneous ideal \(J^*\). For instance, if \(A = k[x_1, \ldots, x_n]\), then \[ R \cong k[x_1, \ldots, x_n, y_1, \ldots, y_n]/(x_i y_j - x_j y_i\mid i,j = 1, \ldots, n). \] In this paper we want to compare the homological properties of the homogeneous ideals \(J\) and \(J^*\) .

MSC:

13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13D45 Local cohomology and commutative rings
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