The core of an ideal in Cohen-Macaulay rings. (English) Zbl 1412.13007

Cumming, Christine

J. Commut. Algebra 10, No. 2, 163-170 (2018).

Let \((R,\mathfrak{m},k)\) be a local Cohen-Macaulay ring with infinite residue field and canonical module \(\omega\). If \(I\) is an ideal of \(R\) with \(\text{grade} I = g > 0\) and analytic spread \(\ell\), and \(J\) is a minimal reduction of \(I\) with reduction number \(r\), the author shows that under certain conditions satisfied by the ideal \(I\), the core of \(I\), that is, the intersection of all the minimal reductions of \(I\), is equal to \((J^{n+1}:I^{n})\) for every non-negative \(n\geq r-\ell +g\).
The proof begins by passing to the trivial extension \(S=R \ltimes \omega\), which is a Gorenstein ring, and showing that, under the conditions satisfied by the ideal \(I\), \(\text{core}(I)S=\text{core}(IS)\). Then one can apply a result of C. Polini and B. Ulrich [Math. Ann. 331, No. 3, 487–503 (2005; Zbl 1089.13005)] that gives a formula for the core of \(I\) when the ring is Gorenstein. The main result of the paper is then obtained by contracting back to \(R\).

Reviewer: Catalin Ciuperca (Fargo)

MSC:

13A30

Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics

Keywords:

core; canonical modules; trivial extension

Citations:

Zbl 1089.13005

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