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

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 $$R$$.

### MSC:

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

### Keywords:

core; canonical modules; trivial extension

Zbl 1089.13005
Full Text:

### References:

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