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On an endomorphism ring of local cohomology. (English) Zbl 1273.13027

Let \(I\) be an ideal of a \(d\)-dimensional complete local ring \((R,m)\). Assume that \(H^d_ I (R)\) is nonzero. The paper deals with the endomorphism ring of the top local cohomology module \(H^d_ I (R)\), by investigating the kernel of the natural map \(R\to \mathrm{End}_R(H^d_ I (R))\). Let us recall the undirected graph \(G(R)\) of \(R\). Vertices are prime ideals \(p\) such that \(\dim R = \dim R/p\), and two distinct vertices \(p, q\) are joined by an edge if \((p, q)\) is of height one. The paper connects the connectedness of \(G(R)\) with the localness of the ring \(\mathrm{End}_R(H^d_ I (R))\). The results inspired by [M. Hochster and C. Huneke, Contemp. Math. 159, 197–208 (1994; Zbl 0809.13003)].

MSC:

13D45 Local cohomology and commutative rings

Citations:

Zbl 0809.13003
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References:

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