Embeddings of certain graded rings into their canonical modules
Let \(A\) be a Cohen-Macaulay local ring and \(\mathcal{F}=\{F_i\}_{i\in\mathbb {Z}}\) a filtration of ideals in \(A\). For each filtration \(\mathcal{F}\) one denotes by \(R=\mathcal{R}(\mathcal{F})\), respectively \(G=\mathcal{G}(\mathcal{F})\), the Rees algebra and the associated graded ring of \(\mathcal{F}\). Suppose in addition that \(F_1\neq A\) and \(\mathcal{R}(\mathcal{F})\) is a Noetherian ring. The authors prove the following criterion: Assume that \(A\) is a Gorenstein local ring and \(G\) is a Cohen-Macaulay ring. Let \(\mathcal{A}=\{p\in V(F_1): \text{ht}_Ap=\dim G_p/pG_p\}\). Then \(\mathcal{A}=\text{Ass}_AG\) and the following conditions are equivalent: \((1)\) \(G\) is a Gorenstein ring; \((2)\) there is a monomorphism of graded modules from \(G(a)\) to \(K_G\), where \(a=\text{a}(G)\) denote the a-invariant of \(G\) and \(K_G\) the canonical module of \(G\); \((3)\) \(G_p\) is a Gorenstein ring with \(a=\text{a}(G_p)\) for all \(p\in\mathcal{A}\). Thanks to this criterion one gets the following theorem: Let \(I\neq A\) be an ideal in \(A\) and let \(k\geq 1\) be an integer with \(I^k\neq (0)\). Suppose that \(G=\mathcal{G}(I)\) is a Cohen-Macaulay ring. Then the following conditions are equivalent: \((1)\) \(G\) is a Gorenstein ring and \(a\equiv -1\bmod k\); \((2)\) \(\mathcal{G}(I^k)\) is a Gorenstein ring and \((0):I^i\subseteq I^{a-i+1}\) for all \(1\leq i\leq a\). When this is the case, we have \(\text{a}(\mathcal{G}(I^k))=(a+1)/k-1\). Suppose that the residue field of \(A\) is infinite, and let \(J\) be a minimal reduction of an ideal \(I\neq A\). Then the authors get the following theorem: Assume that \(A\) is a Gorenstein local ring and let \(I\) be an equimultiple ideal in \(A\) with \(\text{ht}_AI\geq 1\). Then the following two conditions are equivalent: \((1)\) \(\mathcal{G}(I)\) is a Gorenstein ring; \((2)\) \(\mathcal{G}(I)\) is a Cohen-Macaulay ring and \(I^r=J^r:I^r\), where \(r\) is the reduction number of \(I\) with respect to \(J\).
scientific article; zbMATH DE number 1475253
scientific article