Lynch's conjecture and attached primes of the top local cohomology modules
Let \(R\) be a commutative noetherian ring, \(\mathfrak{a}\) an ideal of \(R\), and \(M\) an \(R\)-module. For any \(i \geq 0\), the \(i\)th local cohomology module of \(M\) with respect to \(\mathfrak{a}\) is given by \[H^{i}_{\mathfrak{a}}(M) \cong \underset{n\geq 1}\varinjlim \operatorname{Ext}^{i}_{R}\left(R/ \mathfrak{a}^{n},M\right).\] The cohomological dimension of \(M\) with respect to \(\mathfrak{a}\), denoted by \(\operatorname{cd}(\mathfrak{a},M)\), is the integer \(c\) such that \(H^{c}_{\mathfrak{a}}(M)\neq 0\) while \(H^{i}_{\mathfrak{a}}(M)= 0\) for every \(i > c\). The Strong Lynch's conjecture claims that \[\dim \left(R/ \operatorname{ann}_{R}\left(H^{\operatorname{cd}(\mathfrak{a},R)}_{\mathfrak{a}}(R)\right)\right)= \dim \left(R/ \Gamma_{\mathfrak{a}}(R)\right),\] and the weak Lynch's conjecture claims that \[\operatorname{ht}\left(\operatorname{ann}_{R}\left(H^{\operatorname{cd}(\mathfrak{a},R)}_{\mathfrak{a}}(R)\right)\right)= \operatorname{ht} \left(\Gamma_{\mathfrak{a}}(R)\right).\] When \(R\) is an integral domain, then both conjectures are equivalent to \(H^{\operatorname{cd}(\mathfrak{a},R)}_{\mathfrak{a}}(R)\) being faithful. On the other hand, it is known that both the weak and the strong forms of Lynch's conjecture are false in this generality. However, the conjectures are proved in some cases upon imposing various restrictions on the ring. The authors of this paper establish some new results on Lynch's conjectures and the set of the attached primes of the top local cohomology module \(H^{\operatorname{cd}(\mathfrak{a},M)}_{\mathfrak{a}}(M)\) in the case where \(R\) has prime characteristic and \(\operatorname{cd}(\mathfrak{a},M) = \operatorname{ara}_{M}(\mathfrak{a})\). In order to prove their results, they establish a new relation between the set \(\operatorname{Att}_{R}\left(H^{\operatorname{cd}(\mathfrak{a},M)}_{\mathfrak{a}}(M)\right)\) and the \(M\)-height of the annihilator of the top local cohomology module \(H^{\operatorname{cd}(\mathfrak{a},M)}_{\mathfrak{a}}(M)\).
scientific article; zbMATH DE number 7489069
scientific article