Associated primes of local cohomology of flat extensions with regular fibers and \(\varSigma\)-finite \(D\)-modules (Q397890)

Albeit it is well known that local cohomology modules are, in general, far away from being finitely generated, it is also true that, under certain assumptions, they behave in several aspects like finitely generated ones. The following conjecture, raised by \textit{G. Lyubeznik} [Invent. Math. 113, No. 1, 41--55 (1993; Zbl 0795.13004)], illustrates this fact.NEWLINENEWLINEConjecture (Lyubeznik). For any commutative Noetherian regular ring \(R\) and for any ideal \(I\) of \(R\), the set of associated primes of \(H_I^j (R)\) is finite.NEWLINENEWLINEThis conjecture is known to be true when \(R\) is of prime characteristic (by results of \textit{C. L. Huneke} and \textit{R. Y. Sharp} [Trans. Am. Math. Soc. 339, No. 2, 765--779 (1993; Zbl 0785.13005)]), when \(R\) is either a regular local ring of characteristic zero, or a finitely generated regular algebra over a field of characteristic zero (by results of Lyubeznik), when \(R\) is a regular unramified local ring of mixed characteristic (by results of Lyubeznik), and when \(R\) is a smooth \(\mathbb{Z}\)-algebra (by results of Bhatt, Blickle, Lyubeznik, Singh and Zhang [\textit{B. Bhatt} et al., Invent. Math. 197, No. 3, 509--519 (2014; Zbl 1318.13025)]). It is also worth mentioning that, by counterexamples provided by Katzman, Singh and Swanson, the above conjecture is, in general, false dropping the assumption of regularity.NEWLINENEWLINEHereafter, we suppose that \((R,\mathfrak{m},K)\) is a commutative local Noetherian ring and that \(S\) is a flat extension of \(R\) with regular closed fiber. Hochster asked whether, for any ideal \(I\) of \(S\), the set of associated primes of \(H_{\mathfrak{m}S}^0 (H_I^i (S))\) is finite; he also raised a weaker statement, replacing \(S\) by either \(R[x_1,\ldots ,x_n]\) or \(R[\![x_1,\ldots ,x_n]\!]\).NEWLINENEWLINEIn the paper under review, the author proves the below statements.NEWLINENEWLINE1. If \(S\) is either \(R[x_1,\ldots ,x_n]\) or \(R[\![x_1,\ldots ,x_n]\!]\), and \(I\subseteq S\) is an ideal such that \(\dim\left(R/I\cap R\right)\leq 1\), then the set of associated primes of \(H_{\mathfrak{m}S}^0 (H_I^i (S))\) is finite (see Theorem 1.3).NEWLINENEWLINE2. If \(R\) contains a field, and \(R\subseteq S\) is a flat extension of local rings with regular closed fiber such that the induced extension on completions \(\widehat{R}\subseteq\widehat{S}\) maps a coefficient field of \(R\) into a coefficient field of \(S\), and that \(I\subseteq S\) is an ideal such that \(\dim\left(R/I\cap R\right)\leq 1\), then the set of associated primes of \(H_{\mathfrak{m}S}^0 (H_I^i (S))\) is finite (see Theorem 1.4).NEWLINENEWLINEBy the way, the author introduces the class of \(\Sigma\)-finite \(D\)-modules (see Definition 3.3), a subclass of \(D\)-modules which are interesting in its own right because, although they are, in general, not finitely generated as \(D\)-modules, they have a finite set of associated primes.