Associated primes of local cohomology after adjoining indeterminates (Q2451744)

Let \(R\) be a commutative Noetherian ring with identity. Let \(I\) be an ideal of \(R\) and \(M\) a finitely generated \(R\)-module. In 1992 Huneke asked whether the local cohomology modules \(H_{I}^i(M)\) have finitely many associated primes. Eight years later, \textit{A. K. Singh} [Math. Res. Lett. 7, No. 2--3, 165--176 (2000; Zbl 0965.13013)] showed that \(H_{(x,y,z)T}^3(T)\) has infinitely many associated primes, where \(T\) is the hypersurface \(\displaystyle\frac{\mathbb{Z}[u,v,w,x,y,z]}{(ux+vy+wz)}\). \textit{G. Lyubeznik} [Invent. Math. 113, No. 1, 41--55 (1993; Zbl 0795.13004)] conjectured that if \(R\) is regular, then each local cohomology module \(H_I^i(R)\) has finitely many associated prime ideals. This paper studies the finitiness of associated primes of the local cohomology modules \(H_{I}^i(R),\) in the case \(R=A[t_1, \dots, t_n]\) or \(A[[t_1,\dots, t_n]],\) where \(A\) is a domain finitely generated as an algebra over a field of characteristic zero. Assume that \(A\) has a resolution of singularities, \(Y_0\), which is the blowup of \(A\) along an ideal of depth at least two and is covered by either two or three open affines with \(H^j(Y_0,\mathcal{O}_{Y_0})\) of finite length over \(A\) for \(j>0\). The author proves that \(\mathrm{Ass}_R(H^i_I(R))\) is finite for every \(i\). The assumptions on \(A\) fulfilled by any two or three dimensional normal domain with an isolated singularity which is finitely generated over a field of characteristic zero.