On Hartshorne’s problem and Huneke’s conjecture for generalized local cohomology modules

Abstract

In this work we present new positive cases of the natural generalized versions of both Hartshorne’s problem and Huneke’s conjecture about cofiniteness of local cohomology modules and finiteness of their sets of associated primes. With respect to the classical dimensional approach, we extend several results and also furnish new special cases, which include a new contribution in dimension six. Other positive answers are developed in a complementary approach based on other numerical invariants such as cohomological dimension and Gorenstein projective dimension. As a byproduct, we derive a new criterion for weak cofiniteness of ordinary local cohomology.