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The goal of this paper is twofold: on the one hand, motivated by questions raised by Schenzel, we explore situations where the Hartshorne–Lichtenbaum Vanishing theorem for local cohomology fails, leading us to simpler expressions of certain local cohomology modules. As application, we give new expressions of the endomorphism ring of these modules. On the other hand, building upon previous work by Àlvarez Montaner, we exhibit the shape of Lyubeznik tables of the so-called partially sequentially Cohen–Macaulay rings as introduced by Sbarra and Strazzanti. [ABSTRACT FROM AUTHOR]

We study when an abstract finitely presented group of cohomological dimension has pro- completion of cohomological dimension . Furthermore, we prove that for a tree hyperbolic limit group we have and show an example of a hyperbolic limit group that is not free and is free pro-. For a finitely generated residually free group that is not a limit group, we show that is not free pro-. [ABSTRACT FROM AUTHOR]

For each d ∈ ℕ, we construct a 3-generated group Hd, which is a subdirect product of free groups, such that the cohomological dimension of Hd is d. Given a group F and a normal subgroup N ⊳ F we prove that any right angled Artin group containing the special HNN-extension of F with respect to N must also contain F/N. We apply this to construct, for every d ∈ ℕ, a 4-generated group Gd, embeddable into a right angled Artin group, such that the cohomological dimension of Gd is 2 but the cohomological dimension of any right angled Artin group, containing Gd, is at least d. These examples are used to show the non-existence of certain "universal" right angled Artin groups. We also investigate finitely presented subgroups of direct products of limit groups. In particular, we show that for every n ∈ ℕ there exists δ(n) ∈ ℕ such that any n-generated finitely presented subgroup of a direct product of finitely many free groups embeds into the δ(n)-th direct power of the free group of rank 2. As another corollary we derive that any n-generated finitely presented residually free group embeds into the direct product of at most δ(n) limit groups. [ABSTRACT FROM AUTHOR]