The finiteness dimension of local cohomology modules and its dual notion

Let \fa be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. We explore the behavior of the two notions f_{\fa}(M), the finiteness dimension of M with respect to \fa, and, its dual notion q_{\fa}(M), the Artinianess dimension of M with respect to \fa. When (R,\fm) is local and r:=f_{\fa}(M) is less than f_{\fa}^{\fm}(M), the \fm-finiteness dimension of M relative to \fa, we prove that H^r_{\fa}(M) is not Artinian, and so the filter depth of \fa on M doesn't exceeds f_{\fa}(M). Also, we show that if M has finite dimension and H^i_{\fa}(M) is Artinian for all i>t, where t is a given positive integer, then H^t_{\fa}(M)/\fa H^t_{\fa}(M) is Artinian. It immediately implies that if q:=q_{\fa}(M)>0, then H^q_{\fa}(M) is not finitely generated, and so f_{\fa}(M)\leq q_{\fa}(M).Comment: 14 pages, to appear in Journal of Pure and Applied Algebr

Cohomological Finiteness Conditions in Bredon Cohomology

We show that any soluble group $G$ of type Bredon-\FP_{\infty} with respect to the family of all virtually cyclic subgroups such that centralizers of infinite order elements are of type \FP_{\infty} must be virtually cyclic. To prove this, we first reduce the problem to the case of polycyclic groups and then we show that a polycyclic-by-finite group with finitely many conjugacy classes of maximal virtually cyclic subgroups is virtually cyclic. Finally we discuss refinements of this result: we only impose the property Bredon-\FP_n for some $n \leq 3$ and restrict to abelian-by-nilpotent, abelian-by-polycyclic or (nilpotent of class 2)-by-abelian groups.Comment: Corrected a mistake in Lemma 2.4 of the previous version, which had an effect on the results in Section 5 (the condition that all centralisers of infinite order elements are of type $FP_\infty$ was added

Invariant expectations and vanishing of bounded cohomology for exact groups

We study exactness of groups and establish a characterization of exact groups in terms of the existence of a continuous linear operator, called an invariant expectation, whose properties make it a weak counterpart of an invariant mean on a group. We apply this operator to show that exactness of a finitely generated group $G$ implies the vanishing of the bounded cohomology of $G$ with coefficients in a new class of modules, which are defined using the Hopf algebra structure of $\ell_1(G)$.Comment: Final version, to appear in the Journal of Topology and Analysi

Few smooth d-polytopes with n lattice points

We prove that, for fixed n there exist only finitely many embeddings of Q-factorial toric varieties X into P^n that are induced by a complete linear system. The proof is based on a combinatorial result that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with at most 12 lattice points. In fact, it is sufficient to bound the singularities and the number of lattice points on edges to prove finiteness.Comment: 20+2 pages; major revision: new author, new structure, new result