Cohomological Finiteness Research Articles

Finiteness of Cohomology for Pro-locally Proper Maps

Finiteness of Cohomology for Pro-locally Proper Maps

On the Σ$\Sigma$‐invariants of Artin groups satisfying the K(π,1)$K(\pi,1)$‐conjecture

AbstractWe consider ‐invariants of Artin groups that satisfy the ‐conjecture. These invariants determine the cohomological finiteness conditions of subgroups that contain the derived subgroup. We extend a known result for even Artin groups of FC‐type, giving a sufficient condition for a character to belong to . We also prove some partial converses. As applications, we prove that the ‐conjecture holds true when there is a prime that divides for any edge with even label , we generalize to Artin groups the homological version of the Bestvina–Brady theorem and we compute the ‐invariants of all irreducible spherical and affine Artin groups and triangle Artin groups, which provide a complete classification of the and properties of their derived subgroup.

Cohomology of group theoretic Dehn fillings II

We study the cohomology of group theoretic Dehn fillings. Applying the Cohen-Lyndon property for sufficiently deep Dehn fillings of hyperbolically embedded subgroups H↪hG, obtained by the second named author in [67], we derive a spectral sequence that computes the cohomology of the corresponding Dehn filling quotients G‾. As an application, we establish an isomorphism between the relative cohomology of the group pair (G,H) and its sufficiently deep Dehn filling quotient pair (G‾,H‾). This allows us to generalize the results of Fujiwara and Manning on simplicial volume of Dehn fillings of hyperbolic manifolds to Dehn fillings of Poincaré duality pairs.We also strengthen the results of Olshanskii [58], Dahmani-Guirardel-Osin [27] and Hull [42] on SQ-universality and common quotients of acylindrically hyperbolic groups by adding cohomological finiteness conditions. We apply these results to obtain hyperbolic and acylindrically hyperbolic quotients with special properties.

Cohomology of finite tensor categories: Duality and Drinfeld centers

We consider the finite generation property for cohomology of a finite tensor category C \mathscr {C} , which requires that the self-extension algebra of the unit \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},\mathbf {1}) is a finitely generated algebra and that, for each object V V in C \mathscr {C} , the graded extension group \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},V) is a finitely generated module over the aforementioned algebra. We prove that this cohomological finiteness property is preserved under duality (with respect to exact module categories) and taking the Drinfeld center, under suitable restrictions on C \mathscr {C} . For example, the stated result holds when C \mathscr {C} is a braided tensor category of odd Frobenius-Perron dimension. By applying our general results, we obtain a number of new examples of finite tensor categories with finitely generated cohomology. In characteristic 0 0 , we show that dynamical quantum groups at roots of unity have finitely generated cohomology. We also provide a new class of examples in finite characteristic which are constructed via infinitesimal group schemes.

Discriminantal bundles, arrangement groups, and subdirect products of free groups

We construct bundles $$E_k({{\mathscr {A}}},{{\mathscr {F}}}) \rightarrow M$$ over the complement M of a complex hyperplane arrangement $${{\mathscr {A}}}$$ , depending on an integer $$k \geqslant 1$$ and a set $${{\mathscr {F}}}=\{f_1, \ldots , f_\mu \}$$ of continuous functions $$f_i :M \rightarrow {{\mathbb {C}}}$$ whose differences are nonzero on M, generalizing the configuration space bundles arising in the Lawrence–Krammer–Bigelow representation of the pure braid group. We display such families $${{\mathscr {F}}}$$ for rank two arrangements, reflection arrangements of types $$A_\ell $$ , $$B_\ell $$ , $$D_\ell $$ , $$F_4$$ , and for arrangements supporting multinet structures with three classes, with the resulting bundles having nontrivial monodromy around each hyperplane. The construction extends to arbitrary arrangements by pulling back these bundles along products of inclusions arising from subarrangements of these types. We then consider the faithfulness of the resulting representations of the arrangement group $$\pi _1(M)$$ . We describe the kernel of the product $$\rho _{{\mathscr {X}}}:G \rightarrow \prod _{S \in {{\mathscr {X}}}} G_S$$ of homomorphisms of a finitely-generated group G onto quotient groups $$G_S$$ determined by a family $${{\mathscr {X}}}$$ of subsets of a fixed set of generators of G, extending a result of Theodore Stanford about Brunnian braids. When the projections $$G \rightarrow G_S$$ split in a compatible way, we show the image of $$\rho _{{\mathscr {X}}}$$ is normal with free abelian quotient, and identify the cohomological finiteness type of G. These results apply to some well-studied arrangements, implying several qualitative and residual properties of $$\pi _1(M)$$ , including an alternate proof of a result of Artal, Cogolludo, and Matei on arrangement groups and Bestvina–Brady groups, and a dichotomy for a decomposable arrangement $${{\mathscr {A}}}$$ : either $$\pi _1(M)$$ has a conjugation-free presentation or it is not residually nilpotent.

DUALITY FOR COHOMOLOGY OF CURVES WITH COEFFICIENTS IN ABELIAN VARIETIES

In this paper, we formulate and prove a duality for cohomology of curves over perfect fields of positive characteristic with coefficients in Néron models of abelian varieties. This is a global function field version of the author’s previous work on local duality and Grothendieck’s duality conjecture. It generalizes the perfectness of the Cassels–Tate pairing in the finite base field case. The proof uses the local duality mentioned above, Artin–Milne’s global finite flat duality, the nondegeneracy of the height pairing and finiteness of crystalline cohomology. All these ingredients are organized under the formalism of the rational étale site developed earlier.

A generalization of Serre’s condition $$\mathrm {(F)}$$ ( F ) with applications to the finiteness of unramified cohomology

In this paper, we introduce a condition ( $$\mathrm {F}_m'$$ ) on a field K, for a positive integer m, that generalizes Serre’s condition (F) and which still implies the finiteness of the Galois cohomology of finite Galois modules annihilated by m and algebraic K-tori that split over an extension of degree dividing m, as well as certain groups of etale and unramified cohomology. Various examples of fields satisfying ( $$\mathrm {F}_m'$$ ), including those that do not satisfy (F), are given.

Homological finiteness conditions for a class of metabelian groups

We consider a class of metabelian groups first studied by Baumslag and Stammbach and we show that these groups are consistent with the Bieri-Groves conjecture which relates cohomological finiteness conditions to the Bieri-Neumann-Strebel sigma invariant.

Some results on local cohomology of polynomial and formal power series rings: The one dimensional case

In this paper, we prove several results on the finiteness of local cohomology of polynomial and formal power series rings. In particular, we give a partial affirmative answer for a question of L. Núñez-Betancourt (2014) [14].

Cohomological finiteness conditions and centralisers in generalisations of Thompson’s group V

Abstract We consider generalisations of Thompson’s group V, denoted by V r ⁢ ( Σ ) ${V_{r}(\Sigma)}$ , which also include the groups of Higman, Stein and Brin. We show that, under some mild hypotheses, V r ⁢ ( Σ ) ${V_{r}(\Sigma)}$ is the full automorphism group of a Cantor algebra. Under some further minor restrictions, we prove that these groups are of type F ∞ ${\operatorname{F}_{\infty}}$ and that this implies that also centralisers of finite subgroups are of type F ∞ ${\operatorname{F}_{\infty}}$ .

Equivariant Euler–Poincaré characteristic in sheaf cohomology

Let X be a Hausdorff space equipped with a continuous action of a finite group G and a G-stable family of supports $${\Phi}$$ . Fix a number field F with ring of integers R. We study the class $${\chi = \sum_j (-1)^j [H^j_\Phi (X, \mathcal{E}) \otimes_R F]}$$ in the character group of G over F for any flat G-sheaf $${\mathcal{E}}$$ of R-modules over X. Under natural cohomological finiteness conditions we give a formula for $${\chi}$$ with respect to the basis given by the irreducible characters of G. We discuss applications of our result concerning the cohomology of arithmetic groups.

Finiteness of crystalline cohomology of higher level

We prove the finiteness of crystalline cohomology of higher level. An important ingredient is a “higher de Rham complex” that satisfies a kind of Poincaré lemma.

Cohomological finiteness conditions for Mackey and cohomological Mackey functors

We study cohomological finiteness conditions for groups associated to Mackey and cohomological Mackey functors, proving that the cohomological dimension associated to cohomological Mackey functors is always equal to the F-cohomological dimension, and characterising the conditions Mackey-FPn and cohomological Mackey-FPn.

Bredon cohomological finiteness conditions for generalisations of Thompson groups

We define a family of groups that generalises Thompson’s groups T and G , and also those of Higman, Stein and Brin. For groups in this family we describe centralisers of finite subgroups and show that for a given finite subgroup Q there are finitely many conjugacy classes of finite subgroups isomorphic to Q . We consider groups of type quasi- \underline{\rm F}_\infty . This is a property slightly weaker than possessing a finite type model for the classifying space for proper actions \underline{E}G . We give criteria for the T versions of our groups to be of type quasi- \underline{\rm F}_\infty . We also generalise some well-known properties of ordinary cohomology to Bredon cohomology.

Cohomological finiteness properties of the Brin–Thompson–Higman groups 2V and 3V

AbstractWe show that Brin's generalizations 2V and 3V of the Thompson–Higman group V are of type FP∞. Our methods also give a new proof that both groups are finitely presented.

Cohomological Hasse principle and motivic cohomology for arithmetic schemes

In 1985 Kazuya Kato formulated a fascinating framework of conjectures which generalizes the Hasse principle for the Brauer group of a global field to the so-called cohomological Hasse principle for an arithmetic scheme X. In this paper we prove the prime-to-characteristic part of the cohomological Hasse principle. We also explain its implications on finiteness of motivic cohomology and special values of zeta functions.

On Galois cohomology of unipotent algebraic groups over local fields

In this paper, we give a necessary and sufficient condition for the finiteness of Galois cohomology of unipotent groups over local fields of positive characteristic.

Cohomological finiteness conditions in Bredon cohomology

We show that soluble groups G of type Bredon-FP∞ with respect to the family of all virtually cyclic subgroups of G are always virtually cyclic. In such a group centralizers of elements are of type FP∞. We show that this implies that the group is polycyclic. Another important ingredient of the proof is 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-FPn for some n ⩽ 3 and restrict to abelian-by-nilpotent, abelian-by-polycyclic or (nilpotent of class 2)-by-abelian groups.

Cohomological finiteness conditions for elementary amenable groups

It is proved that every elementary amenable group of type FP? admits a cocompact classifying space for proper actions.

Sheaves on Artin stacks

We develop a theory of quasi–coherent and constructible sheaves on algebraic stacks correcting a mistake in the recent book of Laumon and Moret-Bailly. We study basic cohomological properties of such sheaves, and prove stack–theoretic versions of Grothendieck’s Fundamental Theorem for proper morphisms, Grothendieck’s Existence Theorem, Zariski’s Connectedness Theorem, as well as finiteness Theorems for proper pushforwards of coherent and constructible sheaves. We also explain how to define a derived pullback functor which enables one to carry through the construction of a cotangent complex for a morphism of algebraic stacks due to Laumon and Moret–Bailly. 1.1. In the book ([LM-B]) the lisse-etale topos of an algebraic stack was introduced, and a theory of quasi–coherent and constructible sheaves in this topology was developed. Unfortunately, it was since observed by Gabber and Behrend (independently) that the lisse-etale topos is not functorial as asserted in (loc. cit.), and hence the development of the theory of sheaves in this book is not satisfactory “as is”. In addition, since the publication of the book ([LM-B]), several new results have been obtained such as finiteness of coherent and etale cohomology ([Fa], [Ol]) and various other consequences of Chow’s Lemma ([Ol]). The purpose of this paper is to explain how one can modify the arguments of ([LM-B]) to obtain good theories of quasi–coherent and constructible sheaves on algebraic stacks, and in addition we provide an account of the theory of sheaves which also includes the more recent results mentioned above. 1.2. The paper is organized as follows. In section 2 we recall some aspects of the theory of cohomological descent ([SGA4], V) which will be used in what follows. In section 3 we review the basic definitions of the lisse-etale site, cartesian sheaves over a sheaf of algebras, and verify some basic properties of such sheaves. In section 4 we relate the derived category of cartesian sheaves over some sheaf of rings to various derived categories of sheaves on the simplicial space obtained from a covering of the algebraic stack by an algebraic space. Loosely speaking the main result states that the cohomology of a complex with cartesian cohomology sheaves can be computed by restricting to the simplicial space obtained from a covering and computing cohomology on this simplicial space using the etale topology. In section 5 we generalize these results to comparisons between Ext–groups computed in the lisse-etale topos and Ext–groups computed using the etale topology on a hypercovering. In section 6 we specialize the discussion of sections 3-5 to quasi–coherent sheaves. We show that if X is an algebraic stack and OXlis-et denotes the structure sheaf of the lisse-etale topos, then the triangulated category D qcoh(X ) of bounded below complexes of OXlis-et–modules with quasi–coherent cohomology sheaves satisfies all the basic properties that one would expect from the theory for schemes. For example we show in this section that if f : X → Y is a quasi–compact morphism of algebraic stacks and M is a quasi–coherent sheaf on X Date: November 2, 2005. 1