Cohomological Dimension Research Articles

Injective optimal realizations of finite metric spaces

A realization of a finite metric space (X,d) is a weighted graph (G,w) whose vertex set contains X such that the distances between the elements of X in G correspond to those given by d. Such a realization is called optimal if it has minimal total edge weight. Optimal realizations have applications in fields such as phylogenetics, psychology, compression software and internet tomography. Given an optimal realization (G,w) of (X,d), there always exist certain “proper” maps from the vertex set of G into the so-called tight span of d. In [A. Dress, Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces, Adv. Math. 53 (1984) 321–402], Dress conjectured that any such map must be injective. Although this conjecture was recently disproven, in this paper we show that it is possible to characterize those optimal realizations (G,w) for which certain generalizations of proper maps–that map the geometric realization of (G,w) into the tight span instead of its vertex set–must always be injective. We also prove that these “injective” optimal realizations always exist, and show how they may be constructed from non-injective ones. Ultimately it is hoped that these results will contribute towards developing new ways to compute optimal realizations from tight spans.

Artinianness of Certain Graded Local Cohomology Modules

AbstractWe show that ifR= ⴲn∈ℕ0Rnis a Noetherian homogeneous ring with local base ring (R0, m0), irrelevant idealR+, andMa finitely generated gradedR-module, thenis Artinian forj= 0, 1 wheret= inf﹛i∈ ℕ0:is not finitely generated﹜. Also, we prove that if cd(R+,M) = 2, then for eachi∈ ℕ0,) is Artinian if and only ifis Artinian, where cd(R+,M) is the cohomological dimension ofMwith respect toR+. This improves some results of R. Sazeedeh.

Local cohomology properties of direct summands

In this article, we prove that if R→S is a homomorphism of Noetherian rings that splits, then for every i≥0 and ideal I⊂R, AssRHIi(R) is finite when AssSHISi(S) is finite. In addition, if S is a Cohen–Macaulay ring that is finitely generated as an R-module, such that all the Bass numbers of HISi(S), as an S-module, are finite, then all the Bass numbers of HIi(R), as an R-module, are finite. Moreover, we show these results for a larger class a functors introduced by Lyubeznik [5]. As a consequence, we exhibit a Gorenstein F-regular UFD of positive characteristic that is not a direct summand, not even a pure subring, of any regular ring.

K(π, 1) and word problems for infinite type Artin–Tits groups, and applications to virtual braid groups

Let Γ be a Coxeter graph, let (W, S) be its associated Coxeter system, and let (A, Σ) be its associated Artin–Tits system. We regard W as a reflection group acting on a real vector space V. Let I be the Tits cone, and let E Γ be the complement in I + iV of the reflecting hyperplanes. Recall that Salvetti, Charney and Davis have constructed a simplicial complex Ω(Γ) having the same homotopy type as E Γ. We observe that, if $${T \subset S}$$ , then Ω(Γ T ) naturally embeds into Ω (Γ). We prove that this embedding admits a retraction $${\pi_T: \Omega(\Gamma) \to \Omega (\Gamma_T)}$$ , and we deduce several topological and combinatorial results on parabolic subgroups of A. From a family $${\mathcal{S}}$$ of subsets of S having certain properties, we construct a cube complex Φ, we show that Φ has the same homotopy type as the universal cover of E Γ, and we prove that Φ is CAT(0) if and only if $${\mathcal{S}}$$ is a flag complex. We say that $${X \subset S}$$ is free of infinity if Γ X has no edge labeled by ∞. We show that, if $${E_{\Gamma_X}}$$ is aspherical and A X has a solution to the word problem for all $${X \subset S}$$ free of infinity, then E Γ is aspherical and A has a solution to the word problem. We apply these results to the virtual braid group VB n . In particular, we give a solution to the word problem in VB n , and we prove that the virtual cohomological dimension of VB n is n−1.

On the Iwasawa algebra for pro-l Galois groups

Let l be an odd prime and K/k a Galois extension of totally real number fields with Galois group G such that K/k∞ and \({k/\mathbb{Q}}\) are finite. We give a full description of the algebraic structure of the semisimple algebra \({\mathcal{Q} G = {\rm Quot} (\Lambda G)}\) for pro-l Galois groups G with \(\Lambda G = {\mathbb{Z}_l}\)[[G]] the Iwasawa algebra. We moreover compute the cohomological dimension of the centres of the Wedderburn components of \({\mathcal{Q} G}\) and state some results on the completion of \({\mathcal{Q} G}\) .

Annihilators of Top Local Cohomology

In many important theorems in the homological theory of commutative local rings, an essential ingredient in the proof is to consider We examine these annihilators for an arbitrary local ring when i is the cohomological dimension of I. When the cohomological dimension coincides with the dimension of the ring, we provide an explicit description of In general, we examine its dimension over R and conjecture that for i > 0.

ON THE COHOMOLOGY OF ALMOST-COMPLEX MANIFOLDS

Following [T.-J. Li and W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.17(4) (2009) 651–683], we continue to study the link between the cohomology of an almost-complex manifold and its almost-complex structure. In particular, we apply the same argument in [T.-J. Li and W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.17(4) (2009) 651–683] and the results obtained by [D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math.36(1) (1976) 225–255] to study the cone of semi-Kähler structures on a compact semi-Kähler manifold.

Sheaves of slice regular functions

AbstractSlice regular functions have been introduced in 20 as solutions of a special partial differential operator with variable coefficients. As such they do not naturally form a sheaf. In this paper we use a modified definition of slice regularity, see 21, to introduce the sheaf of slice regular functions with values in in the algebra of quaternions and, more in general, in a Clifford algebra and we study its cohomological properties. We show that the first cohomology group with coefficients in the sheaf of slice regular functions vanishes for any open set in the space of quaternions (resp. the space of paravectors in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^{n+1}$\end{document}). However, we prove that not all the open sets are domains of slice regularity but only those special sets which are axially symmetric, i.e., invariant with respect to rotations that fix the real axis.

The Yagita Invariant of Some Symplectic Groups

If a group G of finite virtual cohomological dimension has p-periodic Farrell cohomology, then the Yagita invariant of this group equals its p-period. The group of symplectic 2(p + 1) × 2(p + 1) matrices over \({\mathbb{Z}}\) has elementary abelian p-subgroups of rank at least 2 and hence \({{\rm Sp}(2(p+1), \mathbb{Z})}\) and \({{\rm Sp}(2(p+1), \mathbb{Q})}\) do not have p-periodic Farrell cohomology. We compute the Yagita invariants of these groups for any odd prime p.

Classification of Rational Homotopy Type for 8-Cohomological Dimension Elliptic Spaces

The different methods used to classify rational homotopy types of manifolds are in general fascinating and various (see [1,7,8]). In this paper we are interested to a particular case, that of simply connected elliptic spaces, denoted X, by discussing its cohomological dimension. Here we will the discuss the case when dimH*( Χ ;Q)=8 and χ(Χ)=0.

Results of generalized local cohomology modules of a-minimax modules

Let R be a commutative Noetherian ring, M a finitely generated R-module, and N a minimax R-module. It is shown that if a is an ideal of R, such that cd(a)=1, where cd is the cohomological dimension of a in R, then H_a^j(M, N) and Ext_R^i(M,H_a^j(N)) are a-cominimax for all i,j. Furthermore, if t is a non-negative integer such that H_a^j(M, N) is a-minimax for all j

On the equality between homological and cohomological dimension of groups

On the equality between homological and cohomological dimension of groups

Special Lifts of Ordinary K3 Surfaces and Applications

We study the cohomological properties of quasi-canonical lifts of an ordinary K3 surface over a nite eld. As applications, we obtain a Torelli type theorem for

The Rational Cohomology of the Mapping Class Group Vanishes in its Virtual Cohomological Dimension

Let Mod_g be the mapping class group of a genus g >= 2 surface. The group Mod_g has virtual cohomological dimension 4g-5. In this note we use a theorem of Broaddus and the combinatorics of chord diagrams to prove that H^{4g-5}(Mod_g; Q) = 0.

Co-Cohen-Macaulay Modules and Generalized Local Cohomology

Let (R,𝔪) be a Noetherian local ring, 𝔞 a proper ideal of R, and M, N two finitely generated R-modules of finite projective dimension m and of finite dimension n, respectively. It is shown that if n ≤ 2, then the generalized local cohomology module [Formula: see text] is a co-Cohen-Macaulay module. Additionally, we show that [Formula: see text] for all i > m+s and [Formula: see text], where s is the cohomological dimension of N with respect to 𝔞.

Artinian and Non-Artinian Local Cohomology Modules

AbstractLet M be a finite module over a commutative noetherian ring R. For ideals a and b of R, the relations between cohomological dimensions of M with respect to a, b, a ⋂ b and a + b are studied. When R is local, it is shown that M is generalized Cohen–Macaulay if there exists an ideal a such that all local cohomology modules of M with respect to a have finite lengths. Also, when r is an integer such that 0 ≤ r < dimR(M), any maximal element q of the non-empty set of ideals ﹛a : (M) is not artinian for some i, i ≥ r} is a prime ideal, and all Bass numbers of (M) are finite for all i ≥ r.

ON THE LOCAL COHOMOLOGY OF MINIMAX MODULES

Let R be a commutative Noetherian ring, a an ideal of R, and M a minimax R-module. We prove that the local cohomology modules <TEX>$H^j_a(M)$</TEX> are a-cominimax; that is, <TEX>$Ext^i_R$</TEX>(R/a, <TEX>$H^j_a(M)$</TEX>) is minimax for all i and j in the following cases: (a) dim R/a = 1; (b) cd(a) = 1, where cd is the cohomological dimension of a in R; (c) dim <TEX>$R{\leq}2$</TEX>. In these cases we also prove that the Bass numbers and the Betti numbers of <TEX>$H^j_a(M)$</TEX> are finite.

On the Integral Cohomology of Bianchi Groups

Extensive and systematic machine computations are carried out to investigate the integral cohomology of the Euclidean Bianchi groups and their congruence subgroups. The collected data give insight into several aspects, including the asymptotic behavior of the torsion in the first homology. Along with the experimental work, some basic properties of the integral cohomology are recorded with an eye toward the liftability issue of Hecke eigenvalue systems.

Delta-discreteG–spectra and iterated homotopy fixed points

Let G be a profinite group with finite virtual cohomological dimension and let X be a discrete G-spectrum. If H and K are closed subgroups of G, with H normal in K, then, in general, the K/H-spectrum X^{hH} is not known to be a continuous K/H-spectrum, so that it is not known (in general) how to define the iterated homotopy fixed point spectrum (X^{hH})^{hK/H}. To address this situation, we define homotopy fixed points for delta-discrete G-spectra and show that the setting of delta-discrete G-spectra gives a good framework within which to work. In particular, we show that by using delta-discrete K/H-spectra, there is always an iterated homotopy fixed point spectrum, denoted (X^{hH})^{h_\delta K/H}, and it is just X^{hK}. Additionally, we show that for any delta-discrete G-spectrum Y, (Y^{h_\delta H})^{h_\delta K/H} \simeq Y^{h_\delta K}. Furthermore, if G is an arbitrary profinite group, there is a delta-discrete G-spectrum {X_\delta} that is equivalent to X and, though X^{hH} is not even known in general to have a K/H-action, there is always an equivalence ((X_\delta)^{h_\delta H})^{h_\delta K/H} \simeq (X_\delta)^{h_\delta K}. Therefore, delta-discrete L-spectra, by letting L equal H, K, and K/H, give a way of resolving undesired deficiencies in our understanding of homotopy fixed points for discrete G-spectra.

Affine cellular algebras

Graham and Lehrer have defined cellular algebras and developed a theory that allows in particular to classify simple representations of finite dimensional cellular algebras. Many classes of finite dimensional algebras, including various Hecke algebras and diagram algebras, have been shown to be cellular, and the theory due to Graham and Lehrer successfully has been applied to these algebras. We will extend the framework of cellular algebras to algebras that need not be finite dimensional over a field. Affine Hecke algebras of type A and infinite dimensional diagram algebras like the affine Temperley–Lieb algebras are shown to be examples of our definition. The isomorphism classes of simple representations of affine cellular algebras are shown to be parameterised by the complement of finitely many subvarieties in a finite disjoint union of affine varieties. In this way, representation theory of non-commutative algebras is linked with commutative algebra. Moreover, conditions on the cell chain are identified that force the algebra to have finite global cohomological dimension and its derived category to admit a stratification; these conditions are shown to be satisfied for the affine Hecke algebra of type A if the quantum parameter is not a root of the Poincaré polynomial.