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Abstract: We prove a generalization of the Edwards–Walsh Resolution Theorem: [Copyright &y& Elsevier]

Let I be an ideal of a commutative Noetherian ring R such that ara(I) = t ≥ 2. The purpose of this article is to show that there exists an I-filter regular sequence y1, ..., yt for R such that Rad(I) = Rad(y1, ..., yt) and cd((y1, ..., yi), R) = i for all 1 ≤ i < t. Also, it is shown that ara(I) ≤ dim R + 1, which is a generalization of a nice result of Kronecker [14]. In addition, some applications are included. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we will compare the connectivity dimension of an ideal I in a polynomial ring P with that of any initial ideal of I. Generalizing a theorem of Kalkbrener and Sturmfels [M. Kalkbrener, B. Sturmfels, Initial complex of prime ideals, Adv. Math. 116 (1995) 365–376], we prove that for each monomial order ≺. As a corollary we have that every initial complex of a Cohen–Macaulay ideal is strongly connected. Our approach is based on the study of the cohomological dimension of an ideal in a noetherian ring R and its relation with the connectivity dimension of . In particular we prove a generalized version of a theorem of Grothendieck [A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), in: Séminaire de Géométrie Algébrique du Bois Marie, 1962]. As consequence of these results we obtain some necessary conditions for an open subscheme of a projective scheme to be affine. [Copyright &y& Elsevier]

The first part of this paper is concerned with the Artinianness of certain local cohomology modules $H^i_{\frak a}(M)$ when M is a Matlis reflexive module over a commutative Noetherian complete local ring R and 픞 is an ideal of R. Also, we characterize the set of attached prime ideals of $H^n_{\frak a}(M)$, where n is the dimension of M. The second part is concerned with the vanishing of local cohomology and generalized local cohomology modules. In fact, when R is an arbitrary commutative Noetherian ring, M is an R-module and 픞 is an ideal of R, we obtain some lower and upper bounds for the cohomological dimension of M with respect to 픞. [ABSTRACT FROM AUTHOR]

We consider the Grothendieck topologies on low semi-lattices, defined by one family, and the corresponding sheaf cohomology. This is a basis to define and study the left and right cohomologies and the left and right dimensions of the Chu spaces. The construction of Chu spaces allows to characterize a large class of quantities, for example, the dimension of a Noether space or the Krull dimension of a ring, the Lebesgue-type dimensions, as well as to compare them with the cohomology dimensions of the corresponding Chu spaces. We prove existence of spectral sequences of the morphisms of the Chu spaces. [ABSTRACT FROM AUTHOR]

Let a be an ideal of a commutative Noetherian ring R with identity. We study finitely generated R -modules M whose a -finiteness and a -cohomological dimensions are equal. In particular, we examine relative analogues of quasi-Buchsbaum, Buchsbaum and surjective Buchsbaum modules. We reveal several interactions between these types of modules that extend some of the existing results in the classical theory to the relative one. [ABSTRACT FROM AUTHOR]

Abstract: We consider the homotopy types of -complexes X with fundamental group π such that and π has one end. Let and . Our main result is that (modulo two technical conditions on ) there are at most orbits of k-invariants determining “strongly minimal” complexes (i.e., those with homotopy intersection pairing trivial). The homotopy type of a -complex X with π a -group is determined by π, w, and the -type of X. Our result also implies that Fox''s 2-knot with metabelian group is determined up to homeomorphism by its group. [Copyright &y& Elsevier]

Abstract: Let R be a commutative Noetherian ring and let M be a non-zero finitely generated R-module. Let I be an ideal of R and t a non-negative integer such that for all . It is shown that the R-modules are I-cofinite and the R-module is finitely generated. This immediately implies that if I has dimension one (i.e., ), then is I-cofinite for all . This is a generalization of the main results of Delfino and Marley [D. Delfino, T. Marley, Cofinite modules and local cohomology, J. Pure Appl. Algebra 121 (1997) 45–52] and Yoshida [K.I. Yoshida, Cofiniteness of local cohomology modules for ideals of dimension one, Nagoya Math. J. 147 (1997) 179–191] for an arbitrary Noetherian ring R. Also, we prove that if R is local and for all , then the R-modules and are weakly Laskerian for all and all . As a consequence, it follows that the set of associated primes of is finite for all , whenever . [Copyright &y& Elsevier]

Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let a be an ideal of R and M denote the intersection of all prime ideals p ϵ SuppR Had (M). It is shown that (Multiple line equation(s) cannot be represented in ASCII text) where for an Artinian R-module A we put (Multiple line equation(s) cannot be represented in ASCII text). As a consequence, it is proved that for all ideals a of R, there are only finitely many non-isomorphic top local cohomology modules H ad (M) having the same support. In addition, we establish an analogue of the Lichtenbaum-Hartshorne vanishing theorem over rings that need not be local. [ABSTRACT FROM AUTHOR]

Abstract: An ideal I of a local Gorenstein ring () is called cohomologically complete intersection whenever for all . Here , , denotes the local cohomology of R with respect to I. For instance, a set-theoretic complete intersection is a cohomologically complete intersection. Here we study cohomologically complete intersections from various homological points of view, in particular in terms of their Bass numbers of , . As a main result it is shown that the vanishing for all is completely encoded in homological properties of , in particular in its Bass numbers. [Copyright &y& Elsevier]

Abstract: We establish cohomological and extension dimension versions of the Hurewicz dimension-raising theorem. [Copyright &y& Elsevier]

Abstract: Let be a commutative Noetherian local ring of dimension d and I an ideal of R. We show that the set of associated primes of the local cohomology module is finite whenever R is regular. Also, it is shown that if is a system of parameters for R, then has infinitely many associated prime ideals for all , where denotes the Matlis dual functor and is the injective hull of the residue field . Finally, we explore a counterexample of Grothendieck''s conjecture by showing that, if , then the R-module is not finitely generated, where . [Copyright &y& Elsevier]

We prove a connectedness result for products of weighted projective spaces. [ABSTRACT FROM AUTHOR]

The study of the cohomological dimension of algebraic varieties has produced some interesting results and problems in local algebra. Let 픞 be an ideal of a commutative Noetherian ring R. For finitely generated R-modules M and N, the concept of cohomological dimension cd픞(M, N) of M and N with respect to 픞 is introduced. If 0 → N' → N → N'' → 0 is an exact sequence of finitely generated R-modules, then it is shown that cd픞(M, N) = max{cd픞(M, N'), cd픞(M, N'')} whenever proj dim M < ∞. Also, if L is a finitely generated R-module with Supp(N/Γ픞(N)) ⊆ Supp(L/Γ픞(L)), then it is proved that cd픞(M, N) ≤ max{cd픞(M,L), proj dim M}. Finally, as a consequence, a result of Brodmann is improved. [ABSTRACT FROM AUTHOR]

We introduce the notion of a (stable) dimension scale d-sc( X) of a space X, where d is a dimension invariant. A bicompactum X is called dimensionally unified if dim F = dimG F for every closed F ⊂ X and for an arbitrary abelian group G. We prove that there exist dimensionally unified bicompacta with every given stable scale dim- sc. [ABSTRACT FROM AUTHOR]

Abstract: Suppose that K is a CW-complex. When we say that a space Y is an absolute co-extensor for K, we mean that K is an absolute extensor for Y, i.e., that for every closed subset A of Y and any map , there exists a map that extends f. Our main theorem will provide several statements that are equivalent to the condition that whenever K is a CW-complex and X is a space which is the topological sum of a countable collection of compact metrizable spaces each of which is an absolute co-extensor for K, then the Stone-Čech compactification of X is an absolute co-extensor for K. [Copyright &y& Elsevier]

Let R be a commutative Noetherian ring,  an ideal of R, and M an R-module. We prove that for a fixed non-negative integer n, the nth local cohomology module [image omitted] is finitely generated if and only if [image omitted] is -cofinite and [image omitted]. This enables us to establish the Noetherian property of local cohomology modules in several cases. Finally, we obtain a new characterization of the cohomological dimensions. [ABSTRACT FROM AUTHOR]

Abstract: We give an explicit description of a certain high order local cohomology module with support in an ideal in terms of some combinatorial information about the minimal primes of the ideal. As a consequence, we give a combinatorial necessary and sufficient condition for an upper bound on the cohomological dimension of the ring with support in the ideal. [Copyright &y& Elsevier]

Abstract: The simplest condition characterizing quasi-finite CW complexes K is the implication for all paracompact spaces X. Here are the main results of the paper: Quasi-finite CW complexes lead naturally to the concept of , where is a family of maps between CW complexes. We generalize some well-known results of extension theory using that concept. [Copyright &y& Elsevier]

For a finitely generated module M over a commutative Noetherian local ring (R,픪), it is shown that there exist only a finite number of non-isomorphic top local cohomology modules ${\rm H}_{{\frak a}}^{\dim (M)}(M)$ for all ideals 픞 of R. It is also shown that for a given integer r ≥ 0, if ${\rm H}_{{\frak a}}^{r}(R/{\frak p})$ is zero for all 픭 in Supp(M), then ${\rm H}_{{\frak a}}^{i}(M)= 0$ for all i ≥ r. [ABSTRACT FROM AUTHOR]

Abstract: We rephrase Gromov''s definition of Markov compacta, introduce a subclass of Markov compacta defined by one building block and study cohomological dimensions of these compacta. We show that for a Markov compactum X, for all but finitely many primes p where is the localization of Z at p. We construct Markov compacta of arbitrarily large dimension having as well as Markov compacta of arbitrary large rational dimension with for a given p. [Copyright &y& Elsevier]

Abstract: The hyperspaces of strongly countable dimensional compacta of positive dimension and of strongly countable dimensional continua of dimension greater than 1 in the Hilbert cube are homeomorphic to the Hurewicz set of all nonempty countable closed subsets of the unit interval . These facts hold true, in particular, for covering dimension dim and cohomological dimension , where G is any Abelian group. [Copyright &y& Elsevier]

Let G be a finitely generated group, and A a ℤ[G]-module of flat dimension n such that the homological invariant Σn(G, A) is not empty. We show that A has projective dimension n as a ℤ[G]-module. In particular, if G is a group of homological dimension hd(G) = n such that the homological invariant Σn(G, ℤ) is not empty, then G has cohomological dimension cd(G) = n. We show that if G is a finitely generated soluble group, the converse is true subject to taking a subgroup of finite index, i.e., the equality cd (G) = hd(G) implies that there is a subgroup H of finite index in G such that Σ∞(H, ℤ) ≠ ∅︀. [ABSTRACT FROM AUTHOR]

Abstract: We extend the definition of quasi-finite complexes from countable complexes to arbitrary ones and provide a characterization of quasi-finite complexes in terms of L-invertible maps and dimensional properties of compactifications. Several results related to the class of quasi-finite complexes are established, such as completion of metrizable spaces, existence of universal spaces and a version of the factorization theorem. Furthermore, we define -spaces in the realm of metrizable spaces and show that some properties of -spaces and -maps remain valid for -spaces and -maps, respectively. [Copyright &y& Elsevier]

Abstract: Let G be a linear algebraic group defined over a field k. We prove that, under mild assumptions on k and G, there exists a finite k-subgroup S of G such that the natural map is surjective for every field extension . We give several applications of this result in the case where k an algebraically closed field of characteristic zero and is finitely generated. In particular, we prove that for every there exists an abelian field extension such that is represented by a G-torsor over a projective variety. From this we deduce that has trivial fixed point obstruction. We also show that a (strong) variant of the algebraic form of Hilbert''s 13th problem implies that the maximal abelian extension of K has cohomological dimension ⩽1. The last assertion, if true, would prove conjectures of Bogomolov and Königsmann, answer a question of Tits and establish an important case of Serre''s Conjecture II for the group . [Copyright &y& Elsevier]

The relationship between the group-theoretic properties of a pro- p-group G and the G-module structure of the group $$H^n (G,\mathbb{F}_q \left[\kern-0.15em\left[ G \right]\kern-0.15em\right])$$ is studied. A necessary and sufficient condition for a pro- p-group G to contain an open Poincare subgroup of dimension n is obtained. This condition does not require that G have finite cohomological dimension and, therefore, applies to groups with torsion. Results concerning the possible values of $$\dim _{\mathbb{F}p} H^n (G,\mathbb{F}_p \left[\kern-0.15em\left[ G \right]\kern-0.15em\right])$$ are also obtained. [ABSTRACT FROM AUTHOR]

Abstract: Let M be a nilpotent CW-complex. We give necessary and sufficient cohomological dimension theory conditions for a finite-dimensional metric compactum X so that every map , where A is a closed subset of X, can be extended to a map . This is a generalization of a result by Dranishnikov [Mat. Sb. (1991)] where such conditions were found for simply-connected CW-complexes M, and Cencelj and Dranishnikov [Canad. Bull. Math. (2001) and Topology Appl. (2002)] where a condition of finitely generatedness was imposed on the nilpotent CW-complex M. [Copyright &y& Elsevier]

Let R be a commutative Noetherian ring and let ?? be an ideal of R. For complexes X and Y of R -modules we investigate the invariant infRG ?? (RHom R ( X , Y )) in certain cases. It is shown that, for bounded complexes X and Y with finite homology, dim Y = dim RHom R ( X , Y ) = proj.dim X + dim( X Y ) + sup X , which strengthen the intersection theorem. Here inf X and sup X denote the homological infimum and supremum of the complex X, respectively. [ABSTRACT FROM AUTHOR]

In the derived category of the category of modules over a commutative Noetherian ring R, we define, for an ideal [afr] of R, we investigate the interplay between the two naturally defined numbers of cohomological dimensions of a complex X in a certain subcategory of the derived category, namely cd([afr], X) = sup{cd([afr], Hℓ(X)) − ℓ|ℓ ∈ [Zopf]} and− infRΓ[afr](X). Here cd([afr], M) = sup{ℓ ∈ [Zopf]| for an R-module M, and infRΓ[afr](X) denotes the homological infimum of the complex RΓ[afr](X). In this paper, it is shown, among other things, that, for any complex X bounded to the left,− infRΓ[afr](X) ≤ cd([afr], X) and equality holds if indeed H(X) is finitely generated. [ABSTRACT FROM AUTHOR]

In this paper a precompact shape theory is investigated. Necessary and sufficient conditions are found for which the precompact shapes of remainders are coinsided. An intrinsically characterization of Čech (co)homology groups of remainders is given. Border cohomological dimension, dimA∞X, and coefficient of border cyclicity, ηA∞X, are defined and the inequality dimA∞X⩽dimA(cX&z.drule;X) and the equality ηA∞X=ηA(cX&z.drule;X) are proved for a space X normally adjoined to its remainder. [Copyright &y& Elsevier]

Suppose that X is a nonempty compact metrizable space and X1⊂X2⊂⋯ is a sequence of nonempty closed subspaces such that for each k∈N, dimZXk&les;k<∞. We show that there exists a compact metrizable space Z, having closed subspaces Z1⊂Z2⊂⋯ , and a surjective cell-like map π :Z→X, such that for each k∈N, (a)dimZk&les;k,(b)π(Zk)=Xk, and(c)π|Zk :Zk→Xk is a cell-like map. Moreover, there is a sequence A0⊂A1⊂⋯ of closed subspaces of Z such that for each k, Zk⊂Ak, dimAk&les;k, π|Ak :Ak→X is surjective, and for k∈N, π|Ak :Ak→X is a UVk−1-map. [Copyright &y& Elsevier]

For every prime p and each n=2,3,…,∞, we constructed in [A.N. Dranishnikov, J.B. West, Topology Appl. 80 (1997) 101–114] an action of G=∏∞i=1(Z/pZ) on a two-dimensional compact metric space X with n-dimensional orbit space. The argument of [A.N. Dranishnikov, J.B. West, Topology Appl. 80 (1997) 101–114] had a gap in Lemma 15 which affected the main lemma of the paper (Lemma 16). In this note we present corrected versions of Lemmas 15 and 16. [Copyright &y& Elsevier]

We prove that for every compactum X and every integer n&ges;2 there are a compactum Z of dim&les;n and a surjective UVn−1-map r :Z→X having the property that:for every finitely generated Abelian group G and every integer k&ges;2 such that dimGX&les;k&les;n we have dimGZ&les;k and r is G-acyclic, or equivalently:for every simply connected CW-complex K with finitely generated homotopy groups such that e-dimX&les;K we have e-dimZ&les;K and r is K-acyclic. (A space is K-acyclic if every map from the space to K is null-homotopic. A map is K-acyclic if every fiber is K-acyclic.) [Copyright &y& Elsevier]

For the cohomology of a sheaf of Abelian groups with involution on a topological space, inequalities that are analogs of the classical Harnack--Thom inequality for the cohomology of a topological space with involution are proved. The general inequalities obtained are applied to reprove some known inequalities and prove new ones. [ABSTRACT FROM AUTHOR]

Let M be a nilpotent CW-complex with finitely generated fundamental group. We give necessary and sufficient cohomological dimension theory conditions for a finite-dimensional metric compactum X so that every map A→M, where A is a closed subset of X can be extended to a map X→M.This is a generalization of a result by Dranishnikov [Mat. Sb. 182 (1991)] where such conditions were found for simply-connected CW-complexes M, and Cencelj and Dranishnikov forthcoming paper [Cannad. Bull. Math.] where such conditions were found for nilpotent CW-complexes M with finitely generated homotopy groups. [Copyright &y& Elsevier]

Let A, B and X be finite-dimensional ANR compacta and let α :A→X and f :A→B be maps such that α restricted to the set Sf={x∈A∣f−1f(x)≠x} is one-to-one. Then the pushout Y of the diagram Xα←Af→B is ANR. We apply this result to a construction of ANRs Mp, p is prime, for which dim(Mp×Mq)≠dimMp+dimMq. [Copyright &y& Elsevier]

Let G be an Abelian group admitting a homomorphism α :Z→G such that the induced homomorphisms α⊗id :Z⊗G→G⊗G and α&ast; :Hom(G,G)→Hom(Z,G) are isomorphisms. We show that for every simplicial complex L there exists an Edwards–Walsh resolution ω :EWG(L,n)→&z.sfnc;L&z.sfnc;. As applications of it we give several resolution theorems. In particular, we haveTheorem. Let G be an arbitrary Abelian group. For every compactum X with c-dimGX&les;n there exists a G-acyclic map f :Z→X from a compactum Z with dimZ&les;n+2 and c-dimGZ&les;n+1.Our methods determine other results as well. If the group G is cyclic, then one can obtain Z with dimZ&les;n. In certain other cases, depending on G, we may resolve X in such a manner that dimZ&les;n+1 and c-dimGZ&les;n. [Copyright &y& Elsevier]