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Let G be a connected and simple graph on the vertex set [ n ]. To the graph G one can associate the generalized binomial edge ideal J m (G) in the polynomial ring R = K [ x i j : i ∈ [ m ] , j ∈ [ n ] ]. We provide a lower bound for the cohomological dimension of J m (G). We also study when J m (G) is a cohomologically complete intersection. Finally, we show that the arithmetical rank of J 2 (G) equals the projective dimension of R / J 2 (G) in several cases. [ABSTRACT FROM AUTHOR]

For a non-compact finite thickness building whose Davis apartment is an orientable pseudomanifold, we compute the supremum of the set of p > 1 such that its top dimensional reduced ℓ p -cohomology is nonzero. We adapt the non-vanishing assertion of this result to any finite thickness building using the Bestvina realization. Using similar techniques, we generalize bounds obtained by Clais on the conformal dimension of some Gromov-hyperbolic buildings to any such building. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of the cohomological dimension filtration with respect to a system of ideals. In particular, a characterization of cohomological dimension filtration of a module by the associated prime ideals of its factors is established. As a main result, we provide a necessary and sufficient condition for an ascending chain of submodules of an R -module M to be the cd-filtration of M, with respect to a system of ideals. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring, I, J be two ideals of R, and M, N be two R-modules. Let S be a Serre subcategory of the category of R-modules. We introduce Serre cohomological dimension of N, M with respect to (I, J), as cdS(I, J, N, M) = sup{i ∈ N0 : HiI,J (N, M) 6∈ S}. We study some properties of cdS(I, J, N, M), and we get some formulas and upper bounds for it. [ABSTRACT FROM AUTHOR]

Let A be a commutative noetherian non-positive DG-ring, ̄ an ideal of Ā : = H 0 (A) , and M ∈ D (A). In this paper, we introduce the notion of cohomological dimension of DG-modules and investigate the interplay between cd A ( ̄ , M) : = sup { cd Ā ( ̄ , H n (M)) + n | n ∈ ℤ } and sup R Γ ̄ (M). It is shown that cd A ( ̄ , M) = sup R Γ ̄ (M) for any 0 ≇ M ∈ D f − (A). As an application, we recover a DG-version of Grothendieck's vanishing and non-vanishing theorems for local cohomology. We also study the cohomological dimension of Koszul DG-modules and get some interesting results. [ABSTRACT FROM AUTHOR]

We prove that any region \Gamma in a homogeneous n-dimensional and locally compact separable metric space X, where n\geq 2, cannot be irreducibly separated by a closed (n-1)-dimensional subset C with the following property: C is acyclic in dimension n-1 and there is a point b\in C\cap \Gamma having a special local base \mathcal B_C^b in C such that the boundary of each U\in \mathcal B_C^b is acyclic in dimension n-2. In case X is strongly locally homogeneous, it suffices to have a point b\in C\cap \Gamma with an ordinary base \mathcal B_C^b satisfying the above condition. The acyclicity means triviality of the corresponding Čech cohomology groups. This implies all known results concerning the separation of regions in homogeneous connected locally compact spaces. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring and I a proper ideal of R. In this paper, we study finitely generated R-modules M with only one non-vanishing local cohomology module H I c (M) where c = c d (I , M) . Let C be a semidualizing R-module. We investigate the conditions under which H I c (M) belongs to either the Auslander class A C (R) or the Bass class B C (R) . [ABSTRACT FROM AUTHOR]

In this paper, first, we generalize a result of Peskine–Szpiro on the relation between the cohomological dimension and projective dimension. Then, we give conditions for the vanishing of local cohomology from local to global and vice versa. Our final goal in the present paper is to examine the set-theoretically Cohen–Macaulay ideals to find some cohomological characterization of these kinds of ideals. [ABSTRACT FROM AUTHOR]

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]

Let I be an ideal of a commutative Noetherian ring R such that the R-modules H I i (U) are I-cofinite, for all finitely generated R-modules U and all i ∈ ℕ 0 . Also, assume that the category of all I-cofinite modules is an Abelian subcategory of the category of R-modules and R-homomorphisms. It is shown that for each pair of I-cofinite modules M and N the R-modules Tor i R (M , N) are I-cofinite for all i ∈ ℕ 0 . Also, some applications of this result will be included. [ABSTRACT FROM AUTHOR]

For a group G, let ℱ n be the family of all subgroups of G containing a subgroup of finite index isomorphic to ℤ r for some r = 0 , 1 , 2 , ... , n . Joecken, Lafont and Sánchez Saldaña computed the ℱ 1 -geometric dimension of 3-manifold groups. As a natural extension of the aforementioned result, the goal of this article is to compute the ℱ n -geometric dimension of 3-manifold groups for all n ≥ 2 . [ABSTRACT FROM AUTHOR]

For a unital commutative Banach algebra A and its closed ideal I, we study the relative Čech cohomology of the pair (Max (A) , Max (A / I)) of maximal ideal spaces and show a relative version of the main theorem of Lupton et al. (Trans Amer Math Soc 361:267–296, 2009): H ˇ j (Max (A) , Max (A / I)) ; Q) ≅ π 2 n - j - 1 (L c n (I)) Q for j < 2 n - 1 , where L c n (I) refers to the space of last columns. We then study the rational cohomological dimension cdim Q Max (A) for a unital commutative Banach algebra and prove an embedding theorem: if A is a unital commutative semi-simple regular Banach algebra such that Max (A) is metrizable and cdim Q Max (A) ≤ m , then (i) the rational homotopy group π k (G L n (A)) Q is stabilized if n ≥ ⌈ (m + k + 1) / 2 ⌉ and (ii) there exists a compact metrizable space X A with dim X A ≤ m such that A is embedded into the commutative C ∗ -algebra C (X A) such that π k (G L n (C (X A))) is rationally isomorphic to π k (G L n (A)) for each k ≥ 1 and π k (G L n (C (X A)) is stabilized for n ≥ ⌈ (m + k + 1) / 2 ⌉ . The main technical ingredient is a modified version of a classical theorem of Davie (Proc Lond Math Soc 23:31–52, 1971). [ABSTRACT FROM AUTHOR]

We define and develop a homotopy invariant notion for the sequential topological complexity of a map f : X → Y , denoted T C r (f) , that interacts with T C r (X) and T C r (Y) in the same way Jamie Scott's topological complexity map T C (f) interacts with T C (X) and T C (Y). Furthermore, we apply T C r (f) to studying group homomorphisms ϕ : Γ → Λ. In addition, we give the characterization of cohomological dimension of group homomorphisms. [ABSTRACT FROM AUTHOR]

For every one-relator monoid M = ⟨ A ∣ u = v ⟩ with u , v ∈ A ∗ we construct a contractible M-CW complex and use it to build a projective resolution of the trivial module which is finitely generated in all dimensions. This proves that all one-relator monoids are of type FP ∞ , answering positively a problem posed by Kobayashi in 2000. We also apply our results to classify the one-relator monoids of cohomological dimension at most 2, and to describe the relation module, in the sense of Ivanov, of a torsion-free one-relator monoid presentation as an explicitly given principal left ideal of the monoid ring. In addition, we prove the topological analogues of these results by showing that all one-relator monoids satisfy the topological finiteness property F ∞ , and classifying the one-relator monoids with geometric dimension at most 2. These results give a natural monoid analogue of Lyndon’s Identity Theorem for one-relator groups. [ABSTRACT FROM AUTHOR]

We prove an injective version of Schanuel’s lemma from homological algebra in the setting of exact categories. [ABSTRACT FROM AUTHOR]

Let q be a non-negative integer. We prove that a perfect field K has cohomological dimension at most q + 1 if, and only if, for any finite extension L of K and for any homogeneous space Z under a smooth linear connected algebraic group over L, the q-th Milnor K-theory group of L is spanned by the images of the norms coming from finite extensions of L over which Z has a rational point. We also prove a variant of this result for imperfect fields. [ABSTRACT FROM AUTHOR]

Let I be an ideal of a commutative Noetherian ring R. It is shown that the R-modules HIj(M) are I-cofinite for all finitely generated R-modules M and all j ∈ ℕ0 if and only if the R-modules ExtRi (N,HIj(M)) and TorRi (N, HIj(M)) are I-cofinite for all finitely generated R-modules M, N and all integers i, j ∈ ℕ0. [ABSTRACT FROM AUTHOR]

Let a be an ideal of a commutative Noetherian ring R and let M be a finitely generated R-module. The purpose of this paper is to establish some new results on Lynch's conjecture and the set of the attached primes of the top local cohomology module H a cd (a , M) (M) for the case that R has prime characteristic and cd (a , M) = ara M (a) . To prove our results, we establish and use a new relation between the set Att R (H a cd (a , M) (M)) and the M-height of the annihilator of the top local cohomology module H a cd (a , M) (M) . Several corollaries of this result are proved. Among other things, we will provide a complete characterization of the attached primes of the top local cohomology module H a cd (a , M) (M) and Rad (Ann R (H a cd (a , M) (M))) . Using this, we show that the ideal Ann R (H a cd (a , M) (M)) has M-height zero. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring, I an ideal of R and M a non-zero R-module. In this paper we calculate the extension of annihilator of local cohomology modules H^t_I(M), t≥0, under the ring extension R⊂R[X] (resp. R⊂R[[X]]). By using this extension we will present some of the faithfulness conditions of local cohomology modules, and show that if the Lynch's conjecture, in [11], holds in R[[X]], then it will holds in R.

Let I be an ideal of a commutative Noetherian ring R such that the R-modules H I i (M) are I-cofinite, for all finitely generated R-modules M and all i ∈ ℕ 0. Let M and N be two I-cofinite R-modules and f : M → N be an R-homomorphism. It is shown that for each finitely generated R-module U the R-module D I (U) ⊗ R coker f is I-cofinite, where D I (−) is the I-transform functor. Also, it is shown that for each I-cofinite R-module X and each finitely generated R-module U the R-module Tor i R (X , D I (U)) is I-cofinite, for all i ∈ ℕ 0. [ABSTRACT FROM AUTHOR]

Let R be a commutative noetherian ring. Let H (R) be the quotient of the Grothendieck group of finitely generated R-modules by the subgroup generated by pseudo-zero modules. Suppose that the R -vector space H (R) R = H (R) ⊗ Z R has finite dimension. Let C (R) (resp. C r (R) ) be the convex cone in H (R) R spanned by maximal Cohen–Macaulay R-modules (resp. maximal Cohen–Macaulay R-modules of rank r). We explore the interior, closure and boundary, and convex polyhedral subcones of C (R) . We provide various equivalent conditions for R to have only finitely many rank r maximal Cohen–Macaulay points in C r (R) in terms of topological properties of C r (R) . Finally, we consider maximal Cohen–Macaulay modules of rank one as elements of the divisor class group Cl (R) . [ABSTRACT FROM AUTHOR]

Let I be an ideal of a commutative Noetherian ring R. It is shown that the R-modules H I i (M) are I-cofinite, for all finitely generated R-modules M and all i ∈ N 0 , if and only if the R-modules H I i (R) are I-cofinite with dimension not exceeding 1, for all integers i ≥ 2 ; in addition, under these equivalent conditions it is shown that, for each minimal prime ideal p over I, either height p ≤ 1 or dim R / p ≤ 1 , and the prime spectrum of the I-transform R-algebra D I (R) equipped with its Zariski topology is Noetherian. Also, by constructing an example we show that under the same equivalent conditions in general the ring D I (R) need not be Noetherian. Furthermore, in the case that R is a local ring, it is shown that the R-modules H I i (M) are I-cofinite, for all finitely generated R-modules M and all i ∈ N 0 , if and only if for each minimal prime ideal P of R ^ , either dim R ^ / (I R ^ + P) ≤ 1 or H I R ^ i (R ^ / P) = 0 , for all integers i ≥ 2 . Finally, it is shown that if R is a semi-local ring and the R-modules H I i (M) are I-cofinite, for all finitely generated R-modules M and all i ∈ N 0 , then the category of all I-cofinite modules forms an Abelian subcategory of the category of all R-modules. [ABSTRACT FROM AUTHOR]

The notion of a V n -continuum was introduced by Alexandroff [1] as a generalization of the concept of n -manifold. In this note we consider the cohomological analogue of V n -continuum and prove that any strongly locally homogeneous generalized continuum X with cohomological dimension dim G ⁡ X = n is a generalized V n -space with respect to the cohomological dimension dim G. In particular, every strongly locally homogeneous continuum of covering dimension n is a V n -continuum in the sense of Alexandroff. This provides a partial answer to a question raised in [12]. An analog of the Mazurkiewicz theorem that no subset of covering dimension ≤ n − 2 cuts any region of the Euclidean n -space is also obtained for strongly locally homogeneous generalized continua X of cohomological dimension dim G ⁡ X = n. [ABSTRACT FROM AUTHOR]

Given a standard graded polynomial ring over a commutative Noetherian ring A, we prove that the cohomological dimension and the height of the ideals defining any of its Veronese subrings are equal. This result is due to Ogus when A is a field of characteristic zero, and follows from a result of Peskine and Szpiro when A is a field of positive characteristic; our result applies, for example, when A is the ring of integers. [ABSTRACT FROM AUTHOR]

Let R be a commutative noetherian ring. In this paper, we study specialization-closed subsets of Spec R. More precisely, we first characterize the specialization-closed subsets in terms of various closure properties of subcategories of modules. Then, for each nonnegative integer n we introduce the notion of n-wide subcategories of R-modules to consider the question asking when a given specialization-closed subset has cohomological dimension at most n. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring with non-zero identity and I be an ideal of R. Let M and N be two finitely generated R-modules. In this paper it is shown that, if cd(I, R) ≤ 1 and pdR(M) < ∞, then HiI(M, N) are I-cofinite for each i ≥ 0. Moreover, it is shown that if q(I, R) ≤ 1 and pdR(M) < ∞, then the Bass numbers of generalized local cohomology modules HiI(M, N) are finite for each i ≥ 0. We also characterize the greatest integer i such that HiI(M, N) is not Artinian and I-cofinite. [ABSTRACT FROM AUTHOR]

Let (R , m , k) be a Noetherian local ring of dimension d ≥ 4 . Assume that 2 ≤ i ≤ d - 2 is an integer and x 1 , ... , x i is a part of a system of parameters for R. Let Υ i denote the set of all prime ideals p of R such that dim R / p = i + 1 , Supp H (x 1 , ... , x i) R i (R / p) = { m } , and dim k Soc R H (x 1 , ... , x i) R i (R / p) = ∞ . In this paper, it is shown that Υ i is an infinite set. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring and Φ be a system of ideals of R. In this paper, we study the annihilators and the set of attached prime ideals of top generalized local cohomology modules with respect to a system of ideals. [ABSTRACT FROM AUTHOR]

The concept of dimension is ubiquitous in Mathematics. In this survey we discuss the interrelations between dimension and injectivity in the categorical sense. [ABSTRACT FROM AUTHOR]

We review the Burghelea conjecture, which constitutes a full computation of the periodic cyclic homology of complex group rings, and its relation to the algebraic Baum–Connes conjecture. The Burghelea conjecture implies the Bass conjecture. We state two conjectures about groups of finite asymptotic dimension, which together imply the Burghelea conjecture for such groups. We prove both conjectures for many classes of groups. It is known that the Burghelea conjecture does not hold for all groups, although no finitely presentable counterexample was known. We construct a finitely presentable (even type F ∞ ) counterexample based on Thompson's group F. We construct as well a finitely generated counterexample with finite decomposition complexity. [ABSTRACT FROM AUTHOR]

We derive three equivalent necessary conditions for a small category to have homological dimension one, generalizing a result of Novikov. As a consequence, any small cancellative category of homological dimension one is embeddable in a groupoid. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring and I be an ideal of R. In this article, we answer affirmatively a question raised by the present author. Also, as a consequence, it is shown that the category of all I-cofinite R-modules C (R , I) cof is an Abelian subcategory of the category of all R-modules, whenever q (I , R) ≤ 1. These assertions also answer affirmatively a question raised by R. Hartshorne, in some special cases. [ABSTRACT FROM AUTHOR]

This is a survey of most important results and unsolved problems about homogeneous finite-dimensional metric ANR-compacta. We also discuss some partial results and possible ways of solutions. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian (not necessary local) ring with identity and a be a proper ideal of R. We introduce a notion of a -relative system of parameters and characterize them by using the notion of cohomological dimension. Also, we present a criterion of relative Cohen-Macaulay modules via relative system of parameters. [ABSTRACT FROM AUTHOR]

In this article, we prove the following generalization of a result of Hartshorne: Let be a regular local ring of dimension 4. Assume that is a regular system of parameters for S and. Then for each finitely generated S-module N with the socle of is infinite dimensional. Also, using this result, for any commutative Noetherian complete local ring , we characterize the class of all ideals I of R with the property that, for every finitely generated R-module M, the local cohomology modules are I-cofinite for all. [ABSTRACT FROM AUTHOR]

Let 𝔞 be an ideal of a local ring (R , 𝔪) with c = cd (𝔞 , R) the cohomological dimension of 𝔞 in R. In the case that c = dim R , we first give a bound for depth D (H 𝔞 c (R)) , where c > 2 and (R , 𝔪) is complete. Later, H 𝔞 c (R) ⊗ R H 𝔞 c (R) , D (H 𝔞 c (R)) ⊗ R D (H 𝔞 c (R)) and H 𝔞 c (R) ⊗ R D (H 𝔞 c (R)) are examined. In the case c = dim R − 1 , the set Att R H 𝔞 c (R) is considered. [ABSTRACT FROM AUTHOR]

Throughout, let R be a commutative Noetherian ring. A ring R satisfies Serre's condition if for all . Serre's condition has been a topic of expanding interest. In this paper, we examine a generalization of Serre's condition. We say a ring satisfies when for all. We prove generalizations of results for rings satisfying Serre's condition. [ABSTRACT FROM AUTHOR]

In this article, we shall investigate the concepts of cofiniteness of local cohomology modules and Abelian categories of cofinite modules over arbitrary Noetherian rings. Then we shall improve some of the well known results in the area. [ABSTRACT FROM AUTHOR]

Let M be a (2 × n) non-generic matrix of linear forms in a polynomial ring. For large classes of such matrices, we compute the cohomological dimension (cd) and the arithmetical rank (ara) of the ideal I_2(M) generated by the 2-minors of M. Over an algebraically closed field, any (2×n)-matrix of linear forms can be written in the Kronecker-Weierstrass normal form, as a concatenation of scroll, Jordan and nilpotent blocks. Badescu and Valla computed ara(I_2 (M)) when M is a concatenation of scroll blocks. In this case we compute cd(I2 (M)) and extend these results to concatenations of Jordan blocks. Eventually we compute ara(I_2(M)) and cd(I_2 (M)) in an interesting mixed case, when M contains both Jordan and scroll blocks. In all cases we show that ara(I_2(M)) is less than the arithmetical rank of the determinantal ideal of a generic matrix.

Let R be a commutative noetherian ring, and 𝒵 a stable under specialization subset of Spec (R). We introduce a notion of 𝒵 -cofiniteness and study its main properties. In the case dim (𝒵) ≤ 1 , or dim (R) ≤ 2 , or R is semilocal with cd (𝒵 , R) ≤ 1 , we show that the category of 𝒵 -cofinite R -modules is abelian. Also, in each of these cases, we prove that the local cohomology module H 𝒵 i (X) is 𝒵 -cofinite for every homologically left-bounded R -complex X whose homology modules are finitely generated and every i ∈ ℤ. [ABSTRACT FROM AUTHOR]

Let G be a group acting freely, properly discontinuously and cellularly on some finite dimensional C W-complex Σ(2 n) which has the homotopy type of the 2 n -sphere 𝕊2 n . Then, that action induces a homomorphism G → Aut(H 2 n (Σ(2 n))). We classify all pairs (G , φ), where G is a virtually cyclic group and φ: G → Aut(ℤ) is a homomorphism, which are realizable in the way above and the homotopy types of all possible orbit spaces as well. Next, we consider the family of all groups which have virtual cohomological dimension one and which act on some Σ(2 n). Those groups consist of free groups and semi-direct products F ⋊ ℤ2 with F a free group. For a group G from the family above and a homomorphism φ: G → Aut(ℤ), we present an algebraic criterion equivalent to the realizability of the pair (G , φ). It turns out that any realizable pair can be realized on some Σ(2 n) with dim Σ(2 n) ≤ 2 n + 1. [ABSTRACT FROM AUTHOR]

We give a survey of old and new results in dimension theory of compact metric spaces. Most of the relatively new results presented in the survey are based on the cohomological dimension approach. We complement the survey by stating the basics of cohomological dimension theory and listing some of its applications beyond the dimension theory. [ABSTRACT FROM AUTHOR]

Let $$(R, \mathfrak {m})$$ be a local ring and M a finitely generated R-module. It is shown that if M is relative Cohen-Macaulay with respect to an ideal $$\mathfrak {a}$$ of R, then $${\text {Ann}}_R(H_{\mathfrak {a}}^{{\text {cd}}(\mathfrak {a}, M)}(M))={\text {Ann}}_RM/L={\text {Ann}}_RM$$ and $${\text {Ass}}_R (R/{\text {Ann}}_RM)\subseteq \{\mathfrak {p}\in {\text {Ass}}_R M|\,\mathrm{cd}(\mathfrak {a}, R/\mathfrak {p})={\text {cd}}(\mathfrak {a}, M)\},$$ where L is the largest submodule of M such that $$\mathrm{cd}(\mathfrak {a}, L)< \mathrm{cd}(\mathfrak {a}, M)$$ . We also show that if $$H^{\dim M}_{\mathfrak {a}}(M)=0$$ , then $${\text {Att}}_R(H^{\dim M-1}_{\mathfrak {a}}(M))= \{\mathfrak {p}\in {\text {Supp}}(M)|\mathrm{cd}(\mathfrak {a}, R/\mathfrak {p})=\dim M-1\},$$ and so the attached primes of $$H^{\dim M-1}_{\mathfrak {a}}(M)$$ depend only on $${\text {Supp}}(M)$$ . Finally, we prove that if M is an arbitrary module (not necessarily finitely generated) over a Noetherian ring R with $$\mathrm{cd}(\mathfrak {a}, M)=\mathrm{cd}(\mathfrak {a}, R/{\text {Ann}}_RM)$$ , then $${\text {Att}}_R(H^{\mathrm{cd}(\mathfrak {a}, M)}_{\mathfrak {a}}(M))\subseteq \{\mathfrak {p}\in {\text {V}}({\text {Ann}}_RM)|\,\mathrm{cd}(\mathfrak {a}, R/\mathfrak {p})=\mathrm{cd}(\mathfrak {a}, M)\}.$$ As a consequence of this, it is shown that if $$\dim M=\dim R$$ , then $${\text {Att}}_R(H^{\dim M}_{\mathfrak {a}}(M))\subseteq \{\mathfrak {p}\in {\text {Ass}}_R M|\mathrm{cd}(\mathfrak {a}, R/\mathfrak {p})=\dim M\}$$ . [ABSTRACT FROM AUTHOR]

Let C be a class of spaces. An element Z ∈ C is called universal for C if each element of C embeds in Z . It is well-known that for each n ∈ N , there exists a universal element for the class of metrizable compacta X of (covering) dimension dim ⁡ X ≤ n . The situation in cohomological dimension over an abelian group G , denoted dim G , is almost the opposite. Our results will imply in contradistinction that for each nontrivial abelian group G and for n ≥ 2 , there exists no universal element for the class of metrizable compacta X with dim G ⁡ X ≤ n . [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring, I ⊆ J be ideals of R and M be a finitely generated R -module. In this paper it is shown that q ( J , M ) ≤ q ( I , M ) + cd ( J , M / I M ) . Furthermore, it is shown that, for any ideal I of R and any finitely generated R -module M with q ( I , M ) ≤ 1 , the local cohomology modules H I i ( M ) are I -cofinite for all integers i ≥ 0 . As a consequence of this result it is shown that, if q ( I , R ) ≤ 1 , then for any finitely generated R -module M , the local cohomology modules H I i ( M ) are I -cofinite for all integers i ≥ 0 . Finally, it is shown that the category of all I -cofinite R -modules C ( R , I ) c o f is an Abelian subcategory of the category of all R -modules, whenever ( R , m ) is a complete Noetherian local ring and I is an ideal of R with q ( I , R ) ≤ 1 . These assertions answer affirmatively two questions raised by R. Hartshorne in [16] , in the some special cases. [ABSTRACT FROM AUTHOR]

Let V and W be orthogonal representations of G with $$V^G= W^G=\{0\}$$ . Let S( V ) be the sphere of V and $$f : S(V ) \rightarrow W$$ be a G-equivariant mapping. We give an estimate for the dimension of the set $$Z_f=f^{-1}\{0\}$$ in terms of $$ \dim V$$ and $$\dim W$$ , if G is the torus $$\mathbb T^k$$ , or the p-torus $$\mathbb Z_p^k$$ . This extends the classical Bourgin-Yang theorem onto this class of groups. Finally, we show that for any p-toral group G and a G-map $$f:S(V) \rightarrow W$$ , with $$\dim V=\infty $$ and $$\dim W<\infty $$ , we have $$\dim Z_f= \infty $$ . [ABSTRACT FROM AUTHOR]

We construct groups G that are virtually torsion-free and have virtual cohomological dimension strictly less than the minimal dimension for any model for E _ G , the classifying space for proper actions of G . They are the first examples that have these properties and also admit cocompact models for E _ G . We exhibit groups G whose virtual cohomological dimension and Bredon cohomological dimension are two that do not admit any 2-dimensional contractible proper G -CW-complex. [ABSTRACT FROM AUTHOR]

We introduce the notion of cohomologically weakly infinite-dimensional spaces and show the acyclicity of the complement Q ∖ X in the Hilbert cube Q of a cohomologically weakly infinite-dimensional compactum X . As a corollary we obtain the acyclicity of the complement results when (a) X is weakly infinite-dimensional; (b) X has finite cohomological dimension. [ABSTRACT FROM AUTHOR]