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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]

Some relations between cohomological dimensions and depths of linked ideals are investigated and discussed by various examples. [ABSTRACT FROM PUBLISHER]

Let E → B be a fibre bundle and let E' → B be a vector bundle. Let G be a compact Lie group acting fibre preservingly and freely on both E and E' – 0, where 0 is the zero section of E' → B. Let f : E → E' be a fibre-preserving G-equivariant map and let Zf = {x ∈ E | f(x) = 0} be the zero set of f. In this paper we give a lower bound for the cohomological dimension of the zero set Zf when a fibre of E → B is a real Stiefel manifold with a free ℤ/2-action or a complex Stiefel manifold with a free 핊1-action. This generalizes a well-known result of Dold for sphere bundles equipped with free involutions. [ABSTRACT FROM PUBLISHER]

Let $R$ be a commutative Noetherian ring with non-zero identity and $fa$ an ideal of $R$. Let $M$ be a finite $R$--module of finite projective dimension and $N$ an arbitrary finite $R$--module. We characterize the membership of the generalized local cohomology modules $lc^{i}_{fa}(M,N)$ in certain Serre subcategories of the category of modules from upper bounds. We define and study the properties of a generalization of cohomological dimension of generalized local cohomology modules. Let $mathcal S$ be a Serre subcategory of the category of $R$--modules and $n > pd M$ be an integer such that $lc^{i}_{fa}(M,N)$ belongs to $mathcal S$ for all $i> n$. Then, for any ideal $fbsupseteq fa$, it is also shown that the module $lc^{n}_{fa}(M,N)/{fb}lc^{n}_{fa}(M,N)$ belongs to $mathcal S$.

LetAbe a central simple algebra over a fieldF. Letk1,…,krbe cyclic extensions ofFsuch thatk1 ⊗F… ⊗Fkris a field. We investigate conditions under whichAis a tensor product of symbol algebras where each fieldkilies in a symbolF-algebra factor of the same degree askioverF. As an application, we give an example of an indecomposable algebra of degree 8 and exponent 2 over a field of 2-cohomological dimension 4. [ABSTRACT FROM PUBLISHER]

We discuss the problem of determining the de Rham, Dolbeault and algebraic cohomological dimension of , focusing on possible strategies of attack and then concentrating on exhaustion functions. In the final section, we explain how these techniques can be employed to provide a nontrivial upper bound for the Dolbeault cohomological dimension of . [ABSTRACT FROM AUTHOR]

We show that for any set of primes $${\mathcal{P}}$$ there exists a space $${M_{\mathcal{P}}}$$ which is a homology and cohomology 3-manifold with coefficients in $${\mathbb{Z}_{p}}$$ for $${p \in \mathcal{P}}$$ and is not a homology or cohomology 3-manifold with coefficients in $${\mathbb{Z}_q}$$ for $${q \not\in \mathcal{P}}$$ . In addition, $${M_{\mathcal{P}}}$$ is not a homology or cohomology 3-manifold with coefficients in $${\mathbb{Z}}$$ or $${\mathbb{Q}}$$ . [ABSTRACT FROM AUTHOR]

We study when an abstract finitely presented group of cohomological dimension has pro- completion of cohomological dimension . Furthermore, we prove that for a tree hyperbolic limit group we have and show an example of a hyperbolic limit group that is not free and is free pro-. For a finitely generated residually free group that is not a limit group, we show that is not free pro-. [ABSTRACT FROM AUTHOR]

Let $(S,\mathfrak{m})$ be an $n$-dimensional regular local ring essentially of finite type over a field and let $\mathfrak{a}$ be an ideal of $S$. We prove that if $\text{depth}\,S/\mathfrak{a}\geqslant 3$, then the cohomological dimension $\text{cd}(S,\mathfrak{a})$ of $\mathfrak{a}$ is less than or equal to $n-3$. This settles a conjecture of Varbaro for such an $S$. We also show, under the assumption that $S$ has an algebraically closed residue field of characteristic zero, that if $\text{depth}\,S/\mathfrak{a}\geqslant 4$, then $\text{cd}(S,\mathfrak{a})\leqslant n-4$ if and only if the local Picard group of the completion $\widehat{S/\mathfrak{a}}$ is torsion. We give a number of applications, including a vanishing result on Lyubeznik’s numbers, and sharp bounds on the cohomological dimension of ideals whose quotients satisfy good depth conditions such as Serre’s conditions $(S_{i})$. [ABSTRACT FROM AUTHOR]

Let E → B be a fiber bundle and E ′ → B be a vector bundle. Let G be a compact group acting fiber preservingly and freely on both E and E ′ − 0 , where 0 is the zero section of E ′ → B . Let f : E → E ′ be a fiber preserving G -equivariant map, and let Z f = { x ∈ E | f ( x ) = 0 } be the zero set of f . It is an interesting problem to estimate the dimension of the set Z f . In 1988, Dold [5] obtained a lower bound for the cohomological dimension of the zero set Z f when E → B is the sphere bundle associated with a vector bundle which is equipped with the antipodal action of G = Z / 2 . In this paper, we generalize this result to products of finitely many spheres equipped with the diagonal antipodal action of Z / 2 . We also prove a Bourgin–Yang type theorem for products of spheres equipped with the diagonal antipodal action of Z / 2 . [ABSTRACT FROM AUTHOR]

Let S = K[x1,...,xm, y1,...,yn] be the standard bigraded polynomial ring over a field K, and M a finitely generated bigraded S-module. In this paper we study the generalized Cohen-Macaulayness and sequentially generalized Cohen-Macaulayness of M with respect to Q = (y1,...,yn). We prove that if I ⊆ S be a monomial ideal with (Q, S/I) ≤ 2, then S/I is sequentially generalized Cohen-Macaulay with respect to Q. [ABSTRACT FROM AUTHOR]