Module-valued functors preserving the covering dimension

Jan Spěvák

Displaying similar documents to “Module-valued functors preserving the covering dimension”

Levi's forms of higher codimensional submanifolds

Andrea D&#039;Agnolo, Giuseppe Zampieri (1991)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Similarity:

Let $X ≅ C^{n}$ , let $M$ be a $C^{2}$ hypersurface of $X$ , $S$ be a $C^{2}$ submanifold of $M$ . Denote by $L_{M}$ the Levi form of $M$ at $z_{0} \in S$ . In a previous paper [3] two numbers $s^{\pm} (S, p)$ , $p \in {({\dot{T}}_{S}^{*} X)}_{z_{0}}$ are defined; for $S = M$ they are the numbers of positive and negative eigenvalues for $L_{M}$ . For $S \subset M$ , $p \in S \times_{M} \dot{T}^{*}_{S} X)$ , we show here that $s^{\pm} (S, p)$ are still the numbers of positive and negative eigenvalues for $L_{M}$ when restricted to $T_{z_{0}}^{C} S$ . Applications to the concentration in degree for microfunctions at the boundary are given.

Maps with dimensionally restricted fibers

Vesko Valov (2011)

Colloquium Mathematicae

Similarity:

We prove that if f: X → Y is a closed surjective map between metric spaces such that every fiber $f^{- 1} (y)$ belongs to a class S of spaces, then there exists an $F_{σ}$ -set A ⊂ X such that A ∈ S and $d i m f^{- 1} (y) ∖ A = 0$ for all y ∈ Y. Here, S can be one of the following classes: (i) M: e-dim M ≤ K for some CW-complex K; (ii) C-spaces; (iii) weakly infinite-dimensional spaces. We also establish that if S = M: dim M ≤ n, then dim f ∆ g ≤ 0 for almost all $g \in C (X,^{n + 1})$ .

On dimensionally restricted maps

H. Murat Tuncali, Vesko Valov (2002)

Fundamenta Mathematicae

Similarity:

Let f: X → Y be a closed n-dimensional surjective map of metrizable spaces. It is shown that if Y is a C-space, then: (1) the set of all maps g: X → ⁿ with dim(f △ g) = 0 is uniformly dense in C(X,ⁿ); (2) for every 0 ≤ k ≤ n-1 there exists an $F_{σ}$ -subset $A_{k}$ of X such that $d i m A_{k} \leq k$ and the restriction $f | (X ∖ A_{k})$ is (n-k-1)-dimensional. These are extensions of theorems by Pasynkov and Toruńczyk, respectively, obtained for finite-dimensional spaces. A generalization of a result due to Dranishnikov and Uspenskij...

Homogeneity and non-coincidence of Hausdorff and box dimensions for subsets of ℝⁿ

Anders Nilsson, Peter Wingren (2007)

Studia Mathematica

Similarity:

A class of subsets of ℝⁿ is constructed that have certain homogeneity and non-coincidence properties with respect to Hausdorff and box dimensions. For each triple (r,s,t) of numbers in the interval (0,n] with r < s < t, a compact set K is constructed so that for any non-empty subset U relatively open in K, we have $(d i m_{H} (U), {\underset{̲}{d i m}}_{B} (U), {\bar{d i m}}_{B} (U)) = (r, s, t)$ . Moreover, $2^{- n} \leq H^{r} (K) \leq 2 n^{r / 2}$ .

Universal acyclic resolutions for arbitrary coefficient groups

Michael Levin (2003)

Fundamenta Mathematicae

Similarity:

We prove that for every compactum X and every integer n ≥ 2 there are a compactum Z of dimension ≤ n+1 and a surjective $U V^{n - 1}$ -map r: Z → X such that for every abelian group G and every integer k ≥ 2 such that $d i m_{G} X \leq k \leq n$ we have $d i m_{G} Z \leq k$ and r is G-acyclic.

Cofiniteness of torsion functors of cofinite modules

Reza Naghipour, Kamal Bahmanpour, Imaneh Khalili Gorji (2014)

Colloquium Mathematicae

Similarity:

Let R be a Noetherian ring and I an ideal of R. Let M be an I-cofinite and N a finitely generated R-module. It is shown that the R-modules $T o r_{i}^{R} (N, M)$ are I-cofinite for all i ≥ 0 whenever dim Supp(M) ≤ 1 or dim Supp(N) ≤ 2. This immediately implies that if I has dimension one (i.e., dim R/I = 1) then the R-modules $T o r_{i}^{R} (N, H_{I}^{j} (M))$ are I-cofinite for all i,j ≥ 0. Also, we prove that if R is local, then the R-modules $T o r_{i}^{R} (N, M)$ are I-weakly cofinite for all i ≥ 0 whenever dim Supp(M) ≤ 2 or dim Supp(N) ≤ 3. Finally, it...

$ℰ_{B}$ -dimension function of bitopological spaces

M. Jelić (1987)

Matematički Vesnik

Similarity:

On the Separation Dimension of $K_{ω}$

Yasunao Hattori, Jan van Mill (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

We prove that $t r t K_{ω} > ω + 1$ , where trt stands for the transfinite extension of Steinke’s separation dimension. This answers a question of Chatyrko and Hattori.

On the Hausdorff Dimension of Topological Subspaces

Tomasz Szarek, Maciej Ślęczka (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

It is shown that every Polish space X with $d i m_{T} X \geq d$ admits a compact subspace Y such that $d i m_{H} Y \geq d$ where $d i m_{T}$ and $d i m_{H}$ denote the topological and Hausdorff dimensions, respectively.

A finiteness theorem for holomorphic Banach bundles

Jürgen Leiterer (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

Let $E$ be a holomorphic Banach bundle over a compact complex manifold, which can be defined by a cocycle of holomorphic transition functions with values of the form $id + K$ where $K$ is compact. Assume that the characteristic fiber of $E$ has the compact approximation property. Let $n$ be the complex dimension of $X$ and $0 \leq q \leq n$ . Then: If $V \to X$ is a holomorphic vector bundle (of finite rank) with $H^{q} (X, V) = 0$ , then $dim H^{q} (X, V \otimes E) < \infty$ . In particular, if $dim H^{q} (X, 𝒪) = 0$ , then $dim H^{q} (X, E) < \infty$ .

Finitistic dimension and restricted injective dimension

Dejun Wu (2015)

Czechoslovak Mathematical Journal

Similarity:

We study the relations between finitistic dimensions and restricted injective dimensions. Let $R$ be a ring and $T$ a left $R$ -module with $A = {End}_{R} T$ . If $_{R} T$ is selforthogonal, then we show that $rid (T_{A}) \leq findim (A_{A}) \leq findim (_{R} T) + rid (T_{A})$ . Moreover, if $R$ is a left noetherian ring and $T$ is a finitely generated left $R$ -module with finite injective dimension, then $rid (T_{A}) \leq findim (A_{A}) \leq fin . inj . \dim (_{R} R) + rid (T_{A})$ . Also we show by an example that the restricted injective dimensions of a module may be strictly smaller than the Gorenstein injective dimension.

Borsuk-Sieklucki theorem in cohomological dimension theory

Margareta Boege, Jerzy Dydak, Rolando Jiménez, Akira Koyama, Evgeny V. Shchepin (2002)

Fundamenta Mathematicae

Similarity:

The Borsuk-Sieklucki theorem says that for every uncountable family ${X_{α}}_{α \in A}$ of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α ≠ β such that $d i m (X_{α} \cap X_{β}) = n$ . In this paper we show a cohomological version of that theorem: Theorem. Suppose a compactum X is $c l c_{ℤ}^{n + 1}$ , where n ≥ 1, and G is an Abelian group. Let ${X_{α}}_{α \in J}$ be an uncountable family of closed subsets of X. If $d i m_{G} X = d i m_{G} X_{α} = n$ for all α ∈ J, then $d i m_{G} (X_{α} \cap X_{β}) = n$ for some α ≠ β. For G being a countable principal ideal domain the above result was proved by Choi...

Szpilrajn type theorem for concentration dimension

Jozef Myjak, Tomasz Szarek (2002)

Fundamenta Mathematicae

Similarity:

Let X be a locally compact, separable metric space. We prove that $d i m_{T} X = i n f d i m_{L} X^{'} : X^{'} i s h o m e o m o r p h i c t o X$ , where $d i m_{L} X$ and $d i m_{T} X$ stand for the concentration dimension and the topological dimension of X, respectively.

The $G$ -graded identities of the Grassmann Algebra

Lucio Centrone (2016)

Archivum Mathematicum

Similarity:

Let $G$ be a finite abelian group with identity element $1_{G}$ and $L = ⨁_{g \in G} L^{g}$ be an infinite dimensional $G$ -homogeneous vector space over a field of characteristic $0$ . Let $E = E (L)$ be the Grassmann algebra generated by $L$ . It follows that $E$ is a $G$ -graded algebra. Let $| G |$ be odd, then we prove that in order to describe any ideal of $G$ -graded identities of $E$ it is sufficient to deal with $G^{'}$ -grading, where $| G^{'} | \leq | G |$ , ${dim}_{F} L^{1_{G^{'}}} = \infty$ and ${dim}_{F} L^{g^{'}} < \infty$ if $g^{'} \neq 1_{G^{'}}$ . In the same spirit of the case $| G |$ odd, if $| G |$ is even it is sufficient to study only those $G$ -gradings...

Incidence coalgebras of interval finite posets of tame comodule type

Zbigniew Leszczyński, Daniel Simson (2015)

Colloquium Mathematicae

Similarity:

The incidence coalgebras $K^{□} I$ of interval finite posets I and their comodules are studied by means of the reduced Euler integral quadratic form $q^{•} : ℤ^{(I)} \to ℤ$ , where K is an algebraically closed field. It is shown that for any such coalgebra the tameness of the category $K^{□} I - c o m o d$ of finite-dimensional left $K^{□} I$ -modules is equivalent to the tameness of the category $K^{□} I - C o m o d_{f c}$ of finitely copresented left $K^{□} I$ -modules. Hence, the tame-wild dichotomy for the coalgebras $K^{□} I$ is deduced. Moreover, we prove that for an interval finite...

Birational positivity in dimension $4$

Behrouz Taji (2014)

Annales de l’institut Fourier

Similarity:

In this paper we prove that for a nonsingular projective variety of dimension at most 4 and with non-negative Kodaira dimension, the Kodaira dimension of coherent subsheaves of $Ω^{p}$ is bounded from above by the Kodaira dimension of the variety. This implies the finiteness of the fundamental group for such an $X$ provided that $X$ has vanishing Kodaira dimension and non-trivial holomorphic Euler characteristic.