Expand all Collapse all | Results 1 - 5 of 5 |
1. CMB 2013 (vol 57 pp. 335)
Alexandroff Manifolds and Homogeneous Continua ny homogeneous,
metric $ANR$-continuum is a $V^n_G$-continuum provided $\dim_GX=n\geq
1$ and $\check{H}^n(X;G)\neq 0$, where $G$ is a principal ideal
domain.
This implies that any homogeneous $n$-dimensional metric $ANR$-continuum is a $V^n$-continuum in the sense of Alexandroff.
We also prove that any finite-dimensional homogeneous metric continuum
$X$, satisfying $\check{H}^n(X;G)\neq 0$ for some group $G$ and $n\geq
1$, cannot be separated by
a compactum $K$ with $\check{H}^{n-1}(K;G)=0$ and $\dim_G K\leq
n-1$. This provides a partial answer to a question of
Kallipoliti-Papasoglu
whether any two-dimensional homogeneous Peano continuum cannot be separated by arcs.
Keywords:Cantor manifold, cohomological dimension, cohomology groups, homogeneous compactum, separator, $V^n$-continuum Categories:54F45, 54F15 |
2. CMB 2011 (vol 54 pp. 619)
Artinian and Non-Artinian Local Cohomology Modules Let $M$ be a finite module over a commutative noetherian ring $R$.
For ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$, the relations between
cohomological dimensions of $M$ with respect to $\mathfrak{a},
\mathfrak{b}$,
$\mathfrak{a}\cap\mathfrak{b}$ and $\mathfrak{a}+ \mathfrak{b}$ are studied. When $R$ is local, it is
shown that $M$ is generalized Cohen-Macaulay if there exists an
ideal $\mathfrak{a}$ such that all local cohomology modules of $M$ with
respect to $\mathfrak{a}$ have finite lengths. Also, when $r$ is an integer
such that $0\leq r< \dim_R(M)$, any maximal element $\mathfrak{q}$ of the
non-empty set of ideals $\{\mathfrak{a} : \textrm{H}_\mathfrak{a}^i(M)
$ is not artinian for
some $ i, i\geq r \}$ is a prime ideal, and all Bass numbers
of $\textrm{H}_\mathfrak{q}^i(M)$ are finite for all $i\geq r$.
Keywords:local cohomology modules, cohomological dimensions, Bass numbers Categories:13D45, 13E10 |
3. CMB 2007 (vol 50 pp. 588)
Cohomological Dimension and Schreier's Formula in Galois Cohomology Let $p$ be a prime and $F$ a field containing a primitive $p$-th
root of unity. Then for $n\in \N$, the cohomological dimension
of the maximal pro-$p$-quotient $G$ of the absolute Galois group
of $F$ is at most $n$ if and only if the corestriction maps
$H^n(H,\Fp) \to H^n(G,\Fp)$ are surjective for all open
subgroups $H$ of index $p$. Using this result, we generalize
Schreier's formula for $\dim_{\Fp} H^1(H,\Fp)$ to $\dim_{\Fp}
H^n(H,\Fp)$.
Keywords:cohomological dimension, Schreier's formula, Galois theory, $p$-extensions, pro-$p$-groups Categories:12G05, 12G10 |
4. CMB 2001 (vol 44 pp. 266)
Extension of Maps to Nilpotent Spaces We show that every compactum has cohomological dimension $1$ with respect
to a finitely generated nilpotent group $G$ whenever it has cohomological
dimension $1$ with respect to the abelianization of $G$. This is applied
to the extension theory to obtain a cohomological dimension theory condition
for a finite-dimensional compactum $X$ for extendability of every map from
a closed subset of $X$ into a nilpotent $\CW$-complex $M$ with finitely
generated homotopy groups over all of $X$.
Keywords:cohomological dimension, extension of maps, nilpotent group, nilpotent space Categories:55M10, 55S36, 54C20, 54F45 |
5. CMB 2001 (vol 44 pp. 80)
Constructing Compacta of Different Extensional Dimensions Applying the Sullivan conjecture we construct compacta of certain
cohomological and extensional dimensions.
Keywords:cohomological dimension, Eilenberg-MacLane complexes, Sullivan conjecture Categories:55M10, 54F45, 55U20 |