Questions tagged [cohomological-dimension]
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13 questions
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Cohomological dimension of functors from fields to vector spaces
Let $K$ be an algebraically closed field. Denote by $\mathcal F_d$ the category of extensions $K\to F$ of transcendent degree $d$.(Objects are pairs $F,j$ consisting of a field $F$ and the embedding $...
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On the homological dimension of a Borel construction
Let $M$ b a closed connected smooth manifold with fundamental group $\Gamma$. Suppose $G$ is a simply-connected Lie group that acts smoothly on $M$. Then the Borel construction $$M//G = M \times_G EG$$...
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Cohomological dimension of continuous étale cohomology of finitely generated fields
Given a finitely generated field $F$ with prime field $k$, we assume $k$ is finite, of characteristic $p$. Fix a prime $\ell$ invertible in $k$.
In the discussion right after [K, Lemma 2.3], the ...
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Counterexamples concerning $\infty$-topoi with infinite homotopy dimension
In "Higher Topos Theory", Lurie introduces three different notions of dimension for an $\infty$-topos $\mathcal{X}$, namely:
Homotopy dimension (henceforth h.dim.), which is $\leq n$ if $n$-...
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Rational cohomological dimension of the lattices in $SL_n(\mathbb{Q}_p)$
What is the rational cohomological dimension of the lattices in $SL_n(\mathbb{Q}_p)$, where $n\geq 3$ ? A reference would be appreciated.
For the definition of "cohomological dimension of a group ...
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Does the singular cohomology for a metric space of finite topological dimension vanish in high dimensions?
It is known that by applying the universal coefficient theorem, the singular cohomology of closed manifold with coefficient $\mathbb{Z}_2$ vanishes in high dimensions. But for a metric space $M$ with ...
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what's the cohomological dimension of a Stein space?
I want to know the "cohomological dimension" of a Stein space.
I know that:
for $X$ differential manifold and for every sheaf $F$ of abelian
groups, I have $H_c^j(X,F)=H^j(X,F)=0$ for $j>...
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What is the intuition for $\mathbb{Q}^{ab}$ having cohomological dimension $1$?
I frequently talk to people who think of finite fields as arithmetic analogs of punctured discs. This makes some sense: the absolute Galois group of a finite field is the profinite completion of $\...
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Continued fractions and projective resolutions
Hello,
This question might be vague and not thought-through enough.
If we have a real positive number $x$, we can start to write it as a continued fraction:
$x = a_0 + \frac{1}{x_1} , \ldots , x_n=...
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cohomological dimension for coarser/finer topologies
Given a sheaf $\mathcal{F}$ with respect to some Grothendieck topology, is the cohomological dimension for this sheaf less than or equal to the cohomological dimension of a finer topology?
Example: $...
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Cohomological dimension of $\mathcal{B}_n$
What is the cohomological dimension of the braid group $\mathcal{B}_n$ on n-strands ?
A reference would be appreciated.
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cohomological dimension of the push-forward functor
Let $\ell$ be a prime number and let $f:X \to Y$ be a morphism of schemes of finite type over the complex numbers (or a regular scheme of dimension at most 1, in which $\ell$ is invertible). How to ...
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Do the homological dimension and cohomological dimension for a group agree?
Or equivalently, if $G$ is a group, do the projective and injective dimension of $Z$ (viewed as a $ZG$-module) agree?
Thanks!
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