The cohomology tag has no wiki summary.

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**1**answer

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### What's a good example/reference for cohomology classes on Springer fibers that aren't restricted from the flag variety

As usual, by Springer fiber, I mean the fixed points $X^u$ of a unipotent element $u$ of the group $G$ on the flag variety $X=G/B$. It's a lovely theorem that when $G=SL_n$, the induced map on ...

**7**

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**1**answer

288 views

### On quadratic forms, Pontryagin Squares, and H^4(K(pi,2),U(1))

I am trying to get a concrete handle on the isomorphism $H^4(K(\pi_2,2),U(1)) \simeq \{$quadratic forms $\pi_2 \to U(1) \}$. This is explained in Eilenberg and Maclane's ...

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**0**answers

98 views

### Realization of a formal duality isomorphism via integration

This is a more precise version of my previous question. Let $X$ be a smooth variety of dimension $n$ over $\mathbb{C}$ and $Z$ a proper sub-scheme. We denote by $\tilde{X}$ the formal completion of ...

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**2**answers

234 views

### Equivariant cohomology and complex non-degenerate bilinear forms

Let $M = GL(n,\mathbb{C})$ be the set of non-degenerate bilinear forms on $\mathbb{C}^n$ (not necessarily symmetric). The general linear group $G = GL(n,\mathbb{C})$ acts on $M$ in the usual way
...

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**0**answers

406 views

### cohomology of a blowup: reference needed

Does anybody know a reference in which the computation of the cohomology of a blow-up is made in detail?

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**1**answer

262 views

### How cohomology group varies with tensor product [closed]

Let $\mathfrak{F}$ be a sheaf of abelian groups on a smooth scheme $X$. Suppose for some $i>1$, there exists a surjective morphism
$H^i(\mathfrak{F}^{\otimes i-1}) \to H^i(\mathfrak{F}^{\otimes ...

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**1**answer

304 views

### Vanishing cohomology of de-Rham Witt complex

Let $X$ be a smooth scheme over $\mathbb{F}_{p}$ for a prime number $p$. As far as I understand,
there is a surjective morphism from
$\Omega^\bullet_{W\mathcal{O}_X} \to W \Omega_{X}^\bullet$ which ...

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**0**answers

114 views

### Compare cohomology of tensor product and exterior product

Let $X$ be a smooth projective variety over a field $K$ of positive characteristic. Let $\mathcal{F}$ be a sheaf of $K-$algebras. Is there any criterion when the natural map from ...

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**2**answers

381 views

### Weights on equivariant cohomology?

Let $X$ be a quasi-projective variety over the complex numbers, equipped with an action of a linear algebraic group $G$.
Is there a natural mixed Hodge structure on its equivariant cohomology?
Is ...

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**1**answer

159 views

### Is there a prime of height $i$ in support of $H^i_I(R)$?

$I$ is an ideal of a local Noetherian ring $R$ and $i>0$ .
Clearly the height of primes in support of $H^i_I(R)$ is at least $i$
The question is if it
contains a prime of height $i$, specially ...

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**1**answer

197 views

### Seifert Fibrations and their associated Spectral Sequence

In a somewhat limited setting, a Seifert Fibre Space is a 3-manifold $M$ with a "nice" decomposition into circles (http://en.wikipedia.org/wiki/Seifert_fiber_space). That is, $M$ is decomposed into ...

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**2**answers

266 views

### Surjectivity of the Gysin morphism

Let $f:X \to Y$ be a closed immersion between smooth projective complex varieties. Suppose that the codimension of (the image of) $Y$ in $X$ is equal to $r \ge 1$. This induces the Gysin morphism ...

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**1**answer

215 views

### Chern Character Isomorphism for non-finite CW complexes, resp. for non-CW complexes

This is a question I asked at Math.SE but got no answers: http://math.stackexchange.com/q/397164/7110/
Atiyah and Hirzebruch showed in their paper "Vector bundles and homogeneous spaces" that ...

**0**

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**1**answer

165 views

### Relation between $H^i_I(-)$ and $H^i_J(-)$ when $I\subset J$

What is the relation between $H^i_I(-)$ and $H^i_J(-)$ (cohomological functors) when $I\subset J$ are ideals of a (local) noetherian ring?

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**3**answers

353 views

### Reference request: construction of Steenrod operations for an odd p

Where in literature can one find a construction of Steenrod
reduced powers (for an odd $p$) that
(1) works for the singular cohomology of arbitrary topological spaces
(or, more generally, for the ...

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**0**answers

179 views

### KK-theory by abelianized correspondences of smooth stacks?

Whith (Kasparov, bivariant) KK-theory I am left with the nagging feeling that the theory is "more fundamental", than has been made explicit, that there is a "more profound" universal characterization ...

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**1**answer

148 views

### A group 3-cocycle, trivial on a pair of generating subgroups?

I'm looking for an example of the following situation:
A group $G$ generated by finite subgroups $H$ and $K$,
a non-trivial 3-cocycle $\omega \in H^3(G, \mathbb{k}^\times)$
such that
the ...

**6**

votes

**1**answer

487 views

### Can group cohomology be interpreted as an obstruction to lifts?

The standard way to view the first and second group cohomologies is this:
The Standard Story
Let $G$ be a group, and let $M$ be a commutative group with a $G$-action. Then the first cohomology has ...

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**1**answer

304 views

### A computation by the Shapiro Lemma

Hi:
When I read the book "An introduction to Homological algebra" by Weibel, the page 206, line 9 says that
"Shapiro's Lemma tell us that
$H_q(S_n(X)\otimes_{Z}A)$ is zero if $q\neq 0$ and is ...

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**1**answer

303 views

### “Cohomology at the infinity”: what does one call it

Suppose $X$ is a "good enough" Hausdorff topological space; we assume that $X$ is not compact. Now, for a natural number $k$ and an abelian group $G$, consider the group ...

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**1**answer

304 views

### BRST cohomology definition

Is there written anywhere a full definition of BRST cohomology? All I have found so far is BRST cohomology in _______.
As far as I can see, BRST cohomology is the ...

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**0**answers

151 views

### Mayer-Vietoris on Fibered Products

Suppose $M \xrightarrow{p} X$ is a surjective submersion of smooth manifolds. Define the fibered product in this case: $$M \times_X M = \{ (m_1, m_2) \in M \times M | p(m_1) = p(m_2) \}$$
and let $U ...

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**0**answers

497 views

### Cech Cohomology of Sections of Holomorphic Bundle over Contractible Space

Is there anything that can be said in general about the Cech cohomology of the sheaf $\mathcal{F}$ of sections of a holomorphic bundle over a contractible space $X$?
I know that such a bundle must ...

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**1**answer

202 views

### cohomology of infinite product of EM spaces

Is it true that any map from an infinite product of Eilenberg-MacLane spaces in to an Eilenberg-MacLane space factor through, upto homotopy, a finite subproduct? Even if we take with field ...

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144 views

### Cohomology with compact support and the nerve of a recouvrement

Let $X$ be a simplicial complexe and we assume it localy finite and finite dimensional. We suppose taht there exist a simplicial complexe $Y$ and a map assigning to each vertex $s\in Y$ a finite ...

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**1**answer

340 views

### A formal group law over oriented bordism

My question is related to the following question by Mark Grant here on math overflow:
Formal group law of unoriented cobordism
There it is stated that $MO_*$ has a formal group law $F_0$, universal ...

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votes

**2**answers

220 views

### A question on composites of pushforward and pullback

Let a finite group $G$ acts on an orientable manifold $X$ freely. Denote $\pi:X\rightarrow Y=X/G$ be the quotient map. This covering map defines two maps between cohomology groups ...

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**1**answer

304 views

### $\pi$-cohomology class — a variant of cohomology class

Let $X$ be a topological space with a triangulation. The triangulation defines a
chain complex in $X$. Let $\mu_d$ be a cochain and $M^d$ be a chain. We use $<
\mu_d, M^d > \in M$ to denote ...

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**1**answer

352 views

### Does the Čech cohomology always yield long exact sequences from short ones?

Does the Čech cohomology always give rise to a long exact sequences given a short exact sequence of sheaves?
Clearly that cannot occur for sheaves on a paracomact (perhaps also Hausdorff, I'm not ...

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**1**answer

199 views

### The locus where a sheaf is supported in a certain dimension

I am trying to understand a particular case of this question.
Let $T$ be an affine scheme of finite type over the ground field C. Let $X \to T$ be a morphism. I'll assume this to be the base change ...

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**0**answers

164 views

### Coherent Sheaf supported in a point

Hi guys, I'm studying Cech cohomology of sheaves and I've the following doubt:
If you have a coherent sheaf $\mathcal{F}$ in a compact complex variety $X$ then the cohomology groups are finite ...

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**1**answer

281 views

### Cohomology of configuration space of a compact manifold

There is a reference or a methode which by it we can calculate the cohomology of a configuration space of a compact manifold simply connected? It is possible to find a spectral sequence converging to ...

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**1**answer

468 views

### Results from Differential Cohomology

I've been working through some notes on differential cohomology for the past few months. I feel like I have a pretty decent grasp on the concepts and its construction, at least for differential ...

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**1**answer

226 views

### Cohomology of Projective Classical Lie Groups

Let $G$ be a compact, connected, simply-connected Lie group with centre $Z(G)$, and consider the Lie group $G/Z(G)$. I believe that for $G$ a classical group, the Lie group $G/Z(G)$ is sometimes ...

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**3**answers

353 views

### Zero-cohomology of birational varieties

Let $f:X\dashrightarrow Y$ be a birational map of smooth projective varieties, i.e., there exist open subsets $U_1, \subset X$ and $U_2 \subset Y$ such that $f|_{U_1} : U_1 \rightarrow U_2$ is an ...

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**3**answers

377 views

### Computing the cardinality of cohomology groups

I hope this question is not unreasonably broad. It is about calculating or at least bounding the cardinality of cohomology groups in case they are finite.
Let us assume we are given a group $G$ and a ...

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**0**answers

158 views

### In the cohomology of Thom spectrum over LoopS^{2} and p-adic characteristic classes

Let $T$ denote the thom spectrum over $\Omega S^{2}$ defined by the map
$1+3: \Omega S^{2} \to BG_{3}$
where $1 +3$ is a unit in $3$-adics.
Here $G_{3}$ is the unit component of ...

**0**

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**1**answer

180 views

### Reference for notation $H^0(C, mK)$

I am reading the draft of "Equations of Riemann Surfaces of Genus 4, 5 and 6 wih Large Automorphism groups" and the author starts using the notation $H^0(C, mK)$ on page 3, without explaining it. As ...

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**0**answers

144 views

### cech cohomology in topos

Hi,
The following result seems to be well known, but I can't come up with a proof.
Suppose that $C$ is a topos and that $F\to G$ is an effective epimorphism in $C$. If $P$ is
any abelian sheaf on ...

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**2**answers

143 views

### Cohomology of Complements by an analytic subset?

Good moring,
Let $\Omega$ be a domain in $\mathbb{C}^n$ and $S\subset\Omega$ an analytic subset of codimension 1. What can we say about the cohomology group $H^1(\Omega\backslash S, \mathbb{Z})$? ...

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**2**answers

161 views

### Borel localization with Mayer-Vietoris sequence

We have this theorem:
Let $U$, $V$ two open sets of manifold $M$, ($U \cup V = M$). If they are $G$ -stable the induced sequence in cohomology
$$ \cdots \to H^k_G (U \cup V) \to H^k_{G}(U)\oplus ...

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**0**answers

217 views

### Real schubert calculus

Studying real enumerative questions I noticed, that the even-even Schubert varieties of the real Grassmannian $Gr:=Gr_{2k}(2n,\mathbb R)$ behave analogously to the complex case.
I call a partition ...

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**1**answer

465 views

### Essential uniqueness of the real-analytic structure on $\mathbb R$

It is well-known that any $C^k$-smooth $1$-manifold homeomorphic to $\mathbb R$ is $C^k$-diffeomorphic to $\mathbb R$. The cases of $k\in{\mathbb N}\cup$ {$\infty$} may all be handled similarly by ...

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93 views

### A certain 'coniveau-like' filtration for cohomology: what can one say about the intersection of $Ker H^i(X)\to H^i(Z)$ for $Z$ running through subvarieties of $X$ of dimension $m$?

Let $X$ be a smooth variety (say, a complex one; denote its dimension by $n$). What can one say about the intersection of $Ker (H^i(X)\to H^i(Z))$ for $Z$ running through (closed, not necessarily ...

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**1**answer

303 views

### Cohomology of quotient space

Let be $X$ a maximal torus in a Lie group $G$. I'd like to calculate the cohomology $H^{*}(G/N(T))$. I know that it is trivial in odd degree and the base-field is even but I haven't a basic method to ...

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211 views

### Cocycles for right- and left- regular representations on $\ell_2(G)$

Using the standard notation, let $\lambda_gf(x)=f(g^{-1}x)$ be the left regular representation and $\rho_gf(x)=f(xg)$ be the right regular one, acting on the space $V$ of complex-values functions on ...

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**1**answer

227 views

### What can we say about the map $\pi^{*}\pi_{*}$ for $\pi:X\rightarrow X/G$?

Let $X$ be a manifold or scheme with a finite group $G$ acting on it freely. Let $\pi:X\rightarrow X/G$ be the natural projection. We have $\pi_{\*}\pi^{\*}=|G|id$ on $H^*(X/G,\mathbb{Z})$. Can we say ...

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**1**answer

188 views

### Endomomorphisms of Chain Complexes of vector spaces and determinants

Let $C_{\ast} : \cdots \to A_{2} \to A_{1} \to A_{0} \to 0$ be a chain complex of finite dimensional vector spaces over a field $K$.
And let $f_{\ast} : C_{\ast} \to C_{\ast}$ and $g_{\ast} : ...

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**0**answers

142 views

### A nice way to verify whether the Neron-Severi group of a smooth affine variety is zero

Let $S$ be a smooth affine variety over an algebraically closed field (this could be the field of complex numbers). Is there an 'easy' way to verify whether $NS(S)=0$? Unfortunately, I don't know how ...

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167 views

### Evens norm as a derived functor

In the theory of cohomology of groups there are several important maps between cohomologies: restriction, inflation, transfer and Evens norm. The first three of them can be clearly seen on the level ...