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13
votes
0answers
325 views

Combinatorics of Quantum Schubert Polynomials

Let $S_n$ be the symmetric group. Let $s_i$ denote the adjacent transposition $(i \ i+1)$. For any permutation $w\in S_n$, an expression $w=s_{i_1}s_{i_2}\cdots s_{i_p}$ of minimal possible length is ...
0
votes
1answer
328 views

Exact sequences of the cohomology induced by fiber bundle

I'm reading section 2.1 of Lawson's book, Spin Geometry. The book states the following fact. Let $X$ be a manifold and $E$ a vector bundle over it. Equip $E$ with a Riemannian structure. Let $P_O$ be ...
3
votes
2answers
210 views

1st cech cohomology groups on ringed sites

Let $(C, O)$ be a ringed site -- i.e., $C$ is a small category with a grothendieck topology $\tau$ and $O$ a sheaf of rings on the site $(C,\tau)$. In this context, for any object $U$ of $C$ one can ...
2
votes
0answers
164 views

“Extended” Weil Cohomology Theories

According to Wikipedia, a Weil cohomology theory is a functor from the category of smooth projective varieties over a field $k$, to graded algebras over a field $K$ of characteristic zero, together ...
0
votes
0answers
69 views

Hit problem and $\left( \mathbb{F}_2 \otimes_{A} P_6 \right)_{10}$

I try to determine the $\left(\mathbb{F}_2 \otimes_{A} P_6 \right)_{10}$ as a $\mathbb{F}_2$- vector space, in which $P_6$ is polynomial algebra $\mathbb{F}_2[x_1,x_2,\dots,x_6]$ and $A$ is Steenrod ...
1
vote
0answers
194 views

What is your expectation of the depth?

Let $S=k[x_1,...,x_9]$ be a polynomial ring over field $k$. Set $q_1=(x_1,x_2,x_5,x_6)$, $q_2=(x_1,x_2,x_6,x_7)$, $q_3=(x_2,x_3,x_7,x_8)$, $q_4=(x_1,x_5,x_6,x_7)$, $q_5=(x_1,x_6,x_7,x_8)$, ...
1
vote
1answer
316 views

A generalization of cochain complex: quasi-cochain complex

It appears that we can generalize cochain complex to quasi-cochain complex, that still allow us to define cohomology. Definition: A quasi-cochain complex is a sequence of commutative monoids $M_n$ ...
5
votes
2answers
257 views

How does Tate cohomology fit into a derived categories framework?

I've read through one class field theory text after another, but there's something very non-intuitive for me about cohomology that makes it hard for me to understand why Tate cohomology was invented. ...
1
vote
1answer
123 views

first cohomology of exterior power of tangent bundle

suppose G is a Grassmannian manifold, and TG is the tangent bundle. By Bott's theorem $H^1(G, T_G)=0$. Is it true that $H^1(G, \bigwedge^i T_G)=$, for i>0. I saw some vanishing result by lepotier, ...
2
votes
2answers
198 views

Confusion about two statements about cohomology of curves with automorphisms

Let $\pi :C\rightarrow \mathbb{P}^{1}$ be a cyclic cover of degree $m$ of $\mathbb{P}^{1}$. So $C$ has an action of $\mathbb{Z}/m\mathbb{Z}$. Let $\xi$ be a primitive $m$-th root of unity. Consider ...
0
votes
0answers
52 views

on relative divisors over artinian rings

Let $X$ a curve over $\mathbb{C}$, $D$ a divisor on $X$, $R$ a local artinian ring of residue field $\mathbb{C}$ Let $A=H^{0}(X_{R},\mathcal{O}(D_{R}))$ the scheme of sections over $Spec(R)$. Let ...
2
votes
2answers
155 views

Preimage of $1 \in H^n(M^n)$ under Chern character

Let $M$ be a closed, oriented manifold of dimension $n$. We know that the Chern character induces an isomorphism $K^\ast(M) \otimes \mathbb{Q} \cong H^\ast(M; \mathbb{Q})$ and now I was wondering how ...
3
votes
1answer
587 views

Pontryagin square on $Y\times S^1$ where $Y$ is three-dimensional

Let $Y$ be an oriented manifold of dimension three, and let $X=Y\times S^1$. We have $$H^2(X,\mathbb{Z}_2)=H^2(Y,\mathbb{Z}_2)\oplus H^1(Y,\mathbb{Z}_2).$$ Pick an element $m\oplus n\in ...
3
votes
0answers
96 views

consistent orientation for functorial pull-push in generalized cohomology

Given a correspondence $X_{\mathrm{in}} \stackrel{i_{\mathrm{in}}}{\leftarrow} X \stackrel{i_{\mathrm{out}}}{\to} X_{out}$ of suitable "spaces" of sorts, and given a (generalized) cohomology theory ...
16
votes
2answers
521 views

KK-theory as a stable infinity-category and KU Mod

The category KK of bivariant operator K-theory (or possibly its E-theory variant) ought to be the homotopy category of something at least close to a stable infinity-category; notably in that it ...
2
votes
0answers
212 views

Global sections of a coherent sheaf in terms of a presentation

Let $A$ be a graded ring satisfying the usual finiteness conditions of EGA II (for example $A_0$ is noetherian, $A_1$ is finite over $A_0$ and $A$ is generated by finitely many elements of $A_1$ as an ...
7
votes
1answer
213 views

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
votes
1answer
258 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 ...
1
vote
0answers
93 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 ...
3
votes
2answers
225 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 ...
1
vote
0answers
223 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?
-2
votes
1answer
238 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 ...
6
votes
1answer
235 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 ...
0
votes
0answers
108 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 ...
6
votes
2answers
344 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 ...
1
vote
1answer
153 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 ...
4
votes
1answer
172 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 ...
0
votes
2answers
208 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 ...
2
votes
1answer
148 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
votes
1answer
140 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?
3
votes
1answer
206 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 ...
9
votes
0answers
153 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 ...
3
votes
1answer
143 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 ...
5
votes
1answer
394 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 ...
1
vote
1answer
246 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 ...
8
votes
1answer
297 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 ...
7
votes
1answer
278 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 ...
1
vote
0answers
134 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 ...
1
vote
0answers
468 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 ...
1
vote
1answer
189 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 ...
1
vote
0answers
135 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 ...
7
votes
1answer
317 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 ...
0
votes
2answers
203 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 ...
1
vote
1answer
296 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 ...
1
vote
1answer
303 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 ...
0
votes
1answer
180 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 ...
0
votes
0answers
116 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 ...
1
vote
1answer
255 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 ...
10
votes
1answer
399 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 ...
3
votes
1answer
197 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 ...