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3
votes
2answers
533 views

Stiefel-Whitney classes of a projective space bundle

Hi! Let $\gamma_1$ denote the twisted line bundle over $S^1$ and add a trivial $(2k-1)$-bundle $\mathbb{R}^{2k-1}$. Consider the projective bundle $P(\gamma_1 \oplus \mathbb{R}^{2k-1})$ over $S^1$. ...
2
votes
0answers
192 views

Cohomology of a projective limit

If we consider an oriented locally finite graphs $X$, and we suppose that $X$ is the projective limit of a familly of oriented locally finite graphs : ...
6
votes
1answer
302 views

Continuous and smooth Lie groupoid cohomology

In the paper by Weinstein and Xu: Extensions of symplectic groupoids and quantization, J. Reine Angew. Math. 417 (1991), there are two versions of Lie groupoid cohomology. The same differential ...
1
vote
1answer
252 views

Coboundary map on the cochain complex of abelian cosimplicial groups?

Maybe I'm looking at the wrong places, but I can't find a definition of the coboundary map on the cochain complex of abelian cosimplicial groups. What I have in mind is something similar to the ...
9
votes
1answer
490 views

The coproduct on the cohomology of a Hopf algebra

If $H$ is a Hopf algebra over a field $k$, the Hopf algebra cohomology $\mathrm{Ext}_H(k,k)$ is —as usual— an algebra, but the Hopfness of $H$ turns it into a Hopf algebra. Is there a reference ...
2
votes
1answer
237 views

Do cohomologically trivial line bundles affect morphisms?

Assume we have a locally free sheaf $R$ of associative $O_S$-algebras of rank $r^2$, possibly noncommutative, where $S=\mathbb{P}^2$ over $\mathbb{C}$. Furthermore given a locally free $O_S$-module ...
1
vote
1answer
346 views

vanishing of cohomology sheaves with supports and values in the multiplicative group

Let $X$ be a locally noetherian regular scheme and $Y$ be a closed subscheme of codimension $d > 0$ in every point. Why does it "immédiatement" (Grothendieck, Groupe de Brauer III, §6, p. 133 f.) ...
3
votes
0answers
186 views

limit of étale cohomology with supports

I would like to know if the following generalisation of Milne Étale Cohomology, Lemma III.1.16, p. 88 holds: (all cohomology groups with respect to the étale topology) Let $Y \hookrightarrow X$ ...
3
votes
0answers
151 views

Branched Covers of a Two-sphere with Infinite Degree

I would like to ask the following question: Let us consider the following branched cover $X_n$ of $\mathbb{P}^1$ that can be thought as a hyper-surface of $\mathbb{P}^1 \times \mathbb{P}^1$: ...
1
vote
1answer
754 views

Penner's formula for volume of the Moduli Space

In his paper "Weil-Petersson Volumes" Penner gives the following formula for the integral of a top-dimensional cohomology class $\omega$ on the moduli space $\mathcal M_g^s$ of $s$-punctured riemann ...
1
vote
1answer
176 views

combination of two duality theorems

For a variety $X$ over a finite field $k$, one combines étale Poincaré duality for $X \times_k \bar{k}$ with duality for $k$ ($H^0(k,M) \times H^1(k,M^\vee) \to H^1(k,\mathbf{Z}/n) = \mathbf{Z}/n$) to ...
6
votes
0answers
351 views

Cohomology of a space with coefficients in a Lie group?

The following is a very naive construction, and I am almost embarrassed to ask questions about it. Suppose I have a connected simple graph $\Gamma = (\Gamma_0,\Gamma_1)$ (i.e., vertex set, edge set) ...
4
votes
1answer
386 views

Formal Group Laws on Ring Spectra?

Given a (graded) ring $R$, to define a formal group law it is equivalent to define a ring homomorphism $\phi:L\to R$ where $L$ is Lazard's ring. Is there any notion of defining a formal group law on ...
5
votes
2answers
700 views

Are there 'cohomology' functors that respect all Eilenberg-Steenrod axioms except homotopy invariance?

What goes wrong in the axiomatic definition of a generalized (co)homology theory if one drops the axiom of homotopy invariance i.e. that homotopic maps should induce the same map in (co)homology? Or ...
2
votes
1answer
330 views

Which (reducible) projective varieties could be presented as 'relatively smooth' hyperplane sections of irreducible (normal) ones?

Are there any restrictions known on a (complex reducible) projective variety $Y$ that can be presented as $Z\cap H$, where $Z$ is a normal (or just irreducible) closed subvariety of a smooth ...
3
votes
1answer
375 views

Are Kazhdan-Lusztig $R$-polynomials the Poincare polynomials of the corresponding affine varieties

Let $x$ and $y$ be two elements of $S_n$. Let $U(x,y)$ be the intersection $(BxB \cap B_{-} y B)/B$ inside the flag variety. Here $B$ and $B_{-}$ are the groups of upper and lower-triangular matrices ...
2
votes
0answers
303 views

understanding Milne's article “Duality in the flat cohomology of a surface”

I have trouble understanding a point of Milne's article "Duality in the flat cohomology of a surface" http://jmilne.org/math/articles/1976a.pdf see the "Alternatively" on p. 177, paragraph before ...
0
votes
3answers
377 views

Cohomology of complexes

I sometimes see cohomologies of complexes of sheaves. What is the definition of these? Say if $\mathcal F^* $ is a complex of sheaves on $C$, what is $H^i(C,\mathcal F^*) $?
3
votes
1answer
305 views

Does the (torsion) Zariski cohomology of a (singular) hyperplane section of a smooth projective variety vanish (in small degrees)?

Let $P$ be a smooth connected projective variety (say, over complex numbers); $H$ is its smooth hyperplane section. What can be said about the Zariski cohomology of $H$ with constant coefficients? It ...
4
votes
0answers
302 views

Determinant line does not depend on the differential

Let $\pi:X\rightarrow S$ be a proper morphism of schemes over $\mathbf{C}$ and $\mathcal{E}$ a vector bundle on $X$ with a relative flat connection $\nabla_{X/S}$. Is it true that the determinant line ...
5
votes
3answers
565 views

Continuous cohomology of semi-simple Lie group.

Let $G$ be a real connected semi-simple Lie group. Let $M$ be a finite dimensional representation of it. Are there general criteria when the continuous cohomology groups $H_{cont}^q(G,M)$ vanish? A ...
3
votes
2answers
664 views

Group Cohomology of Symmetric Powers

Let $G$ be a finite group and $V$ be an integral representation of $G$, i.e. a free abelian group of finite rank with $G$-action. Now consider the symmetric power $Sym(V)$ of $V$ over $\mathbb{Z}$, ...
4
votes
4answers
782 views

Cohomology groups of homogeneous spaces

Is there a general method to calculate the cohomology groups of homogeneous spaces ($G/H$), such as $\frac{U(4)}{U(2)\times U(2)}$, $\frac{U(5)}{U(2)\times U(3)}$, $U(4)/U(2)$, etc. If yes, could you ...
3
votes
2answers
481 views

Alexander duality theorem for CW-complexes and stable homotopy theory

In Adams, J.F. Infinite Loop Spaces Princ. Univ. Press. page 9 he states Alexander duality theorem Theorem:[Alexander Duality] $$ H^r(X,G)=H_{n-r+1}(S^n-X,G)$$ for finite CW-complexes with a "nice ...
3
votes
1answer
244 views

Large modules with non-trivial cohomology

Let $p$ be a prime and $F$ algebraic closer of $F_p$. I want to know if it is possible to construct family of groups $\{G_i\}_{i=1}^{\infty}$ and a family of simple modules $V_i$ over $F[G_i]$ of ...
11
votes
0answers
304 views

Cohomological interpretations of quadratic form invariants over rings?

The standard approach to classifying of quadratic forms over $\Bbb Q$ is to use the Hasse (local-global) principle together with a system of standard invariants of quadratic forms over the local ...
6
votes
2answers
293 views

Cohomology of fixed point subspaces

Suppose $M$ is a smooth manifold and $\phi : M \to M$ is a homeomorphism whose fixed point set is a smooth submanifold $M_{\phi}$. Is there any relation between the cohomology ring of $M_{\phi}$ and ...
1
vote
1answer
229 views

Vectorial extensions of abelian varities

Hi, this is a very vague question, but I'm also glad about vague answers... One knows that for an abelian variety over the complex numbers, one has a canonical exact sequence $0\rightarrow ...
4
votes
2answers
488 views

Hodge theory and varieties defined over subfields of the complex numbers

This question is related to the question: Is there a $k$-structure for Hodge modules over a $k$-variety?. Suppose $K$ is a subfield of $\mathbb{C}$ and $M$ is a holonomic $D$-module "of geometric ...
2
votes
0answers
183 views

The Mayer-Viertoris exact sequence as a (Zariski) descent spectral sequence.

For certain 'spaces' $U,V$ (they are certain Henselizations of subvarieties) I would like to compute (certain etale) cohomology of $U\cup V$ in terms of the corresponding cohomology of the diagram ...
0
votes
0answers
246 views

spectral sequence of cohomology with compact support

Let k be a field, and A, B is subscheme of C over k, F is a sheaf over C. Question 1. what is the relation between $H_c(A\cap B,k)$, $H_c(B,F)$, and $H_c(A\cap B,F)$ ? Question 2. Let A be a strata ...
10
votes
1answer
542 views

Is de Rham cohomology of affine schemes over discrete valuation rings finitely generated (modulo torsion)?

Let $U$ be a open affine subscheme of a smooth, proper scheme $X$ over $\mathbf{Z}_p$. Over $\mathbf{Q}_p$ we know that $\mathrm{H}^i(U \times \mathbf{Q}_p/\mathbf{Q}_p)$ is finite-dimensional (where ...
3
votes
2answers
634 views

Brauer group of projective space

I've read that $\text{Br} \mathbb{P}^n_k$ (here $\text{Br}$ is the cohomological Brauer group, i.e. $H^2_{ét}(-,\mathbb{G}_m)$) is just isomorphic to $\text{Br} k$. As proof of this fact seems to be ...
3
votes
1answer
212 views

Deforming ample line bundles vs cohomology group

Let X be a smooth projective variety over the complex numbers, of dimension at least two. $D$ is an ample divisor on X. Then we know for $m>>0$, $H^i(mD)=0$. Now suppose $E$ is another divisor ...
10
votes
3answers
927 views

Is there a good definition of (topological) K-Theory over arbitrary spaces?

Hi (this is my very first question here, so please don't hurt me...) for some time now i've been looking for a sufficiently aesthetical definition of (topological) K-theory of arbitrary spaces, yet ...
8
votes
4answers
556 views

$Sq^1$ cohomology of spaces

For any space $X$, the first Steenrod square cohomology operation $$Sq^1\colon H^\ast(X;\mathbb{Z}_2)\to H^{\ast +1}(X;\mathbb{Z}_2)$$ is a derivation, meaning that $Sq^1\circ Sq^1 = 0$ and ...
3
votes
4answers
631 views

Where to start with research regarding maslov index/class

Hi, I am a physicist and currently doing my bachelor thesis about geometric quantization. In the book by Bates and Weinstein I encountered the Maslov index, which seems to be very important :-). But ...
1
vote
0answers
286 views

Does n-multiplication maps of cohomology groups vanish if it vanishes at the 0th cohomology?

In general, we know that a morphism $f=(f ^ {q})$ between universal (cohomological) $\delta$ functors $S=(S ^ {q}),T=(T ^ {q})\ $vanishes if and only if $f ^ {0} \ \colon \ S^{0} \to T^{0}$ ...
4
votes
0answers
180 views

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: ...
5
votes
1answer
496 views

On Brown representability theorem

The classical Brown representability theorem is for set valued functors. Is there a version for abelian group valued functors, and ring valued functors? In other words say we have an abelian group ...
16
votes
1answer
582 views

How are these algebraic and geometric notions of homotopy of maps between manifolds related?

Let $M$ and $N$ be smooth manifolds, and $f,g: M \to N$ smooth maps. Denote by $(\Omega^\bullet M,\mathrm d_M)$ and $(\Omega^\bullet N, \mathrm d_N)$ the cdgas of de Rham forms in each manifold, and ...
2
votes
0answers
273 views

Do non-ordinary Bredon cohomology theories extend?

As shown by Lewis, May, and McClure (MR0598689), the ordinary equivariant Bredon cohomology theory $H^*_G(-; M)$ extends to an $RO(G)$-graded cohomology theory precisely when the coefficient system ...
0
votes
1answer
225 views

Generalized Picard group (reflexive fractional ideals, principal ideals)

Given $\mathcal{O}=k[[u,v]]$ with maximal ideal $\mathfrak{m}$ and an $\mathcal{O}$-algebra $A$, free as an $\mathcal{O}$-module of rank $n^2$. $A$ is genertaed by two elements $x,y$ with $x^n=u$, ...
4
votes
1answer
909 views

Cohomology theory for symplectic manifolds

Suppose I have a symplectic manifold $(M,\omega)$ and a line bundle $\mathcal L$ with a connection with curvature $\omega$ (or perhaps it's more standard to say $\frac i{2\pi}\omega$; anyway, the ...
2
votes
1answer
290 views

Sequences of groups, exact not just in étale but also in the Zariski topology

Let $A=k[[x,y]]$, with say $k=\mathbb{C}$ and $X = \operatorname{Spec}(A)$. Let $B$ denote a free $A$-algebra of rank $e^2$; actually, we have $B=A[u,v]$ with $u^e=x$, $v^e=y$ and $uv = \xi_e vu$, ...
1
vote
1answer
231 views

Cohomology of the general linear group on punctured spectra of 2-dim. power series rings

Let $(A,\mathfrak{m})=k[[x,y]]$ with $char(k)=0$ and $K=Quot(A)$. Set $X=Spec(A)$, $U=Spec(A)\backslash \lbrace \mathfrak{m} \rbrace$ the pointed spectrum. Furthermore given an $A$-algebra $B$, which ...
5
votes
1answer
617 views

A simple proof of the Weyl algebra's rigidity.

I am wondering if there is a nice presentation of the Hochschild cohomology of $A_n$ the Weyl algebra. It is known that $H^m(A_n,A_n)=0$ for $m>0$, and thus it is rigid. A proof can be found in ...
10
votes
1answer
654 views

Torsion for Lie algebras and Lie groups

This question is about the relationship (rather, whether there is or ought to be a relationship) between torsion for the cohomology of certain Lie algebras over the integers, and torsion for ...
13
votes
4answers
2k views

Singular Homology/Cohomology as a derived functor?

Hello, Learning some Alg.geometry and Sheaf theory, I got used to the notion that cohomology arises naturally as a derived functor of some sort. This has led me thinking, singular cohomology, from ...
1
vote
1answer
288 views

Proof that higher cech cohomology groups vanish for fine sheaves.

So I've been trying to understand a proof of the fact that $H^p(M, \mathcal F)=0$ whenever $p\geq 1$ from page 42 of the book "Principles of Algebraic Geometry" by Griffiths and Harris. The proof is ...