This tag is for questions relating to cohomology groups and cochain complexes.

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2
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2answers
55 views

Cohomology groups of coherent sheaves for very small and very big twists.

Let $\mathcal{F}$ be nonzero coherent sheaf over the projective space $\mathbb{P}_k^n$. The Serre vanishing Theorem says that $h^i \mathcal{F}(d)=0$ for $i>0$ and $d\gg 0$. I am wondering if it is ...
3
votes
1answer
61 views

Euler characteristic, genus and cohomology: a deep connection?

For a smooth projective curve $V$ over the complex numbers, the algebraic genus, defined as the dimension of the linear system $L(\omega)$, where $\omega$ is the canonical divisor, coincides with the ...
3
votes
0answers
29 views

non-abelian Galois cohomology

Let $1 \to A \to B \to C \to 1$ be a short exact sequence of (not necessarily abelian) $G$-modules. Passing to non-abelian cohomology, we have the exact sequence of pointed sets $$ 1 \to A^G \to B^G ...
0
votes
1answer
41 views

Restricting the DeRham cohomology class of a submanifold to a coordinate neighborhood.

Suppose $M$ is an $n$-manifold and $A$ a $k$-dimensional submanifold, both compact and oriented. Let the deRham cohomology class of $A$ be denoted $[\phi_A]$. The class is defined by ...
0
votes
0answers
12 views

Cohomology of Cech for contractible spaces [on hold]

Prove that $H^p(X,G)=0$ for X contractible space and G any group.
1
vote
2answers
80 views

Factor sets and group extensions (Homological algebra- Hilton and Stammbach VI.10.1)

Show that an extension $$A\xrightarrow{i} E\xrightarrow{p} G$$ may be described by a factor set, as follows. Let $s:G\rightarrow E$ be a secion so that $ps=1_G$. Every elmenet of $E$ is of the form ...
1
vote
1answer
35 views

How does one actually take the dual of a chain complex?

I know the following about the chain complex used for computing the homology groups of the torus $S^1 \times S^1$: The complex is $0 \to^{\delta_3} \mathbb{Z}[U] \oplus \mathbb{Z}[L] \to^{\delta_2} ...
2
votes
0answers
40 views

Question about Tate resolution and cohomology groups of nonzero coherent sheaves.

Let $\mathcal{F}$ be a nonzero coherent sheaf on the projective space $\mathbb{P}_{k}^m$. I am trying to show that for every integer $d$ there is $j$ for which $h^j\mathcal{F}(d-j) \neq 0$. My ...
1
vote
0answers
37 views

$H^1(X, G_1 \times G_2)$ in terms of simpler first cohomologies

Let $X$ be a variety over a field $k$ of characteristic 0 (not necessarily with $k = \bar{k}$). Let $G_1, G_2$ be linear algebraic groups over $k$. My question is: Can I write $H^1(X, G_1 \times ...
1
vote
0answers
9 views

$H^q(\mathfrak{g},K;V)$ is equal to $Ext_{\left(\mathfrak{g},K\right)}^q\left(\mathbb{C},V\right)$?

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Let $K$ be a closed subgroup of $G$ with corresponding Lie subalgebra $\mathfrak{k}$. Let $V$ be a $\left(\mathfrak{g},K\right)$-module. Then, I ...
3
votes
0answers
47 views

Short proof of Borsuk-Ulam's

By examining the singular cohomology ring with $\mathbb{Z}/2\mathbb{Z}$ coefficients, it is easy to see that if $n>m$ that there can be no map $f:\mathbb{R}P^{n}\to \mathbb{R}P^m$ that induces ...
0
votes
0answers
8 views

1-forms as a direct sum

Let us define the spaces $C_0, C_1$ and $C_2$ of differential $0,1,2$ forms respectively on the sphere $S^2.$ Is it true that $C_1$ is the direct sum of $d(C_0) \oplus \delta^* (C_2)$? I think this ...
2
votes
1answer
61 views

Are generalized cohomology theories, spectra, and infinite loop spaces all the same thing up to homotopy?

More specifically, John Baez mentions here that the following 3 things are equivalent (up to some technicalities). the isomorphism classes of complex line bundles over $X$ the homotopy classes of ...
0
votes
1answer
29 views

Basic question: $H^1$ and $H^{0,1}$

Please could you explain why for a smooth projective variety over $\mathbb{C}$ (or - if you prefer the analytic world - compact complex manifold) $T$ we have $H^1(\mathcal{O}_T)\simeq H^{0,1}(T)$ as ...
4
votes
1answer
70 views

Compute the cohomology of projective schemes

In Hartshorne's book, Section 3.5, the cohomology of projective spaces is computed. How to compute the cohomology of projective schemes? Maybe the general case is complicated, please look at the ...
0
votes
0answers
12 views

antiholomorphic involution action on resolved orbifold of tori

Consider the space $\tilde X= T^2 \times T^2 \times T^2 / G$ where $T^2$ is a torus and $G=Z_2 \times Z_2$ acts as $\theta_1:(z_1,z_2) \mapsto (-z_1,-z_2)$, $\theta_2:(z_2,z_3) \mapsto (-z_2,-z_3)$ ...
2
votes
1answer
19 views

Morphism $H^1(k, G) \to H^1(X,G)$

I have a $k$-variety $X$, and some algebraic group $G$ (not necessarily commutative) over $k$. Here, $k$ is a field of zero characteristic. I know that the functor $H^1(-,G)$ is contravariant, meaning ...
1
vote
0answers
5 views

coboundary operators in relative lie algebra cohomology

I am starting to read relative lie algebra cohomology. We define the coboundary operator $d$ from $Hom_K(\wedge^q\mathcal{g}/\mathcal{k}, V)$ to $Hom_K(\wedge^{q+1}\mathcal{g}/\mathcal{k}, V)$ as ...
0
votes
1answer
31 views

turning a map into a fibration

In Allen Hatcher's book Spectral Sequence page 29 Example 1.18, What means "turning the map into a fibration" and convert a map into a fibration"? Given a map $f:X\to Y$, $f$ is not necessarily a ...
2
votes
2answers
49 views

Confusion about cohomology and universal coefficients theorem.

I want to check that my understanding is correct about cohomology. Let $X$ be a topological space $G$ be an abelian group. The universal coefficients theorem, as stated in hatcher, says that the ...
2
votes
1answer
44 views

Hartshorne Corollary 9.4

I am reading the section of Flatness from Hartshorne. I have a doubt in the proof of the corollary of the following proposition : $\textbf{Proposition 9.3}$ Let $f:X\longrightarrow Y$ be a separated ...
1
vote
1answer
61 views

cohomology of suspension

Let $X$ be a topological space. Let $\Sigma$ be suspension. Does $H^n(X;\mathbb{Z})\cong H^{n+1}(\Sigma X;\mathbb{Z})$ isomorphic or not? Does $H^n(X;\mathbb{Z}_2)\cong H^{n+1}(\Sigma ...
3
votes
0answers
51 views

Hasse's theorem on elliptic curves over finite fields

Suppose $\mathcal{E}$ is an elliptic curve defined over $\mathbb{Q}$, Then Hasse's theorem states that for any large characteristic $p$ there exists an algebraic number $\lambda_p$ of modulus ...
3
votes
1answer
46 views

homology of smash product of Eilenberg-Maclane spaces

Let $K_n=K(\mathbb{Z},n)$ be the Eilenberg-Maclane space. Prove: (1). $K_m\wedge K_n$ is $(m+n-1)$-connected. (2). $H_{m+n}(K_m\wedge k_n;\mathbb{Z})= H_{m+n}(K_m\times k_n;\mathbb{Z})$. How to ...
3
votes
2answers
58 views

Homology of $n$-sheeted covering space

Let $X$ be the Klein bottle, that is $X=\mathbb{R}^2/G$ with $$G=\langle a,b\mid a^{-1}b ab=1\rangle,$$ acting via $a: \mathbb{R}^2\to \mathbb{R}^2, (x,y)\mapsto (x+1,y)$, $b: \mathbb{R}^2\to ...
0
votes
0answers
11 views

stable splittings of projective space

On Hatcher's book Algebraic Topology, page 468 Prop. 4I.3, For prime number $p$, can we decompose $\mathbb{C}P^\infty$ in a similar way?
6
votes
1answer
90 views

Cohomology of $S^2\times S^2/\mathbb{Z}_2$

The product of two spheres admits a diagonal $\mathbb{Z}_2$-action, $(x,y)\mapsto (-x,-y)$. I'm trying to compute the integral singular cohomology ring of the orbit space $X$ of this action. $X$ is ...
0
votes
0answers
12 views

dual hopf algebras

Let $X$ be an H-space with product $\mu$. Let diagonal map $\Delta: x\mapsto (x,x)$. Let $F$ be a field. (1). Then by Kunneth formula, $H_*(X\times X;F)=H_*(X;F)\otimes H_*(X;F)$. (2). Hence $$ ...
0
votes
1answer
46 views

maps between spheres, torus and projective plane [closed]

How to solve these questions by direct and valid argument? Various methods are wanted. Thanks.
0
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1answer
39 views

homology and cohomology with coefficients of ring and field [closed]

(1). Let $R$ be a ring. Let $X$ be a topological space. Then $H^n(X;R)$ is a module over $R$. Also $H_n(X;R)$ is a module over $R$.Is this statement correct? (2). Let $F$ be a field. Let $X$ be a ...
4
votes
1answer
146 views

Thom–Gysin long exact sequence

I have read about the following exact sequence of cohomology: Let $V$ be an algebraic variety over $\mathbb{C}$. If $U\subset V$ is an open subvariety, then there is a long exact sequence for ...
2
votes
2answers
44 views

Reference request for equality of torsion of H1 and H2

I have heard that for a surface $X$ (algebraic? smooth? compact?) the torsion part of $H_1(X,\mathbb{Z})$ is the same as that of $H_2(X,\mathbb{Z})$. Please could you give me a correct statement? I ...
3
votes
0answers
18 views

examples of k-invariants of spectra

The homotopy groups of commonly used topological spectra (like KO, S, MO, MSO, etc) are easy to find in literature, even appearing on Wikipedia's List of Cohomology Theories; however, I have had some ...
1
vote
0answers
19 views

Cohomology of intersection of hyperplanes

let $X = H_1 \cap ... \cap H_d$ be a compact submanifold of $\mathbb{P}_N$ where the $H_i$ are hyperplanes. I want to compute $H^q(X, \mathcal{O}_{\mathbb{P}_N}(m)|X)$. I am pretty unexperienced in ...
0
votes
0answers
25 views

The generator of compact cohomology of punctured plane

Can anyone give detailed derivation of the generator of compact cohomology of $H^1_c (R^2-\{0\}$). (It is homotopic to a circle so it is isomorphic to $R$ but I want computation of its generator, ...
0
votes
1answer
33 views

Poincare dual of a point

Why is the closed Poincare dual of a point in $R^n$ trivial but the compact dual is a "bump"? Please provide very detailed answer.
5
votes
2answers
987 views

Translation Request - Grothendieck's Tohoku Paper

I've been learning sheaf cohomology, and was interested in reading Grothendieck's Tohoku paper. However, I don't read French. I've done a semi-extensive google search, and the majority of links end ...
1
vote
0answers
22 views

cohomology of unordered configuration spaces of sphere

Let $F(X,n)$ be the configuration space of order $n$. Let $F(X,n)/\Sigma_n$ be the unordered configuration space of order $n$. What is $H^*(F(S^2,n)/\Sigma_n;\mathbb{Z}_2)$? I did not find the answer ...
0
votes
0answers
3 views

Let ${\Phi \in H^n(E, E - B; \mathbb{Z}/2)},$ restrict to $E$, and pullback to $B$, where $B \subset E$.

I am reading about characteristic classes, and I came upon this statement. Let ${\Phi \in H^n(E, E - B; \mathbb{Z}/2)},$ restrict to $E$, and pullback to $B$, where $B \subset E$. Can someone ...
0
votes
1answer
88 views

Exact sequences, mostly linear algebra and notation confusion.

Problem: Let $0 \to A^0\stackrel{d^0}{\to} A^1 \stackrel{d^1}{\to} \dots \stackrel{d^{n-1}}{\to} A^n \to 0 $ be a chain complex and assume each $A^i$ is finite-dimensional. Prove that the sequence ...
2
votes
2answers
100 views

Abelian category without enough injectives

What is an example of an abelian category that does not have enough injectives? An example must exist, but I haven't been able to find one. If possible, a brief explanation of why the abelian ...
0
votes
0answers
27 views

Twisted cohomology as sections of bundle of Eilenberg-Maclane spaces

Let $X$ a space, and $E$ an multiplicative cohomology theory represented by a ring spectrum $K$, i.e. $E^\bullet(X)=[X,K]$. Also let $A$ be a local system of abelian groups. Cohomology with local ...
5
votes
1answer
86 views

Bounding the cohomology of a smooth projective variety

Let $X/\mathbb C$ be a smooth projective variety. Suppose it is smoothly embedded in $\mathbf P^n$ as the zero locus of an ideal generated by homogeneous polynomials $f_1, f_2, \dots, f_r$ in $n+1$ ...
0
votes
0answers
35 views

example of use of (co)homology

I'm learning cohomology, but there is very few examples in the book I'm reading. I read the definition of Ext and Tor but don't know how to use these. Are there some examples of proposition such that ...
1
vote
1answer
46 views

Does the cup product on de Rham cohomology induce a nondegenerate bilinear form?

I have small issue I came across in the following. Suppose $M$ is a compact, oriented manifold of dimension $4n+2$. I want to prove that the de Rham cohomology group $H^{2n+1}(M)$ are even ...
6
votes
1answer
185 views

Prove Poincare duality theorem with Morse theory.

First let us consider a smooth n-manifold. And find a Morse function f. Now let's consider -f. A singular point of f with index k is a singular point of -f with index n-k. Thus we have a canonical ...
1
vote
1answer
89 views

Spaces with different homotopy type

I want to show that the spaces $ S^1 \vee S^1 \vee S^2$ and $S^1 \times S^1 $ do not have the same homotopy type. I calculated their homologies and cohomologies and they turn out to be equal. So I ...
2
votes
2answers
96 views

No symplectic structure on $S^{2n},\ n>1$

I am trying to show that there is no symplectic structure on the $2n$-dimensional sphere $S^{2n}$, where $n>1$. I've tried following these steps: (a) Given a compact $2n$-dimensional symplectic ...
2
votes
0answers
62 views

cohomology ring of grassmannian

Let $G_k(\mathbb{R}^n)$ be the grassmannian consisting of all $k$-subspaces in $\mathbb{R}^n$. How to compute the cohomology ring $$H^*(G_k(\mathbb{R}^n);\mathbb{Z})$$ and what is the result?
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vote
0answers
26 views

What maps of $k$-algebras $A\to B$ induce finite maps $\mathrm{Ext}_B^*(k,k)\to\mathrm{Ext}_A^*(k,k)$?

Let $k$ be an algebraically closed field, and let $A$ and $B$ be finitely generated $k$-algebras. A map $\varphi:A\to B$ of $k$-algebras induces a map ...