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

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28 views

Why is it sufficient to prove for $H^0$

I am reading the corestriction homomorphism on cohomology. To prove a proposition (for example the composition of corestriction and restrcition is the multiplication by the index), it says that it is ...
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1answer
77 views

Apparent contradiction when using Mayer Vietoris

I am using the Mayer Vietoris sequence to compute the de Rham cohomology of a twice punctured plane. I computed it first by choosing my open cover to be $U=R^2-{pt}$ and $V=R^2-{pt}$ (the points are ...
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18 views

Show that $con_H^g$ is the identity map

Let $G$ be a group,$H$ a subgroup of $G$ and $A$ be $G$-module. Let $g \in G$ and $H^g=g^{-1}Hg$. Let $Z^2(G,A)$ be the set of $2$-cocycles. Given $f \in Z^2(H,A)$, let $f^g \in Z^2(H^g,A)$ be defined ...
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0answers
69 views

Derived push-forward of projective sheaf

Let S,X be schemes and $s \in S$ be a closed point. Let $D(X)$ be the derived category of complexes of sheaves. Let $$i_s: X \cong {s} \times X \hookrightarrow S \times X$$ be the natural embedding. ...
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2answers
86 views

Exercises - “From calculus to cohomology”

I am reading Madsen's book From calculus to cohomology and I've found it doesn't have any (explicit) exercises at the end of each section. I'd like to know a few books where I can find some problems ...
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1answer
1k views

Cohomology of projective plane

How I can compute cohomology de Rham of the projective plane $P^{2}(\mathbb{R})$ using Mayer vietoris or any other methods?
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0answers
39 views

Compactness and Poincare duality

I am reading Appendix B in Fulton's Young Tableaux about Borel-Moore homology. In particular, I'd like to understand why for compact manifolds the Borel-Moore homology groups are isomorphic to ...
1
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2answers
57 views

Inductive definition of group cohomology?

At the start of Atiyah and Wall's section on group cohomology (in the Cassels-Frhlich collection of Algebraic Number Theory notes) they, of course, define group cohomology (actually, a 'cohomological ...
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1answer
69 views

Dolbeault cohomology and analytic regularity

Let $M$ be a complex analytic $n$-manifold. The Dolbeault cohomology complex is defined using a quotient space of smooth differential forms. My question is : would it make a big difference if we were ...
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1answer
23 views

Why $\operatorname{im} \delta_{-1}$ contains two functions and $f\in C^{0}(X, \mathbb{F}_2)$ is the characteristic function?

In the paper, on page 21, line 15-20. It is said that $B^0=\operatorname{im} \delta_{-1}$ is the one dimensional space containing two functions and $f\in C^{0}(X, \mathbb{F}_2)$ is the characteristic ...
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0answers
21 views

The Existence of Pure Resolutions, Given a Degree Sequence?

I have been trying to understand the proof of the following theorem for the last month, I read some basics of sheaves theory and their cohomology, but still can't get the idea of this important ...
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0answers
47 views

When is $\mathcal{H}om$ functor exact in the category of presheaves

Let $C$ be a projective $\mathbb{C}$-scheme of pure dimension $1$. Suppose that $C$ is local complete intersection in $\mathbb{P}^3$. Let $C_1$ be an irreducible component of $C$, also of pure ...
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1answer
105 views

cohomology groups of K(Zp x Zp, 1)

I have a question regarding the cohomology groups of the Eilenberg-MacLane space $K(\mathbb{Z}_p \times \mathbb{Z}_p,1)$. For $n$ > $2$, is there a way to show that $H^n(K(\mathbb{Z}_p \times ...
2
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1answer
62 views

Exact and closed forms with a vanishing Riemann tensor

I need a result to prove that an closed form is equally exact. I work under the assumption that the Riemann tensor vanishes everywhere on the manifold. (It is the context of general relativity.) The ...
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0answers
72 views

Exact sequences and spectral sequences

We have the well-known theorem for cohomological spectral sequences as follows: Theorem: Let $(E_r , d_r )$ be a third quadrant spectral sequence and let $E^{p,q}_2‎\Rightarrow‎ H^n(Tot(M)$. a) If ...
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1answer
60 views

Show that image of $res$ lies in $H^n(H,A)^{G/H}$

Let $G$ and $G^{\prime}$ be groups, $A$ and $A^{\prime}$ be $G$-module and $G^{\prime}$-module respectively, $C^n(G,A)$ be set of all maps from $G \times \cdots \times G$ ($n$ times) to $A$, $d_n ...
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3answers
234 views

What's the point of spectra?

I'm familiar with the definition of a spectrum, the one due to Adams, however, I'm not really sure why someone would want to define such a thing. I know they allow one to generalize homology and ...
6
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1answer
177 views

Cup Product Structure on the Projective Space

I am reading about cup products and am stuck on this exercise in Hatcher (3.2.5). Taking as given that $H^*(\mathbb{R}P^\infty,\mathbb{Z}_2)\simeq\mathbb{Z}_2[\alpha]$, how does one show ...
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0answers
56 views

Hochschild Serre Spectral sequence

I am reading Hochschild Serre Spectral sequence from Group Theory I by M Suzuki Page 213. I will be thankful to you if you make me understand following points to me. Following is the sequence $0 ...
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1answer
71 views

Equivariant cohomology: $H^{*}_{S^{1}}(S^{2})$

We want calculate $H^{*}_{S^1}(S^2)$. We can choose two open sets $U= S^{2} \setminus p_{+}$ and $V = S^{2} \setminus p_{-}$, where $p_{+}$ and $p_{-}$ are the north pole and south pole of $S^{2}$. ...
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63 views

lift of antiholomorphic involution of Riemann surface to its Jacobian's cohomology

Start from a connected closed Riemann surface $\Sigma_g,$ obtained as the (symmetric) covering of an open and/or unoriented surface $\Sigma,$ namely $\Sigma=\Sigma_g/\Omega,$ where $\Omega$ is an ...
5
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1answer
106 views

How to prove $H^2(\mathfrak{g}, J(\mathfrak{g}))\neq0$, where $J(\mathfrak{g})$ is the augmentation ideal of $\mathfrak{g}$?

$\mathfrak{g}$ is a finite-dimensional semisimple Lie algebra over a field $k$ with $\mathrm{char}k=0$. $J(\mathfrak{g})$ is the augmentation ideal of $\mathfrak{g}$. That is, the kernel of ...
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2answers
709 views

how to compute the de Rham cohomology of the punctured plane just by Calculus?

I have a classmate learning algebra.He ask me how to compute the de Rham cohomology of the punctured plane $\mathbb{R}^2\setminus\{0\}$ by an elementary way,without homotopy type,without ...
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0answers
126 views

Top de Rham cohomology

I just realized that I never really understood why $H_{dR}^n(M, \mathbb{R}) = \mathbb{R}$ if $M$ is compact and $H_{dR}^n(M, \mathbb{R}) = \{0\}$ if $M$ is not compact (provided that's true?). I'm ...
3
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1answer
74 views

de Rham cohomology of $\mathbb{R}P^n$ via action by $SO(m+1)$

In lecture, my teacher proved the theorem that given a smooth $G$-action by a compact, connected Lie group on a manifold $M$, the de Rham cohomology of the $G$-invariant differential forms $H^p_G(M)$ ...
2
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1answer
63 views

About existence of Morse functions

Let's consider 4-manifold $M$, $\partial M = \partial M_1 + \partial M_2 = S^1 \times S^2 + \mathbb{RP}^3$. Is it true that there exist a Morse function $$f\colon M^4 \to [0,1],\quad f^{-1}(0) = ...
0
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1answer
58 views

how can I show $H^1(g , Hom_C(g,M))=0$?

For a simple Lie algebra $g$ and a finite dimensional vector space $M$ with a trivial $g-$action, how can I show $H^1(g , Hom_C(g,M))=0$?
4
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0answers
34 views

Computing cohomology of product space with product-ring coefficients

I'm interested in the following problem: Let $X$ and $Y$ be finite CW complexes and $R$ and $S$ rings. Suppose you are given the cohomology rings $H^* (X; R)$ and $H^* (Y; S)$. Is there an easy way ...
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135 views

Morita-invariance of Hochschild (co)homology.

Ok, I'm reading this paper by Christian Kassel on associative algebras and Hochschild (co)homology and on page 19 he says that Hochschild homology is Morita-invariant, by which he means that if $R$ ...
2
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1answer
281 views

Closed but not exact one-form on $S^2$

I would like to know whether there is any nice prescription to give an example of a closed but not exact one-form on $S^2$ (not the $3$-ball). I assume to take some points out of this surface, e.g. 3. ...
2
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0answers
60 views

Derived functors and coboundary operator

I understand that one can define the cohomology of an object $A$ in terms of a complex (non-zero in positive degrees) in some Abelian category, together with differentials, such that the composition ...
3
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1answer
111 views

Cohomology groups

I have some questions. 1) I tried to compute the cohomology group of $S^3$ with coefficients in $\mathbb{Z}/2\mathbb{Z}$ but I don't know if my result $$ H^k(S^3,\mathbb{Z}/2\mathbb{Z}) = ...
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26 views

Problem with cohomology (II)

Let $G$ be a group and $K$ be subgroup of $G$, Let $A$ be $G$ module. Let $X=\{X_n\}$ be free resolution of $\mathbb{Z}$. Let $\delta$ represents collectively the homomorphism induced in Hom sequence ...
0
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1answer
46 views

Problem with cohomology (I)

I have some doubts regarding cohomology. As title suggests I will ask these one by one. Let $G$ be a group and $A$ be $G$-module. Let $C^n(G,A)$ denote the set of all maps from $G \times \cdots ...
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24 views

Conservativeness on a graph

I'm trying to build a conservative vector field out of something smaller than $\mathbb{R}^2$ to understand how the "conservative" property of differences-of-scalar-fields leads to Green's theorem. (In ...
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2answers
91 views

Cohomology groups for the following pair $(X,A)$

Let $X=S^1\times D^2$, and let $A=\{(z^k,z)\mid z\in S^1\}\subset X$. Calculate the groups and homomorphisms in the cohomology of the exact sequence of the pair $(X,A)$. I know that theorically one ...
7
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1answer
109 views

Strange case of Serre's duality

$\newcommand{\O}{\mathcal{O}}$ Let $X$ be a smooth projective curve and $D$ and effective divisor on it. The normal bundle of $D$ is defined as $$ \O_D(D)\; = \; \O_x(D)\;\otimes_{\O_X}\, \O_D$$ where ...
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2answers
99 views

Cohomological definition of the Chow ring

Let $X$ be a smooth projective variety over a field $k$. One can define the Chow ring $A^\bullet(X)$ to be the free group generated by irreducible subvarieties, modulo rational equivalence. ...
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0answers
61 views

Corestriction map in lie algebra cohomology

Given a lie algebra $\mathfrak{g}$ over a field $k$, we can define the cohomology groups of $\mathfrak{g}$ as follows: $$H^n(\mathfrak{g},k):=\mathrm{Ext}_{U(\mathfrak{g})}^n(k,k)$$ where ...
11
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2answers
182 views

When is a map essential in Čech cohomology?

I read a nice survey of parts of game theory, Foundations of Strategic Equilibrium, by Hillas and Kohlberg. Something where I stumble is the discussion of Mertens stability. There is a definition that ...
2
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1answer
76 views

Flag varieties and representation theory

I've recently been reading about flag varieties and their cohomology. I'm mainly interested in representation theory, and I've heard that flag varieties are important objects, especially in Lie ...
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0answers
25 views

Simple question on splitting of cohomology groups.

From the exponential exact sequence, I have $$ 0 \rightarrow H^2(X,\mathbb{C})/H^2(X,\mathbb{Z})\rightarrow H^2(X,\mathbb{C}^\times) \rightarrow Tor(H^3(X,\mathbb{Z})) \rightarrow 0. $$ for some ...
3
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1answer
81 views

Global section of pull-back of structure sheaf of projective scheme

Let $X$ be a smooth projective variety and $Z_1, Z_2$ two smooth projective divisors in $X$. Is it true that the natural restriction morphism from $H^0(\mathcal{O}_X(-Z_1-Z_2))$ to $H^0(\mathcal{O}_X ...
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1answer
74 views

simple question about cohomology group

Let's consider compact 4-manifold $M^{4}$ and point $P \in M$. Then (use Mayer-Vietoris) inclusion $i\colon M\setminus P \to M$ induce isomorphism $i^{*}\colon H^2(M) \to H^2(M\setminus P)$. Let's ...
1
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1answer
57 views

Differential forms as functionals on curves

Please give me a reference to a book or lecture notes where the following stuff is studied. Let $M$ be a Riemann surface with boundary $\partial M$ (but not necessarily, any smooth $n$-dimensional ...
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0answers
48 views

Definition of the relative de-Rham cohomology and its generalization

Let $M$ be a smooth manifold and $N$ be its smooth submanifold. We say that two closed forms $\omega_1$ and $\omega_2 \in \Lambda^k(M)$ are equivalent if their difference is an exact form from ...
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1answer
75 views

Cohomological ($p$-)dimension of a pro-$p$ group

I have a question concerning the cohomological dimension and $p$-dimension of a pro-$p$-group. Let's first recall the definitions of that The cohomological dimension $cd \ G$ of a pro-finite group ...
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37 views

Do the cyclic or Hochschild homologies satisfy the addition axiom of Eilenberg Steenrod?

Do the cyclic or Hochschild homologies satisfy the addition axiom of ES? If so please provide a reference or proof (reference is preferable).
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45 views

Relative de Rahm cohomology computation for two disjoint circles embedded in R^2

Consider a submanifold $Y$ of $\mathbb{R}^2$ formed by two disjoint embedded copies of $S^1.$ Compute $H^{\bullet}_{dR}(\mathbb{R}^2,Y).$ In this case the long exact sequence splits, and we can ...
7
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1answer
81 views

Finite generation of Tate cohomology groups

Let $G$ be a finite group, and let $F$ be a complete resolution for $G$. In other words, $F$ is an acyclic chain complex of projective $\mathbb{Z}G$-modules together with a map ...