# Tagged Questions

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

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### Dolbeault cohomology on torus

Let $T=\mathbb{C}/\Gamma$ where $\Gamma$ is a lattice of $\mathbb C$. Given that $H_{dR}^1(T)=\mathbb{C}^2$. Prove that $H^{1,0}_\bar{\partial}(T)=\mathbb{C}$. I have no idea what to do. Can someone ...
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### If M is a non-orientable closed connected 3 manifold prove H1(M) is an infinite group.

This is an example from a question sheet (non-assessed) of a university class. If M is a non-orientable, closed, connected 3 manifold, prove $H_1(M;\mathbb{Z})$ is an infinite group. I know that since ...
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### second hochschild cohomology and extension

i started learning the theorem that says there is a one to one correspondence between Ext(A,M) and H^2(A,M). however, the proof is not clear. I managed to show that there is a well-define map U from ...
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### Why do we have $H_2(X,\mathbb Z)\cong\mathbb Z$ for the quintic threefold $X\subset\mathbb P^4$?

Let us work over $\mathbb C$. In this article by S. Katz, it is stated that for a quintic threefold $X\subset \mathbb P^4$ one has $$H_2(X,\mathbb Z)\cong\mathbb Z.$$ Can anyone help me to see why ...
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### $H^k(\mathbb{C}P^2 \times \mathbb{C}P^2, \mathcal{O}^*(\mathbb{C}P^2 \times \mathbb{C}P^2))$

I'd like to try to compute Cech cohomology groups $H^k(\mathbb{C}P^2 \times \mathbb{C}P^2, \mathcal{O}^*(\mathbb{C}P^2 \times \mathbb{C}P^2))$, but I don't know how can I do it. In my notes the author ...
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### Property of G-modules involving the invariant elements under the G-action

I am stuck at some basic fact I would like to prove. I tried proving it using $G-$orbits and cardinalities, but without success. Let $p$ be some prime number, $G$ be a finite $p-$group and $A$ a ...
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### Explititly evaluating the Poincaré duality

Let $M$ be a closed 2-dimensional manifold (a surface). Assume that I have a more or less explicit expression for a Čech 2-cocyle $h_{ijk} \in H^2(M, G)$. I want to know the expression of $h_{ijk}$ as ...
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### Isomorphism in cohomology is an isomorphism in homology

Let $f:X \to Y$ be a continuous map between topological spaces and $R$ some coefficients. From the universal coefficient theorem for homology we immediatly get, that if $H_*(f,\mathbb{Z})$ is an ...
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### Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer ...
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### Algebraic Variety compact cohomology and singular cohomology.

Given an algebraic variety $X$ defined over the complex numbers and its compact cohomology $H^i_c(X)$, under what conditions it is possible to compute its singular cohomology $H^i(X)$.
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### Cohomology with coefficients in a commutative ring, how are the chain groups defined?

I have been studying a course in algebraic topology that follows Hatcher's textbook on the subject. I have some queries as to how certain things are defined. The first part of the text defines the ...
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### Does additivity of (equivariant) cohomology hold at the algebra level?

The additivity property of many (co)homology theories is that if $X = \bigsqcup_{i \in I} X_i$ then $H^*(X) = \bigoplus_{i\in I} H^*(X_i)$. This is usually either an axiom of the theory, can be proven ...
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### Uniqueness of the cohomological functor

This question is from the chapter 'Cohomology of Groups' by Atiyah and Wall in Cassels' and Frohlich's book 'Algebraic Number Theory'. Let $G$ be a group. Theorem 1 on page 95 says that there is a ...
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### When does cohomology take pullbacks to pushouts?

I've encountered a simple situation where one has a pullback diagram of topological spaces and taking cohomology takes it into what I believe is a pushout diagram in the category of rings. I'm not ...
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### A G-isomorphism of certain Hom groups

This question is from 'Cohomology of Groups' by Atiyah and Wall, p.95 of Cassels' and Frohlich's book 'Algebraic Number Theory'. Let $G$ be a group and $A={\rm Hom}_{\mathbb Z}(\mathbb Z[G],X)$ where ...
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### Transgression homomorphism on cohomology

I have one confusion about the transgression homomorphism which I found in two different books. I am unable to show that they really are same. The first description of transgression homomorphism I ...
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### Vanishing of higher direct images of a composition

In a paper I am studying we have the following situation. Let $S$ be the spectrum of a Dedekind domain, and let $X$, $Y$ and $Z$ be scheme of finite type over $S$, where $X$ and $Y$ are smooths and ...
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### Book of Pullbacks and Pushouts

what books can I consult for properties of pullback and pushouts in algebraic topology? I need to understand the theory of homotopy in algebraic topology and I started to study pullbacks and push ...
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### Sheaf cohomology of $\mathbb{P}^3$

Let $\mathbb{P}$ denote the projective space over $\mathbb{C}$. In some lecture notes I found the claim that $$h^0(\mathbb{P}^3, \mathcal{O}(2)) = 10$$ Do you know why this is the case? In ...
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### map to product of eilenberg-maclane spaces

Given a space $X$, and an Eilenberg-MacLane space $K(G,n)$ (hereafter referred to as $K$), and two maps $f: X \to K$ and $g:X \to K$, let $f \times g:X \to K \times K$ map $x \in X$ to $(f(x),g(x))$. ...
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### Intersection form and poincaré duality

Let $M$ be a $2n$-dimensional compact connected oriented smooth manifold and let $A$, $B$ be two $n$-dimensional submanifolds that intersect transversally. Denote by $A \cdot B$ the sum of the ...
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### Global sections of a twisting of a structure sheaf of a projective scheme

Let $X$ be a projective Noetherian scheme over $\mathbb{C}$. Is it true that $H^0(\mathcal{O}_X(-t))=0$ for any $t>0$?
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### Group structure on pointed homotopy classes [X,S^1]

Let $[X,S^1]$ denote the set of pointed homotopy classes of maps $f:X\to S^1$. I need to show that, when $S^1$ is viewed as a subset of $\mathbb{C}$, complex multiplication induces a group structure ...
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### Vanishing of $R^1f_*\mathcal O_X$

I am probably missing something obvious here, but none the less, here goes: Is the following statement (or perhaps some minor modification of it) true and if so, why: $R^1f_*\mathcal O_X = 0$ for a ...
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### what can you say about the degree of $f:\mathbb{C}P^n \to \mathbb{C}P^n$

Any thoughts on this problem: If $M$ and $N$ are simply-connected, $n$-dimensional manifolds, then $H^n(M;\mathbb{Z}) \cong \mathbb{Z} \cong H^n(N;\mathbb{Z})$. A map $f:M \to N$ induces a map ...
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### Making modular representation theory and cohomology 'compelling' and 'accesible'

I'm currently putting together an application for a dissertation completion fellowship offered through my university. A part of the application includes 500-1000 words describing my dissertation. ...
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### de Rham cohomology of $\mathbb R^2 \setminus \mathbb Z^2$

I am trying to calculate the cohomology of $X = \mathbb R^2 \setminus \lbrace \mathbb Z \times \mathbb Z \rbrace = \lbrace (x,y) \in \mathbb R^2 : x \text{ and } y \not \in \mathbb Z \rbrace.$ ...
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### Cochains: terminology

Let a real, smooth manifold $M$ be given. Let $C_k(\mathbb Z, M$) denote the set of $k$-chains with integer coefficients, and let $C_k(\mathbb R, M)$ denote the set of $k$-chains with real ...
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### Cohomology of covering space

Let $B$ be a base space and $E$ be a covering space of $B$ what is the relation between $H^2(B,\mathbb{Z})$ and $H^2(E,\mathbb{Z})$.?
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### Duality in algebraic de Rham cohomology

I am trying to prove that the following is a short exact sequence $$0 \rightarrow H^0(X,\Omega_X) \rightarrow H^1_{\text {dR}}(X/k) \rightarrow H^1(X,\mathcal O_X) \rightarrow 0,$$ where $k$ is an ...
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### 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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### 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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### 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 ...
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 ...