# Tagged Questions

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

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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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### Trivial Cohomology Group->Lower-Dimensional Homotopy?

Calculating the (de-Rham) cohomology of a tee connector (Picture), I got $H^0=R,H^1=R^2,H^2=0$. Furthermore, just from looking at it, I assume the tee connector is homotopic to a circle with an arc ...
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### Intuition for chains and cochains

I'd like to get some "geometric," "physical," (or other form of) intuition for chains, cochains, and their relationship to integration on manifolds at an elementary level. In particular, it would be ...
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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 $$H^k(S^3,\mathbb{Z}/2\mathbb{Z}) = ... 0answers 67 views ### Derived functors and coboundary operator I understand that one can define the cohomology of an object A in terms of a complex (non-zero in positive degrees) in some Abelian category, together with differentials, such that the composition ... 0answers 27 views ### Problem with cohomology (II) Let G be a group and K be subgroup of G, Let A be G module. Let X=\{X_n\} be free resolution of \mathbb{Z}. Let \delta represents collectively the homomorphism induced in Hom sequence ... 1answer 46 views ### Problem with cohomology (I) I have some doubts regarding cohomology. As title suggests I will ask these one by one. Let G be a group and A be G-module. Let C^n(G,A) denote the set of all maps from G \times \cdots ... 0answers 26 views ### Conservativeness on a graph I'm trying to build a conservative vector field out of something smaller than \mathbb{R}^2 to understand how the "conservative" property of differences-of-scalar-fields leads to Green's theorem. (In ... 2answers 146 views ### Cohomology groups for the following pair (X,A) Let X=S^1\times D^2, and let A=\{(z^k,z)\mid z\in S^1\}\subset X. Calculate the groups and homomorphisms in the cohomology of the exact sequence of the pair (X,A). I know that theorically one ... 0answers 74 views ### Corestriction map in lie algebra cohomology Given a lie algebra \mathfrak{g} over a field k, we can define the cohomology groups of \mathfrak{g} as follows:$$H^n(\mathfrak{g},k):=\mathrm{Ext}_{U(\mathfrak{g})}^n(k,k)$$where ... 0answers 27 views ### Simple question on splitting of cohomology groups. From the exponential exact sequence, I have$$ 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 ...
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 ...