# Tagged Questions

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### Hypercohomology: finding a resolution for the de Rham complex of $\mathbb{CP}^1$

Let $\mathbb{P}^1$ be the complex projective line. Using the standard affine cover, $\mathcal{U} = \lbrace U,U' \rbrace, \ \$ we can define some quasi-coherent sheaves on $\mathbb{P}^1$. We can ...
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### $i^{-1} F$ a sheaf if and only if $\varinjlim_{ U \subseteq X \text{ open}, ~ x,y \in U } F(U) \to F_x \times F_y$ is an isomorphism.

Let $X$ be a topological space containing two closed points $x,y$ and let $i : \{x,y\} \to X$ denote the inclusion map. Notice that $\{x,y\}$ carries the discrete topology. Let $F$ be a sheaf on $X$. ...
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### Leray's theorem for cech and derived sheaf cohomology.

My question is about the hypothesis of Leray's theorem. This theorem says that if $\mathcal{U}$ is an open cover of a topological space $X$, and $\mathcal{F}$ is a sheaf over $X$ and if ...
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### The inverse image of a sheaf

By definition, the inverse image of the sheaf $\mathcal{F} : \mathrm{Ouv} (Y) \to \mathrm {Set}$ is the sheaf associated to the presheaf $f^{-1} \mathcal{F} : \mathrm{Ouv} (X) \to \mathrm{Set}$ ...
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### Soft sheaves adapted to $f_!$

I'm reading Gelfand-Manin, Homological Algebra. I understand that the class of soft sheaves is sufficiently large, because every injective sheaf is soft. Now to see that this class is adapted to ...
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### 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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### The Simplicial Flabby Resolution of a Sheaf

I study sheaf cohomology by Demailly's book and I have a trouble. Am I right, that inductive formula at end of page 198 $$\mathcal{A}^{[q]}=(\mathcal{A}^{[q-1]})^{[0]}$$ is wrong? I think that ...
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### Composition of derived functors and comparison between hypercohomology and sheaf cohomology

I had a few questions about compositions of derived functors, the comparison between hypercohomology, and sheaf cohomology and the following theorem from the Gelfand, Manin homological algebra book: ...
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### Is it true, that $H^1(X,\mathcal{K}_{x_1,x_2})=0$? - The cohomology of the complex curve with a coefficient of the shaeaf of meromorphic functions…

Let X be complex curve (complex manifold and $\dim X=1$). For $x_1,x_2\in X$ we define the sheaf $\mathcal{K}_{x_1,x_2}$(in complex topology) of meromorphic functions vanish at the points $x_1$ and ...
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### Computing the hypercohomology of a complex of acyclic sheaves

Let $K^{\bullet}$ be a cochain complex of sheaves of finite-dimensional vector spaces, I wanted to compute $\mathbb{H}^{\bullet}(X,K^{\bullet})$ = the hypercohomology of the complex $K^{\bullet}$, the ...
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### Why didn't Cartan-Eilenberg develop homological algebra on sheaf theory?

Cartan-Eilenberg created homological algebra on modules over rings. I wonder why they didn't develop it also on sheaves over ringed spaces. Grothendieck and Godement did that soon after(or almost at ...
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### Flabby sheaves and exact sequences of sheaves - Question about proof

I was going through this proof from Rotman's 'Introduction to homological algebra' (Pages 381-382) and I just can't seem to make sense of it, am not super well-versed in this so I don't know if it's ...
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### Question about proof that Flabby sheaves are acyclic

Can anybody help me understand this proof? In Rotman's 'An introduction to homological algebra' in Proposition 6.75 (iii). Flabby sheaves $\mathcal{L}$ are acyclic (Page 381), in the proof it says ...
Is the presheaf of continuous functions $f:A\rightarrow B$ from a topological space $A$ to another topological space $B$ a "complete presheaf"? Can't find this, anyone have a reference?