1
vote
2answers
87 views

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 ...
0
votes
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 ...
3
votes
0answers
55 views

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 ...
3
votes
0answers
109 views

A question of extension of vector bundles.

Fix $p \in \mathbb{P}^1$. Let $X=\mathbb{P}^1\times \mathbb{P}^1$, $C_1=\mathbb{P}^1\times \{p\}$ and $C_2=\{p\}\times \mathbb{P}^1$. Since $\mathrm{Ext}^1(\mathcal{O}_{C_2},\mathcal{O}_{C_1})\cong ...
0
votes
1answer
181 views

Will $i^*$ pull back injectives to injectives?

Let $i: Z \rightarrow X$ be a closed embedding. Need $i^*$ of an injective sheaf of abelian groups be injective? Need $i^*$ of a flabby sheaf be flabby? Thanks :D! Also (maybe should be a separate ...
1
vote
1answer
150 views

Is there a quasi-isomorphism between a complex of sheaves and its Godement resolutions?

I have a doubt, I read somewhere that the Godement resolution of a sheaf $\mathcal{F}$ is a quasi-isomorphism $\mathcal{F} \rightarrow C^\bullet(\mathcal{F})$. Just right off the bat when I read ...
3
votes
0answers
149 views

Adapted classes of objects and left (right) exact functors

I had a question about adapted classes of objects, I was confused by the definition and how it relates to left exact functors. Let $\mathcal{A}$ be an abelian category with enough injectives, let $F: ...
7
votes
0answers
240 views

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: ...
1
vote
1answer
83 views

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 ...
5
votes
0answers
181 views

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 ...
6
votes
0answers
344 views

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 ...
2
votes
1answer
173 views

Flabby sheaves and comparison of topologies

Let $A^p$ be a group of sheaves on a topological space $X$, let $F$ be the global sections functor $F(A^p) = A^p(X)$. I have to compute the cohomology of the complex $0\rightarrow A^1(X) \rightarrow ...
4
votes
1answer
423 views

Confused about Hypercohomology terminology and meaning

check this: Given a sheaf complex $F^\bullet$, let's say I want to compute the hypercohomology of this complex, if we consider the bicomplex of sheaves $C^\bullet(F^\bullet) = (C^p(F^q))\quad ...
4
votes
1answer
161 views

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 ...
3
votes
1answer
180 views

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 ...
1
vote
1answer
100 views

Is the presheaf of continuous functions on a topological space a “complete presheaf”?

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?
3
votes
2answers
260 views

Is the sheaf of locally constant functions flasque?

Quick question, is the sheaf of locally constant functions flasque?