All Questions
Tagged with abelian-categories sheaf-theory
28 questions
0
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0
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41
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Proposition (0.1.4.3) in Tamme's Introduction to Etale Cohomology
Proposition 1.4.3, page 7 in Tamme's book, says (among other things) that: given two categories $\cal C$ and $\cal C'$, with $\cal C'$ abelian and AB-$3$, if $\cal C'$ has a generator $Z$ then also ...
- 35
1
vote
1
answer
96
views
Why sheaves of abelian groups form an abelian category
I know it is an elementary result. But some details confuse me.
In Vakil's The Rising Sea (version 2022) Theorem 2.6.2 writes:
Sheaves of abelian groups on a topological space $X$ form an abelian ...
4
votes
0
answers
142
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Why is $ \mathfrak{Mod}(A_{Y}/f) $ a thick subcategory of $\mathfrak{Mod}(A_{Y})$?
Let $f: Y \to X$ be a continuous map and $\mathfrak{Mod}(A_{Y}/f)$ be the full subcategory of $\mathfrak{Mod}(A_{Y})$ (categories of $A_{Y}$-modules, with $A$ a fixed ring) whose sheaves $\mathcal{F}$ ...
- 452
3
votes
1
answer
65
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Do quasi-coherent sheaves form a reflective subcategory?
Let $X = $ Spec $A$ be an affine scheme.
I know that there is an inclusion of categories from $A$-modules to sheaves of $\mathcal O_X$-modules on $X$, which is exact and fully faithful.
It seems to me ...
- 394
1
vote
0
answers
63
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When can we define the stalks of a sheaf?
So i am reading rotman's book on homological algebra, and in it he states in Theorem 5.91, the following:
Let $X$ be a topological space and $\mathcal{A}$ an abelian category and consider the category ...
- 167
2
votes
0
answers
73
views
Injective module is stalkwise injective
Let $X$ be a Hausdorff locally compact topological space, $R$ be a sheaf of $\mathbb{C}_X$-algebras on $X$. If necessary, we can assume that $(X,R)$ is a complex manifold. If $M$ is an injective ...
- 1,360
2
votes
1
answer
76
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What is the difference between $\mathbb C_X$-modules and sheaves of $\mathbb C$-vector spaces?
In the literature I have encountered the notions "category $\mathbb C_X-Mod$ of $\mathbb C_X$-modules" and "category $Sh_{\mathbb C}(X)$ of sheaves of $\mathbb C$-vector spaces". ...
- 982
5
votes
2
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253
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Why is the ambiguity of the target category when first defining sheaves not a serious issue?
I am taking notes on (pre)sheaves of topological spaces. This is my first time studying the subject. Some sources define them as functors to $Set$. Others as functors to $Ab$. Or sometimes $Mod_R$. Or ...
7
votes
1
answer
181
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Is the category of abelian presheaves on a topos closed?
Take the category of presehaves of abelian groups on a topos $\mathcal{C}$. That is, an object of our category is a functor $F: \mathcal{C}^{\operatorname{op}} \to \operatorname{Ab}$.
We have a clear ...
- 3,278
4
votes
1
answer
242
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Definition of a presheaf on a topological space with values in an abelian category
As far as I know if $C$ be a category, then a presheaf on $C$ is simply a functor $F:C^{op}\longrightarrow Set$. Now, let $X$ be a topological space and let $O(X)$ be the set of opens of $X$. Now, if ...
- 1,470
1
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0
answers
31
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Locally free sheaves of modules form thick subcategory of sheaves of modules
I was wondering, if the following is true: If we have a constant sheaf of rings $K_{X}$ ($K$ a field) over a topological space $X$ and an exact sequence of $K_{X}$-modules
$M_{1}\xrightarrow{f} M_{...
- 437
2
votes
1
answer
519
views
If $\textbf{C}$ is an abelian category,thenthe category of presheaves of abelian groups on $\textbf{C}$ is abelian.Is $\textbf{C}$ abelian necessary?
I am looking at an exercise which asks that, if, $\textbf{C}$ is abelian, then the category of presheaves of abelian groups on $\textbf{C}$ is abelian. The proof of this proceeds as you would expect, ...
1
vote
1
answer
39
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The global section functor of abelian sheaf over a topological space of dimension zero
Let $X$ be a topological space of dimension $0$. I learn from certain text claiming the functor $\Gamma(X,\;\cdot\;)$ gives rise to a categorical equivalence between the category of sheaf of abelian ...
- 797
0
votes
0
answers
241
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Lemma of dévissage from Mumford's AGII
In Mumford's & Oda's Algebraic Geometry II, on page 81 the authors give a proof for 6.12: Lemma of dévissage.
After a careful reading of the proof, I failed to understand an argument:
Theorem 6....
user526728
4
votes
2
answers
475
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Category of sheaves on a topological space is not $\mathbf{AB4^{*}}$
I want to show that the category of sheaves on a topological space ($\mathbf
{Sh}(X)$) is not $\mathbf{AB4^{*}}$. Recall that $\mathbf{AB4^{*}}$ means that the product of epimorphisms is an ...
- 378
0
votes
0
answers
64
views
Why do the finitely generated subsheaves of a sheaf form a directed system?
Suppose we have a ringed space $(X, \mathcal{O}_{X})$ with sheaf of $\mathcal{O}_{X}$-modules $\mathcal{F}$. Let $I$ be an indexing set and suppose we have a collection of subsets $U_{i} \subseteq X$ ...
- 3,693
2
votes
0
answers
40
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Bijection between $\mathcal{G}\subset \mathcal{L}\subset \mathcal{F}$ and $\mathcal{L}\subset \mathcal{F}/\mathcal{G}$
Let $\mathcal{G}\subset \mathcal{F}$ be two sheaves (say of $A$-module with a suitable ring $A$) on a topological space $X$. Is there an increasing correspondance between the sheaves $\mathcal{L}$ ...
- 1,548
25
votes
1
answer
1k
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Category of quasicoherent sheaves not abelian
Wikipedia mentions that the category of quasicoherent sheaves need not form an abelian category on general ringed spaces. Is there a `naturally occurring' example of this failing, even for locally ...
- 8,114
3
votes
1
answer
240
views
Boundedness assumption for isomorphism between some derived functors in Kashiwara and Schapira
Let $\mathscr{R}$ be a sheaf of ring on a topological space, and denote $\mathbf{D}^+(\mathscr{R})$ (resp. $\mathbf{D}^-(\mathscr{R})$, $\mathbf{D}^b(\mathscr{R})$) to be the bounded below (resp. ...
- 1,653
6
votes
2
answers
2k
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Is the category of sheaves of objects from an abelian category abelian?
Suppose we have an abelian category $\mathcal{A}$ and a topological space $X$. Let $\mathscr{F}^{\mathcal{A}}$ be the category of all sheaves of $\mathcal{A}$ objects over $X$. Is $\mathscr{F}^{\...
- 2,255
16
votes
0
answers
296
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Applications of $Ext^n$ in algebraic geometry
I have been doing a project about $\operatorname{Ext}^n$ functors for my commutative algebra class. I used the approach via extensions of degree n. Basically I have shown the long exact sequence ...
- 3,795
1
vote
0
answers
231
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When are sheafification and the embedding of sheaves into presheaves exact functors?
Let $\text{PreSh}(X, \mathsf{A})$ and $\text{Sh}(X, \mathsf{A})$ be the categories of $\mathsf{A}$-valued presheaves and sheaves on $X$, respectively. Here, $\mathsf{A}$ is a category such that we ...
- 11k
0
votes
0
answers
51
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Are there right-deformations for abelian sheaves?
A sufficient condition for the existence of a point-set derived functor is the existence of a deformation of the corresponding functor. For modules, such a deformation always exists (see section 2.3). ...
- 14.2k
10
votes
2
answers
3k
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Why can we use flabby sheaves to define cohomology?
In my algebraic geometry class, we defined sheaf cohomology using flabby sheaves, and the functor on the category of sheaves on a space $X$:
$$
D: \mathcal F \mapsto D\mathcal F
$$
where
$$
D\...
- 25.4k
10
votes
1
answer
1k
views
Example of epimorphisms such that the product is not an epimorphism in the category of sheaves
I've heard that in the category of sheaves over a topological space $X$, products of epimorphisms are not epimorphisms. I think that it's equivalent to saying that $\mathbf{Sh}(X)$ does not satisfy ...
- 5,423
0
votes
0
answers
205
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Exactness of functors as "iff"; conjecture about bifunctors
The definition of (right-/left-) exact functors is that they preserve (right-/left-) exactness of SESs. However, for some certain nice functors, as $\def\Hom{\text{Hom}\,}\Hom (A,-)$ and $A\otimes-$ ...
- 7,856
3
votes
2
answers
524
views
Does the adjugant of additive functors between abelian categories preserve the abelian structure of the hom-set?
I think the following is a counter-example. I noticed it when trying to prove that the sheafification functor induces isomorphism on the stalks (Vakil 2.4M).
As in Vakil (2.6.3) the stalk functor is ...
- 7,856
6
votes
0
answers
654
views
Mistake in contradiction argument to show that sheafification commutes with cokernel
The sheafification functor is a left-adjoint to the forgetful functor. Hence it commutes with colimits. The cokernel is a colimit. Hence the cokernel of a sheafified morphism is the same as the ...
- 7,856