All Questions
Tagged with sheaf-theory ag.algebraic-geometry
493 questions
1
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0
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333
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Does n-multiplication maps of cohomology groups vanish if it vanishes at the 0th cohomology?
In general, we know that a morphism $f=(f ^ {q})$ between universal (cohomological) $\delta$ functors $S=(S ^ {q}),T=(T ^ {q})\ $vanishes if and only if $f ^ {0} \ \colon \ S^{0} \to T^{0}$ vanishes....
- 945
11
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5
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8k
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When is the push-forward of the structure sheaf locally free
Let $f:X\longrightarrow Y$ be a morphism of noetherian schemes. Under what conditions is $f_\ast \mathcal{O}_X$ a locally free $\mathcal{O}_Y$-module?
Example 1. Suppose that $f$ is affine. Then $f_\...
CommunityBot
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5
votes
2
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3k
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morphisms of affine schemes question
So, in chapter 2, section 2 of Hartshorne, (prop 2.3), he describes how if $\varphi : A\rightarrow B$ is a homomorphism of rings, then you get a morphism of (affine schemes):
$\newcommand{\Spec}{\...
- 57.6k
3
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1
answer
463
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For an l-adic sheaf (F_n), why is the complex F_n of finite Tor dimension?
Let $X$ be a variety and let $\mathcal{F}=(\mathcal{F}_n)_{n\geq 0}$ be a (constructible) $\ell$-adic sheaf on $X$. Let $K_n$ be the object in the derived category $D(X,\mathbf{Z}/\ell^{n+1})$ of ...
- 9,497
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2
answers
700
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Subsheaf of quotient of quasi coherent sheaves
We know that any submodule of a quotient module $\frac{M}{N}$ is of the form $\frac{K}{N}$, where $K$ is a submodule of $M$ containing $N$.
Now here is a question: Let $\cal F$ and ${\cal G}$
be quasi ...
- 42.9k
0
votes
1
answer
382
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The behavior of pure sheaves under functor Hom( F, -)
We know that a submodule A of B is pure if and only if the functor $Hom(M, -)$ is exact on the sequence
$ 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$
for every finitely presented module ...
- 15.2k
16
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0
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4k
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Sheaf cohomology and inverse limits
In proving the formal function theorem, Grothendieck uses a rather technical lemma in EGA 0-III.13:
Lemma: Let $\mathcal{F}_n$ be an inverse system of sheaves on a space $X$ with surjective ...
- 25.6k
34
votes
4
answers
15k
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When will the pushforward of a structure sheaf still be a structure sheaf?
Let $f:X\rightarrow Y$ be a morphism of schemes.
When $PicY\rightarrow PicX$ is an embedding and $f_{*}\mathscr{O}_{X}$ is invertible, it is the structure sheaf of $Y$.
In the proof of Zariski's Main ...
- 20.5k
5
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0
answers
374
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Sheaf Cohomology on Zariski-Riemann Spaces
Can sheaf cohomology on the Zariski-Riemann spaces give some sort of classification for field extensions (even just for function fields)? If not, are there any significant or useful results (e.g. for ...
- 468
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1
answer
800
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What kind of colimits are preserved by a certain Yoneda embedding?
(This question is related to this one)
Let $k$ be a field and consider the category $Sch/k$ of schemes over $k$, say also separable and of finite type. The Yoneda embedding
$$
Y:Sch/k \to Pre(Sch/k)
$...
CommunityBot
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4
votes
1
answer
604
views
Extension of a first order deformation of a sheaf
Given a coherent and torsion free sheaf $F$ on a smooth projective scheme $S$.
Then we have a bijection between $Ext^1_S(F,F)$ and deformations of $F$ over $k[\epsilon]$, $\epsilon^2=0$.
Assume all ...
- 66.3k
1
vote
1
answer
660
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when a section descends?
Let $C$ be a (reduced, possibly reducible, complex) projective singular curve. Let $\nu: C'\to C$ a finite surjective birational morphism. (For example the normalization, but could be some ...
- 42.9k
4
votes
2
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570
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If one wants to work with $Q_l$-adic sheaves, should the scheme be of finite type over a 1-dimensional one?
In section 6 of his 'Adic Formalism' T. Ekedahl states that $l$-adic sheaves 'behave nicely' for finite type separated schemes over $S$ that is regular of dimension $\le 1$. Is the dimension ...
- 1,576
52
votes
7
answers
5k
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What does a projective resolution mean geometrically?
For R a commutative ring and M an R-module, we can always find a projective resolution of M which replaces M by a sequence of projective R-modules. But as R is commutative, we can consider the affine ...
- 19.6k
5
votes
2
answers
985
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Chern classes in flat families
Given a smooth projective variety $X$ over an algebraically closed field $k$. Now given a another projective variety $T$ and a coherent $O_{X\times T}$-module $F$, which is flat over $T$.
Given $r,s \...
- 42.9k
18
votes
2
answers
4k
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Locally constant sheaves for the étale topology, lack of intuition about "étale-localness"
I have started studying some étale cohomology and I am trying to build up some intuition about the concept of local for the étale topology. I can understand some nice examples (like Kummer exact ...
CommunityBot
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3
votes
1
answer
736
views
About direct image of ideal sheaves
Let $\mu:X'\rightarrow X$ be a birational morphism of normal complex projective varieties.
Consider the two ideal sheaves $I_1= \mu_*\mathcal{O}_{X'}(-\sum d(E)E)$, $I_2=\mu_*\mathcal{O}_{X'}(-\sum(...
- 1,811
3
votes
1
answer
844
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A form of cohomology and base change
Let $f \colon X \to Y$ be a proper morphism of (Noetherian) schemes, $\mathcal{F} \in \mathop{Coh}(X)$. Let $i_Z \colon Z \hookrightarrow Y$ be a closed subscheme and take the inverse image $W := X \...
- 4,540
11
votes
3
answers
6k
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Serre's FAC versus Hartshorne as an introduction to sheaves in algebraic geometry
I just found an English translation of Serre's FAC at Richard Borcherds' Algebraic Geometry course web page. I really want to read it sometime. I am beginner in Algebraic Geometry, just started ...
- 12.4k
6
votes
1
answer
1k
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When are non-quasi-coherent sheaves used?
Non-quasi-coherent sheaves of $\mathcal O_X$ modules on a scheme seem like a wild concept to me; are they actually used for something?
- 8,717
1
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1
answer
405
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Realizing a restriction as direct/inverse image of sheaves
Consider the inclusion $j$ of ${\mathbb{R}}$ as the real axis of ${\mathbb{C}}$. On ${\mathbb{C}}$ I have a real polynomial algebra ${\mathbb{R}}[x,\bar{x}]$, where $\bar{x}$ denotes conjugation. ...
- 45.7k
4
votes
1
answer
3k
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Ringed and locally ringed spaces
A pair $(X,O_X)$ is a ringed space if $X$ is a topological space and $O_X$ is a sheaf of rings. If every stalk $O_{X,x}$ is a local ring, then we say that $(X,O_X)$ is a locally ringed space.
In the ...
- 45.7k
7
votes
2
answers
796
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Restriction of Ext sheaves
Let $f \colon X \to Y$ be a map of schemes, $\mathcal{F}, \mathcal{G}$ two coherent sheaves on $Y$. I'm interested in conditions which guarantee an isomorphism
$$f^{*} \mathcal{E}xt^i(\mathcal{F}, \...
CommunityBot
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36
votes
6
answers
9k
views
What is the Zariski topology good/bad for?
In a comment to this question the quotation "The Zariski Topology is the 'Wrong' topology for Algebraic Geometry" appears.
Well, so some spontaneous questions arise:
1) What is Zariski topology ...
CommunityBot
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1
vote
1
answer
629
views
The fiber of the sheaf of invariants
Let us suppose the the group $G:=\mathbb{Z}/2\mathbb{Z}=(1,i)$ freely act on a smooth projective variety/$k$ $X$ and denote by $Y$ the G.I.T. quotient $X/G$. Let $\pi:X\longrightarrow Y$ the quotinet ...
- 669
5
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1
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631
views
Does the concept of a basis for a topology on a category exist?
If we want to define a sheaf F on a topological space X and we have a basis B for the topology of X, what we can do is to define objects and restrictions for guys in B, check that they satisfy the "B-...
- 1,993
7
votes
0
answers
205
views
sheaves on thickened nodal cubics
Suppose F is an algebraically closed field (of any characteristic) and that h in F[x,y,z]
is an irreducible cubic form defining a plane curve C with a node. A lot is known about
sheaves on C; for ...
- 5,422
4
votes
3
answers
3k
views
Internal hom of sheaves
Consider a topos, i.e. the category $Shv$ of sheaves on a Grothendieck site $T$ with values in abelian groups. The category $Shv$ is symmetric monoidal with $\otimes$, the tensor product in every ...
- 19.8k
5
votes
1
answer
723
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Sheaf condition and representability in the category Top
This is a rather nice question I got from this user via private communication.
Let $\mathcal{C} = Top$ the category of topological spaces. Let $\mathcal{C}^\prime$ be the category $Funct(\mathcal{C}^{...
CommunityBot
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17
votes
2
answers
1k
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What are the merits of the different finiteness conditions on quasi-coherent sheaves?
It's my understanding that there's no disagreement about the right way to define a quasi-coherence for a sheaf $F$ of $O_X$-algebras (over a scheme, locally ringed space, or even locally ringed topos)....
- 39.3k
8
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2
answers
4k
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Closed subschemes and pulling back the structure sheaf via the inclusion map
I would just like a clarification related to closed subschemes.
If $(X,{\cal O}_X)$ is a locally ringed space and $A\subset X$ is any subset with the subspace topology then $i^{-1}{\cal O}_X$ will be ...
- 63.1k
5
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1
answer
3k
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Question about hypercohomology / spectral sequence of a complex of "almost-acyclic" sheaves
I have a very particular situation involving a (non-exact) complex $K$ of coherent sheaves on a nonsingular projective variety $X$, and I need to compute the hypercohomology of the complex. The ...
- 27k
8
votes
1
answer
562
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Category of copresheaves over commutative monoids
Let C be a symmetric monoidal category. Let Comm(C) be the category of commutative monoids in C. Consider the topos X = CoPSh(Comm(C)) of covariant functors from Comm(C) to the category Set of sets.
...
- 2,399
4
votes
1
answer
383
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How to characterize Abelian sheaves that are quasi-coherent?
Let $X$ be a scheme. Suppose you are given a sheaf of Abelian groups $\mathcal{A}$ over $X$. How can you determine if $\mathcal{A}$ is the underlying Abelian sheaf of a sheaf of $O_X$-modules? In ...
- 23.8k
4
votes
1
answer
549
views
Does this condition reduce to the correct notion of irreducibility on schemes?
Consider the category of sheaves (of sets) on the affine étale site. It's a well known fact that a morphism of schemes is a Zariski-open immersion if and only if it is an étale monomorphism, so we ...
- 19.6k
36
votes
3
answers
4k
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What is the right version of "partitions of unity implies vanishing sheaf cohomology"
There are several theorems I know of the form "Let $X$ be a locally ringed space obeying some condition like existence of partitions of unity. Let $E$ be a sheaf of $\mathcal{O}_X$ modules obeying ...
- 156k
3
votes
3
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687
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Nature of Invertible Sheaves in which there are no global sections.
EDIT: Let me try to make the question clearer.
Consider the invertible sheaves $\mathcal{O}(d)$ over the projective space $\mathbb{P}^n$ where $d\in \mathbb{Z}$. Now, if $d>0$, among many ...
- 2,132
24
votes
4
answers
6k
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What is a section?
This question comes out of the answers to Ho Chung Siu's question about vector bundles. Based on my reading, it seems that the definition of the term "section" went through several phases of ...
CommunityBot
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6
votes
2
answers
790
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Do quotients of representable sheaves represent quotients?
Here's the context for the question: Proposition 4.6 of Freitag and Kiehl's book on etale cohomology shows that a sheaf (of sets) $\mathcal{F}$ (on the site Et(X)) is constructible if and only if it ...
- 11.3k
15
votes
2
answers
2k
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Total Spaces of Quasicoherent Sheaves
You can construct a total space of a quasicoherent sheaf on an scheme by taking relative spec of the symmetric algebra of the dual sheaf. For locally free sheaves, you get vector bundles, and every ...
CommunityBot
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6
votes
2
answers
1k
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Explicit Direct Summands in the Decomposition Theorem
Let f:X→Y be a semismall resolution of singularities. Then the pushforward of the constant sheaf on X is a semisimple perverse sheaf on Y. Under these conditions, I know how to calculate the ...
- 15.1k
11
votes
2
answers
2k
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Finiteness conditions on simplicial sheaves/presheaves
Could someone give an overview, or just some examples, of "finiteness conditions" for simplicial sheaves/presheaves and/or simplicial schemes? Any answer or comment about this would be interesting, ...
- 13.6k
2
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1
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406
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Are there any criteria for a presheaf which is an etale sheaf to be a sheaf in the fppf topology?
I am happy to hear answers to variants too. For instance, my situation I actually have a sheaf in the smooth topology.
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