All Questions
Tagged with ag.algebraic-geometry sheaf-theory
493 questions
-1
votes
1
answer
259
views
Pure Quotient and pure sub-object
Let $\mathcal{C}$ be the category of modules over a ring.
Let also $\mathcal{F}$ be a class of objects in $\mathcal C$ closed under pure subobject (pure quotient) and direct limit. Is $\mathcal{F}$ ...
- 25.8k
0
votes
0
answers
85
views
$\mathcal{F}$-- twists of Lie algebras
I am trying to figure out with Drindfel's Opers. Let us consider Lie group $G$ and $G$ -- bundle $\mathcal{F}$ on the smooth algebraic curve $Y$. Can anybody help me and clarify definition of the $\...
- 181
1
vote
1
answer
143
views
Sheaves on the site of $\pi$-sets
Let $\pi$ be a group, and let $\mathcal{C}$ be the site whose underlying category is that of $\pi$-sets (with $\pi$-linear maps as morphisms). The covers are jointly surjective families of such $\pi$-...
- 73.2k
4
votes
2
answers
2k
views
Modules, Sheaves and Vector bundles
Given a graded ring $S$ and a graded S-module $M$ we can carry out a construction in order to get $\tilde{M}$, which is a sheaf over the scheme $\mathrm{Proj}~ S$. With this in view, I have an ...
CommunityBot
- 1
0
votes
0
answers
307
views
local systems, duals, cohomology
Let $U=\mathbb{P}^1-\{p_1, \ldots, p_n\}$ be a Zariski open subset of the projective line. Consider a rank $r$ local system of complex vector spaces $V$ on $U$ and assume that the monodromy ...
- 11
10
votes
1
answer
777
views
Why is there no stack of $\ell$-adic sheaves on a curve?
One of the main players in the categorical geometric langlands correspondence is the moduli stack of rank n integrable connections on a complex curve. The reason for considering such objects is that ...
- 17.8k
4
votes
0
answers
166
views
Homotopy-theoretic measure of operations on sheaves failing to be sheaves
Here's something I've been wondering about for a few weeks:
Consider a topological space $X$ and a sheaf of rings $\mathscr O_X$ on $X$. Suppose $\mathscr{F}$ and $\mathscr{G}$ are $\mathscr O_X$ ...
CommunityBot
- 1
10
votes
3
answers
2k
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Representable Presheaf
I have a very quick question. Is there an easy example of a representable presheaf on a site that is not a sheaf? This certainly can't happen on a small FPPF site so I would expect a counterexample to ...
- 135
0
votes
1
answer
175
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Stability of $T_{\mathbb{P}^2}$ and $\Omega_{\mathbb{P}^2}$?
How can one prove that the tangent bundle $T_{\mathbb{P}^2}$ and its dual $\Omega_{\mathbb{P}^2}$ are stable vector bundles with respect to $\mathcal{O}_{\mathbb{P}^2}(1)$? Similarly, is it true that $...
CommunityBot
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4
votes
1
answer
259
views
$f^{-1}\mathcal I \cdot \mathcal O_X$ vs $f^\ast \mathcal I$
Let $X$ ad $Y$ be (noetherian) schemes and let $\mathcal I \subseteq \mathcal O_Y$ be a sheaf of ideals on $Y$. Let $f \colon X \to Y$ be a morphism of schemes. In general the sheaf $f^\ast \mathcal I$...
- 39.3k
3
votes
1
answer
245
views
Families of local rings coming from a locally ringed space
Let $X$ be a topological space. For every point $x \in X$ let $R_x$ be a local ring. Under what (necessary / sufficient / necessary and sufficient) conditions is there a sheaf ${\cal O}_X$ such that $(...
- 5,407
14
votes
1
answer
2k
views
Hypercohomology of a complex via Cech cohomology
Let $X$ be a reasonable topological space. If $\mathcal{F}$ is a sheaf of abelian groups then Cech cohomology gives us a method to compute the cohomology groups $H^p(X, \mathcal{F})$ - the main input ...
- 3,396
16
votes
4
answers
2k
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Coboundaries and Gluing in Cech Cohomology - Intuition?
I'm trying to develop an intuition for Cech cohomology geometrically, but am currently failing. A lot of people seem to say that the groups $H^n$ measure obstructions to gluing local sections to get ...
- 34.7k
2
votes
0
answers
180
views
on geometric Satake and functions
Let $G(F)/G(O)$ the affine grassmanian with $F=k((t))$ where $k$ is a finite field.
For $\lambda$ a dominant cocharacter, we have by Cartan decomposition the schubert strata $\overline{Gr^{\lambda}}$....
- 3,472
19
votes
6
answers
4k
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Understanding Adjointness of Sheaves in Algebraic Geometry
Pushforward and pullback are very basic operations in algebraic geometry, as is the adjointness between them. I worked out a very careful of adjointness of sheaves (below) when I was working out of ...
- 148k
1
vote
1
answer
1k
views
Sheaf Hom and the functor Hom
Let $\varepsilon: 0\to A\to B \to C\to 0$ be an exact sequence of ${\cal O}_X$-modules with $X$ a quasi-compact space. $\varepsilon$ is called pure if the induced sequence
$0\rightarrow Hom(F,A)\...
- 23k
0
votes
0
answers
859
views
restriction and pullback of representable etale sheaf along closed immersion
I find that the restriction and pullback of representable etale sheaf along closed immersion are very confusing. I think they are different in general, I hope some experts can confirm my understanding ...
- 997
7
votes
3
answers
2k
views
Are the global sections of a vector bundle a projective module?
Given a scheme $X$ with structure sheaf $\mathcal{O}_X$, we can associate to each $\mathcal{O}_X$-module $\mathcal{F}$ its global sections $\Gamma(\mathcal{F})$, which gets the structure of a $\Gamma(\...
- 3,101
0
votes
0
answers
130
views
morphisms in the construction of the moduli space of curves by mumford
Hi fellow mathematicians,
I'm just studying mumfords proof of the existence of a coarse modulispace for curves of genus $g$in his great GIT book. On page 102 (in the proof of proposition 5.2) he ...
- 9
3
votes
0
answers
306
views
Does this property of subgroups (or sheaves of ideals) already have a name?
I'm constructing an example of a group which has a particular property on its subgroups, and the property looks like something that might have been considered before.
Fix a group $G$ and a pair of ...
- 35.5k
1
vote
0
answers
101
views
How can one compute the cohomology of $i'^*C$, for $i':\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\setminus \{0\} $?
For an (etale or 'topological', constructible bounded) complex of sheaves $C$ on $X'=\mathbb{A}^{N}\setminus \{0\} $, $i'$ being the embedding $\mathbb{A}^{N-1}\setminus \{0\}\to \mathbb{A}^{N}\...
- 16.9k
13
votes
2
answers
3k
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Interpreting $f^*f_*$
For a morphism of schemes $f: X\rightarrow Y$, one often considers the function $f^*f_*$ on sheaves. For example, this appears in adjunction for sheaves of $\mathcal{O}_X$-modules, the projection ...
- 39.3k
2
votes
1
answer
569
views
Inclusion of logarithmic de-Rham complex into differentials
Let $X$ be a complex manifold and $D$ a normal crossing divisor. Let $U = X - D$ and $j: U \rightarrow X$ the natural map. Voisen observes that there is a natural inclusion $$\Omega^k_X(\log D) \...
- 42.9k
6
votes
1
answer
1k
views
Tensor product of sheaves separated?
Let $(X, O_X)$ be an arbitrary scheme and $\mathcal{F}$ a presheaf of $O_X$-modules. The presheaf $\mathcal{F}$ is said to be separated if the natural map to it's sheafification $\mathcal{F}^+$ is an ...
- 4,111
6
votes
2
answers
385
views
cohomology and $j_!$
I have a projective variety $X$ and an open immersion $j : U \to X$.
Say I have a sheaf, locally free in my case of interest, $\mathcal{S}$ on $U$. Is there any reasonable relationship between $H^i(X,...
- 23.5k
7
votes
1
answer
451
views
Coverage, itself considered as a presheaf
A coverage $J$ on a category $C$ assigns to an object $U$ of $C$ a set of covering families $J(U)$. The covering families are required to be stable under pullback, which amounts to requiring that for ...
- 66.8k
2
votes
0
answers
175
views
Does mapping cylinder category have enough injectives?
Let $A, B$ be two abelian categories, and $\tau : A \to B$ a left exact functor.
We define a category $C$ as follows:
objects: triples $(M, N, \varphi)$ where $M\in A, N\in B$ and $\varphi: N\to \...
- 945
2
votes
1
answer
258
views
Carving out subsheaves of local hom-sheaves of stacks of categories
Recall from my previous question the definition of a local hom-sheaf of a stack of categories. I am interested in stacks of categories such that the underlying stack of groupoids is a moduli stack.
...
CommunityBot
- 1
10
votes
4
answers
4k
views
Why is an injective quasi-coherent sheaf's restriction to an open subset still an injective object?
X is a Noetherian scheme, F is an injective object in the category of quasi-coherent sheaves on X. U is an open subset of X. Why F's restriction on U is still an injective object in the category of ...
CommunityBot
- 1
0
votes
1
answer
672
views
left adjoint to restriction functor
We know that for an immersion $j:U \to X$ the restriction functor $j^*:{\cal O}_X-mod \to {\cal O}_u-mod$ has a left adjoint $j!$.
I am looking for some condition to deduce that $j!$ takes its values ...
- 63.1k
5
votes
2
answers
1k
views
trying to understand the support of the sheaf of relative differentials
So I'm trying to understand a proof of Belyi's theorem from http://eprints.soton.ac.uk/29785/1/b45h1koe.pdf
specifically lemma 3.4.
The setup is as follows: Let $X/\mathbb{C}$ be a curve, and let $t ...
- 11.6k
14
votes
2
answers
904
views
What's the easiest example of a morphism of topoi that is not from that of a site?
A topos is defined to be a category that's equivalent to the category of sheaves on a site. Morphisms between topoi is defined by a pair of adjoint functors that behave like pull-back/push-forward of ...
- 5,670
5
votes
0
answers
336
views
Do 'change of coefficients' functors for sheaves commute with the four functors (formalism)?
For a morphism $f$ of varieties over a field of characteristic $\neq l$ I can consider the functors $Rf_*$, $f^\ast$, $f_!$, and $f^!$ both for the corresponding derived categories of 'all' (...
- 16.9k
8
votes
2
answers
684
views
Is $H^i(X,F)$ finitely generated over $\Gamma(O_X)$ if $F$ is coherent?
Suppose $\mathcal{X}$ is a smooth quasi-projective variety over $\mathbb{C}$ (I apologize if these hypotheses have little to do with the question at hand). Let $\mathcal{F}$ be a coherent sheaf on $\...
- 7,338
21
votes
2
answers
11k
views
Elementary short exact sequence of sheaves
This question arised when I was trying to use this answer to understand Reid's "Young Person's guide to Canonical Singularities". In particular page 352 when computing the blow-up $Y\rightarrow A^2/\...
CommunityBot
- 1
5
votes
1
answer
1k
views
Euler characteristic and inclusion-exclusion
Define the Euler characteristic of a scheme to be the Euler characteristic of its structure sheaf. I remember being told that for curves, this invariant satisfies inclusion-exclusion. That is, if $C_1,...
- 42.9k
9
votes
1
answer
4k
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surjective morphism of schemes or epimorphism of sheaves?
I have a technical question coming from reading Toen's master course on stacks.
If we view schemes as locally ringed spaces then there we could define a morphism to be surjective if it the underlying ...
- 1,889
6
votes
1
answer
2k
views
Is the stalk of the (co)limit of sheaves equal to the (co)limit of the stalks?
More precisely, if $\mathcal F_i$ is a system of sheaves, is it the case that
$$
(\lim \mathcal F_i)_p = \lim ((\mathcal F_i)_p)
$$
and similarly for colimits? I can see how to get a map
$$
(\lim ...
- 63.1k
4
votes
1
answer
250
views
When do adjunctions preserve equivalence?
Let $\mathcal{C}'$ and $\mathcal{D}'$ be categories so that $\mathcal{C} \subseteq \mathcal{C}'$ and $\mathcal{D} \subseteq \mathcal{D}'$ are full subcategories. Suppose the forgetful functors $F_{\...
- 1,576
0
votes
0
answers
186
views
Do infinitesimal neighbourhoods help to compute the inverse images of coherent sheaves?
Let $i:Z\to X$ be a closed embedding of (projective) varieties; $S$ is a coherent sheaf on $X$. How could one compute $H^*(Z,i^\ast S)$ (I don't know whether I should write $H^\ast (Z,i^{-1}S)$ ...
- 16.9k
1
vote
0
answers
249
views
On inverse images with respect to Zariski-etale topology.
For a variety $X$ I define its Zariski-etale site as follows: the category is the category of etale $X$-schemes, and the coverings are Zariski ones. Note that this topology is more coarse than the ...
- 16.9k
7
votes
3
answers
2k
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Cohomology with compact support for coherent sheaves on a scheme
Is there a notion (for schemes or just locally ringed spaces) of cohomology with compact support? I guess there is for algebraic schemes over $\mathbf{C}$, but what about schemes in general? Does ...
- 419
2
votes
0
answers
358
views
What are the easiest cases of base change (for sheaves on sites)?
I have a closed embedding of schemes $i:X'\to X$, and for each of them I consider three Grothendieck topologies for the category of the corresponding (relatively) \'etale schemes: the \'etale one, the ...
- 16.9k
2
votes
0
answers
414
views
Do inverse images respect flabby sheaves?
Let $i:Y\to X$ be a closed embedding of varieties, and let $S$ be a flabby \'etale (or Nisnevich) sheaf of abelian groups on $X$. Is $i^*S$ flabby also? I am mostly interested in the case when $S=i_{x*...
- 16.9k
2
votes
1
answer
429
views
Nisnevich points
Here is a probably stupid question : If $F$ is a sheaf on the big Nisnevich site, then is the morphism $F(X) \to \amalg F(x)$ injective where the sum is over ALL the points of $X$ (not just the closed ...
CommunityBot
- 1
0
votes
1
answer
295
views
sequence of sheaves for studying intersection
I'm studying intersection of curves with a fixed plane cubic, the first case I consider is of course lines, in particular lines intersecting the cubic at only one point. The problem is quite easy and ...
- 1,811
1
vote
0
answers
236
views
Can etale $X$-schemes be lifted to $Y$, where $X$ is closed in $Y$?
For a closed embedding (of varieties) $X\to Y$ let $U/X$ be etale. Is is true that there necessarily exists an etale $U'/Y$ such that $U'_X=U$? If this is wrong in general, are there any assumptions ...
CommunityBot
- 1
8
votes
1
answer
1k
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Does one need to sheafify when defining the inverse image of a sheaf with respect to an embedding?
This seems to be a rather simple (stupid?:)) question; yet I was not able to find an answer quickly.
For a morphism $f:X\to Y$ of schemes (or topological spaces) and an (etale or topological) sheaf $...
- 721
4
votes
1
answer
1k
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Vector space structure on the tangent bundle of a scheme and relation to the tangent sheaf
First a word of warning: I am not a trained algebraic geometer, so it is possible (likely) that these questions are inappropriate for MO, if so: my apologies.
Said this: As far as I understand the ...
- 45.7k
3
votes
0
answers
877
views
The "pullback presheaf" and the proper base change theorem in topology
Let $f:X\rightarrow Y$ be a continuous map of topological spaces and let $\mathcal{F}$
be a sheaf (say of abelian groups to fix the idea) on $Y$. Define the following rule on open sets of $X$:
$$
V\...
- 7,609