All Questions
Tagged with sheaf-theory ag.algebraic-geometry
493 questions
0
votes
1
answer
188
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Support of a coherent sheaf over a fiber product scheme
I'm trying to prove the following fact which I don't know if it is true since I am not able to find a counterexample:
Let $X,S$ be two $K$-scheme of finite type with $K$ an algebraically closed field....
- 395
0
votes
2
answers
700
views
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 ...
- 55
0
votes
1
answer
750
views
Equivariant Sheaf: Explanation on Stalks
I have a question about the explanation of the data defining a so called equivariant sheaf $F$ on a scheme X from wiki: https://en.wikipedia.org/wiki/Equivariant_sheaf. Let denote by $\sigma: G \...
- 6,038
0
votes
1
answer
788
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Global to local for Ext groups and Sheaves
Let $X$ be a projective variety. The sheaf $\mathcal{E}xt^{1}(\Omega_{X},\mathcal{O}_{X})$ is supported on $Sing(X)$.
Now, there should be a theorem (perhaps by Schlessinger) that says that if $X$ ...
- 8,998
0
votes
1
answer
519
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A functorial isomorphism in derived category
This question is a direct continuation of Question 1 in this post: Two basic questions on derived categories
Let $f\colon \mathcal{A}\to\mathcal{B}$ be a left exact functor between two abelian ...
- 21.8k
0
votes
1
answer
351
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Birkhoff decomposition vanishing of the Chern numbers
Birkhoff decomposition vanishing of the Chern numbers of the holomorphic line bundles of the Birkhoff-Grothendieck decomposition, is some statement I read off in One of Connes papers. Without going ...
user39066
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 $...
- 1,663
0
votes
1
answer
672
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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 ...
- 13
0
votes
1
answer
382
views
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 ...
- 55
0
votes
2
answers
331
views
Vakil exercise on sheaf associated to the divisor of rational section
This is exercise 15.4.G. of Vakil's notes.
Let $\mathscr{L}$ be an invertible sheaf on an irreducible normal scheme $X$ with $s$ a rational section of $\mathscr{L}$. We want that $\mathscr{O}_X(\text{...
- 29
0
votes
1
answer
607
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Minimal Destabilizing Quotient
For a pure $d$-dimensional sheaf $E$ on a projective algebraic variety over a field $k$, one has the Harder-Narasimhan filtration $$0\subset E_1\subset E_2\subset...\subset E_{l-1}\subset E_l:=E,$$ ...
- 2,108
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 ...
- 379
0
votes
0
answers
98
views
A question about the sheaf supported on the zero section
Let $X$ be the total space of the cotangent sheaf on $\mathbb{P}^{2}$ and $i \colon \mathbb{P}^{2} \hookrightarrow X$ be thezero section. Suppose that $E$ is a coherent sheaf on $X$ which is set-...
0
votes
0
answers
123
views
Counit map surjective
Let $X \to Y$ a (set theoretically) surjective morphism of schemes, $L$ a line bundle/invertible sheaf on $X$ (maybe more generally a locally free coh sheaf, but let's stick firstly on invertible ...
- 6,038
0
votes
0
answers
116
views
How can I calculate $\chi(\mathscr{O}(P))$
Let be X a reduced and irreducible curve over a field $L_0$. Let $L$ an extension of $L_0$ and set
\begin{gather*}
\overline{X}=L \otimes X.
\end{gather*}
Assume $\overline{X}$ also irreducible. Now, ...
- 69
0
votes
0
answers
133
views
Higher direct images of locally constant etale sheaf under smooth proper map locally constant
Let $f:X \to Y$ a surjective smooth proper map between Noetherian schemes and $F$ a locally constant sheaf on small etale site of $X$.
Question: Refering to Donu Arapura's answer here, how to see that ...
- 6,038
0
votes
0
answers
57
views
Lifting of quadrics containing hyperplane section for projectively normal curves
Let $C \subset \mathbb{P}^r$ be a projective curve (over $k=\mathbb{C}$), smooth, irreducible and nondegenerate of degree $d$, ie the embedding line bundle $\mathcal{O}_C(1)=(\mathcal{O}_{\mathbb{P}^r}...
- 6,038
0
votes
0
answers
156
views
A stalk criterion for unit map to be an isomorphism on étale site
Let $f: X \to Y$ be a morphism of schemes and $\mathcal{F}$ sheaf of sets/Abelian groups on the small étale site $Y_{ét}$. Assume we manage somehow to show thatat every geometric point $\overline{y} \...
- 6,038
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votes
0
answers
91
views
Why does the associated sheaf vanish?
I am learning local cohomology from Hartshorne’s book Local Cohomology.
My question is about understanding a line in the proof of proposition 1.11 in this book.
The set-up for proposition 1.11 is that ...
- 639
0
votes
0
answers
247
views
Is $\mathbb{C}^*$ not irreducible, or is every locally constant sheaf on $\mathbb{C}^*$ constant?
I am running into contradiction from the following set of definitions, propositions, and assumptions. Can anyone spot where I'm off?
Definition A sheaf $\mathcal{F}$ on a topological space $X$ is ...
- 511
0
votes
0
answers
343
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Stalks of Sheaves
I saw a statement in a paper like what follows:
Let $X=\text{Spec} A$ be an affine scheme, and let $\mathscr{F}$ be a sheaf of $\mathscr{O}_X$-modules on $X$. For each geometric point $x$ of $X$ we ...
- 1
0
votes
0
answers
100
views
Quotient of $\text{Proj}(A)$ by the action of a finite group
Let $X$ be $ \operatorname{Proj}(A)$ for some graded ring A, and let $G$ be a finite group acting on $A$ with morphisms of graded rings; consequently $G$ acts on $X$.
I know I can write $X = \bigcup_{...
- 1
0
votes
0
answers
391
views
Subsheaves of constant sheaves
Let $X$ be a connected topological space. I am looking for examples of a locally constant subsheaf (of $\mathbb{C}$-vector spaces) of a constant sheaf (of $\mathbb{C}$-vector spaces) on X, which is ...
- 2,323
0
votes
0
answers
157
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Hyperplane which does not contain any associated point of qc sheaf $\mathcal{F}$
I have a question about an argument on $m$-regularity
from 'Fundamental Algebraic Geometry' by Fantechi on page 114, Chapter
5.2: Castelnovo-Mumford regularity. The statement is:
Let $k$ be a field ...
- 6,038
0
votes
0
answers
185
views
Recipe for resolving a coherent sheaf
Let $X$ be a complex manifold and let $V\subset X$ be a subvariety. Let $F\rightarrow V$ be a holomorphic vector bundle over $V$ and let $\mathcal{S}=\Gamma(F)$ be the sheaf of holomorphic section of $...
- 789
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votes
0
answers
413
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When are the cotangent and tangent sheaves isomorphic?
Let $X$ be an $S$-scheme. Under what conditions, if any, is the cotangent sheaf $\Omega_{X/S}$ isomorphic to the tangent sheaf $\Theta_{X/S}$ as $\mathcal{O}_X$- modules? For example, given a ...
- 327
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votes
0
answers
180
views
Theorem on Formal Schemes
I have few questions about the proof of Thm 7.5 from P. Wagreich's paper "Elliptic Singularities of Surfaces" (page 452):
Denote by $\mathcal{N}$ the kernel of $A \to H^0(O_Z)$ (sorry, I haven't ...
- 6,038
0
votes
0
answers
657
views
Mistake in Hartshorne's Exercise II.1.1?
This is really an elementary question, but let me state it. Exercise 1.1 of the second Chapter of Hartshorne's Algebraic Geometry ask to prove that the sheaf associated to the presheaf sending every ...
- 149
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votes
0
answers
165
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Examples of degree zero, rank one reflexive sheaves without r-th roots
Let $X$ be a normal, projective surface (or more generally a variety) over $\mathbb{C}$ (i.e., $X$ is irreducible). Fix a polarisation on $X$. I am looking for examples of rank one, degree zero (...
- 1,593
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votes
0
answers
144
views
Induced Morphism on Fibre Product
Let $X$ be a proper $k$-scheme and $k \subset k'$ a field extension. Consider the fibre product \ base change $X' = X \otimes _k k'$.
Let $\mathcal{F} \in Coh(X)$ and $p: X' \to X$ the canonical ...
- 6,038
0
votes
0
answers
182
views
Analytic-Local Germs of "General" Sections
Let $C$ be an algebraic curve over an algebraically closed field $k$ of characteristic $0$, and let $\mathcal{L}$ be a base-point-free line bundle on $C$. Furthermore, let $p \in C$ be a smooth point, ...
0
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0
answers
193
views
About the definition of flat twisted sheaves
Flat twisted sheaves are mentioned in Căldăraru's thesis (Lemma 2.1.2 for example), but I'm confused about how they should be defined. I have in mind some possibilities, given an $\alpha$-twisted ...
- 2,079
0
votes
0
answers
303
views
Sheaves invariant for group actions two equivalent definitions?
Given a (topological) group acting on $X$ a topological space continuously.
Then we have the category $Sh_G (X)$, it's the full subcategory of $Sh(X)$ consisting on the sheaves $\mathcal{L}$ such ...
- 335
0
votes
0
answers
265
views
Explicit adjunction formula and local top form
I am reading the section 4.2 in Kollar-Mori, where they provide the explicit equations for Du Val Singularities. In the course of the proof, they reduce to studying the equation $x^2+f(x,y)=0$ in a ...
- 625
0
votes
0
answers
191
views
First sheaf cohomology $H^1(\mathscr{O}_D, \mathbb{D})=0$
Given a finite divisor$$D=p_1+\dots +p_m -q_1 -\dots -q_n$$on the unit disk $\mathbb{D}$, does it necessarily follow that the first sheaf cohomology group equals zero, i.e.$$H^1(\mathscr{O}_D, \mathbb{...
user76513
0
votes
0
answers
239
views
Cohomology group vs sheaf of cohomology group
Suppose $F$ is a coherent sheaf on a smooth (algebraic or complex) variety $X$. Then we can consider the cohomology groups $$H^p(X,F)$$ for all $i$. Now, let we consider the sheaf $$\mathcal{H}^p(X,F)$...
- 401
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
0
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0
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307
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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
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
0
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0
answers
130
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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
0
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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
votes
1
answer
177
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When morphism of complexes is homotopic to 0?
Let $f\colon A^\bullet\to I^\bullet$ be a morphism of bounded below complexes in an abelian category. Assume all $I^i$ are injective objects. Assume also that $f$ induces the zero map on cohomology.
...
- 21.8k
-1
votes
1
answer
259
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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 $\mathcal{F}$ ...
- 3