All Questions
Tagged with sheaf-theory ag.algebraic-geometry
493 questions
2
votes
0
answers
131
views
$m$-regularity of sheaves
This is Lemma 1.4 on Green and Lazarsfeld's Some results on the syzygies of finite sets and algebraic curves. Let $X$ be a closed subscheme of $\mathbb{P}^r$. Suppose the ideal sheaf $\mathcal{I}$ of $...
- 439
4
votes
2
answers
416
views
Is any constant Zariski sheaf already a Nisnevich sheaf?
Lat $A$ be a set and $\underline{A}$ the associated constant Zariski sheaf on the category $Sm/S$ of schemes which are smooth over $S$ for a fixed base scheme $S$. Is $\underline{A}$ already a (...
- 1,906
3
votes
1
answer
382
views
Is the perfection (perfect closure) presheaf a sheaf?
The perfection of a ring $A$ of prime characteristic $p$ is the perfect ring $A_\rm{pf}=$ lim{$A\to A\to ...$} where all maps are Frobenius. It does not commute with products, as was shown by YCor in ...
- 5,429
1
vote
1
answer
2k
views
Pullback map on global sections surjective
Let $X,Y$ be two irreducible, projective $k$-schemes. $k$ is assumed to be algeraically closed. Consider a dominant morphism $f: X \to Y$ between them which is not an isomorphism!
Let $\mathcal{L}$ ...
- 39.3k
6
votes
1
answer
761
views
The Yoneda pairing, hypercohomology, and cup product
Let $\mathcal{F}$ and $\mathcal{G}$ be coherent analytic sheaves on $\mathbb{P}^n$. Let $\mathcal{F}_\bullet$ be a locally free resolution of $\mathcal{F}$. In Principles of Algebraic Geometry by ...
- 294
6
votes
1
answer
494
views
Prove category of constructible sheaves is abelian
Let $X$ be a nice enough topological space, perhaps a complex algebraic variety with its analytic topology. I'm hoping someone could help me prove that the category $\text{Constr}(X)$ of ...
- 148k
4
votes
1
answer
675
views
Jordan–Hölder sequence for $\mu$-semi stable sheaves
Let $X$ be a smooth variety over $\mathbb{C}$, and let $\omega \in \operatorname{Pic}(X)_\mathbb{R}$ be an ample class.
I would like to know if any $\mu_\omega$-semistable sheaf $E \in \operatorname{...
CommunityBot
- 1
3
votes
1
answer
195
views
Finitely generated sheaf of algebras over geometric points
I would like to ask if the following is true or not: Let $S$ a scheme and $X$ a $S$-scheme which is proper and flat. Let $\mathcal{F}$ a sheaf of $\mathcal{O}_{X}$-algebras over $X$. Let's suppose ...
- 2,928
4
votes
1
answer
231
views
Morphisms of flat families of sheaves
$X$: projective scheme over a scheme $S$.
$E, F$: $\mathscr{O}_X$-modules, flat/$S$
$\phi$: $E \rightarrow F$ : morphism s.t. $\phi_t$: $E_t \rightarrow F_t$ is zero morphism for all $t \in S$
Then,...
- 479
3
votes
0
answers
202
views
What's wrong with higher dimensional nearby cycles?
Suppose we have a complex algebraic variety $X$ with a map $f: X \to \mathbb{C}$ with $Y=f^{-1}(0)$. Let $\overset{\sim}{\mathbb{C}}$ be the universal cover of $\mathbb{C}-\{0\}$ and consider the ...
- 3,019
31
votes
3
answers
4k
views
Sheaf description of $G$-bundles
Now, among algebraic geometers, at least, it is well known that there is an equivalence between locally free $\mathcal{O}_X$-modules of rank $n$ and vector bundles of rank $n$. So, equivalently, ...
- 63.9k
3
votes
1
answer
315
views
What to call a morphism of sites inducing an equivalence on categories of sheaves?
Is there a standard term for a morphism of sites $(C,J)\to(C',J')$ which induces an equivalence on sheaf categories $\operatorname{Sh}(C',J')\xrightarrow\sim\operatorname{Sh}(C,J)$? The nLab entry ...
- 64k
0
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
1
vote
0
answers
262
views
Devissage lemma (Mumford's & Oda's AG II)
This question is part II of my proof reading of Lemma of devissage
from Mumford's & Oda's Algebraic Geometry II, findable on page 81; Theorem 6.12:
Theorem 6.12 (“Lemma of devissage”). Let $K$...
- 6,038
5
votes
0
answers
268
views
Coherent cohomological dimension and affine morphisms
For simplicity, all varieties in this question are quasiprojective varieties over an algebraically closed field of characteristic $0$.
The coherent cohomological dimension $cd(X)$ of a variety $X$ is ...
- 59
4
votes
1
answer
216
views
$G$-torsor for topological space compared to that for sheaf of groups
I just read about the definitions about torsor of sheaf of groups and get a bit confused.
How does the notion of $G$-torsor for a topological space compared to that of a sheaf of groups? Is there a ...
- 1,906
18
votes
0
answers
548
views
Donaldson-Thomas Theory and "Quantum Foam" for Mathematicians
Let $X$ be a smooth, projective Calabi-Yau threefold. From an algebro-geometric perspective, the Donaldson-Thomas invariants $\text{DT}_{\beta, n}(X)$ are virtual counts of ideal sheaves on $X$ with ...
- 1,701
3
votes
0
answers
81
views
Image of Obstruction Map for Relative Quot-scheme
Let $f: X \to S$ be a projective morphisms between finite-type projective schemes, and $\mathcal{O}_{X}(1)$ an $f$-ample line bundle. Given an $S$-flat coherent $\mathcal{O}_{X}$-module $\mathcal{H}$ ...
- 1,701
2
votes
1
answer
374
views
Local question and descent category for a quasi-coherent sheaf on $\mathbb{G}_m$-gerbe
Update: I removed what I thought was unecessary and tried to be more straightforward in the hope to get an answer.
Context:
Suppose I have a $\mathbb{G}_m$-gerbe $\mathcal{G}$ over a scheme $X$ with ...
- 45.7k
3
votes
0
answers
285
views
What is the logical progression in algebraic tools for studying spaces (varieties -> schemes, sheaves, topos etc.)?
Some algebraists (Cartier, Weil, Atiyah, etc.) sometimes speak of geometry as a long history of essentially asking the same question—"what is space, and how would one describe a space uniquely". ...
- 35.5k
2
votes
0
answers
645
views
Direct image functor commuting with infinite direct sum of sheaves
Normally I would think this kind of question doesn't belong on overflow, but I haven't been able to find an answer anywhere else, so perhaps it is not so trivial.
Let $f: X \rightarrow Y$ be a ...
- 453
2
votes
0
answers
293
views
Direct sums of invertible sheaves commuting with global sections and the functor of points approach
I am looking at the Stacks Project's treatment of the functor of points for projective space.
Let's restrict to the case that $S$ is a graded ring, generated by $S_{1}$ as an $S_{0}$ algebra. The ...
- 453
5
votes
1
answer
1k
views
Purity and skyscraper sheaves
In "The Geometry of moduli spaces of sheaves" a coherent sheaf $\mathcal{F}$ is defined to be pure of dimension $d$ if dim$(\mathcal{E})=d$ for all non-trivial proper subsheaves $\mathcal{E} \subset \...
- 5,857
3
votes
1
answer
467
views
Restriction of Ext sheaves on closed subschemes
Let $f:X\rightarrow C$ be a morphism, where $C$ is a smooth curve. For $t\in C$ let $i_t:X_t = f^{-1}(t)\rightarrow X$ be the inclusion of the fiber of $f$ over $t$, and let $\mathcal{F}$ a coherent ...
- 23.3k
3
votes
0
answers
375
views
Equivariant sheafs and $G$ actions on modules
I am reading Simpson's paper on The Hodge filtration on nonabelian cohomology. In particular Chapter 5 (p.24) and I am confused about the notion of a group acting on an equivariant sheaf.
The set up ...
- 296
1
vote
0
answers
343
views
Cokernel of section of a general coherent sheaf
Given a scheme $X$ and an $\mathcal{O}_{X}$-module $\mathscr{E}$, we know that a section $s \in H^{0}(X, \mathscr{E})$ is equivalent to a morphism $s :\mathcal{O}_{X} \to \mathscr{E}$. It is the ...
- 1,701
7
votes
2
answers
627
views
Vanishing of higher direct image of finite morphisms relative to the fppf topology
Let $f:X \to Y$ be a finite morphism of schemes.
Let $\mathcal{F}$ be a sheaf of abelian groups on the the etale site of $X$ then we know that $R^{i}f_{*} \mathcal{F} = 0$. Is this statement also true ...
- 2,928
2
votes
0
answers
315
views
Any vector bundle can be twisted to have sections
Let $X$ be an integral scheme proper over $\mathbb{C}$. Let $V$ be a locally free $\mathcal{O}_X$-module of positive finite rank. Does there necessarily exist a locally free $\mathcal{O}_X$-module of ...
CommunityBot
- 1
4
votes
0
answers
205
views
Sheaf-type property for Derived Categories?
Suppose $X$ is a finite dimensional complex space (I'm happy to restrict to $X$ being a scheme of finite type over $\mathbb C$ as well). I'm wondering if the following sheaf-like properties hold for ...
- 356
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
1
vote
1
answer
362
views
Coherent locally free sheaves on projective varieties
Let $k$ be a field, $X$ be a connected smooth projective $k$-scheme. Let $f:X\rightarrow X$ be a finite $k$-morphism that is surjective on the underlying topological spaces. Suppose $f$ has degree $n$....
- 66.3k
2
votes
1
answer
506
views
Ext sheaves as extension by zero of locally free sheaves
Let $X$ be a complex projective manifold and let $\phi \colon E \hookrightarrow F$ an injection of locally free sheaves. Then we have a sequence of coherent sheaves
$$
0 \to E \to F \to F/E \to 0
$$
...
- 39.3k
26
votes
1
answer
4k
views
Affine scheme on spec(A) of a ring A as the sheafification of a pre-sheave on spec(A)?
It is obvious that there is a parallel between the definition of structure sheaf of $\operatorname{Spec}(A)$
versus the sheafification of a pre-sheaf.
The definition of the sheaf $\mathscr F^+$ ...
CommunityBot
- 1
9
votes
1
answer
400
views
The (co)tangent sheaf of a topological space
Let $X$ be a topological space (assume additional assumptions if needed) and denote by $\mathcal O _X$ its sheaf of $\Bbbk$-valued continuous functions where $\Bbbk$ is $\mathbb{R}$ or $\mathbb{C}$ ...
- 55.9k
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 \...
- 4,111
2
votes
1
answer
1k
views
Cohomology of tangent sheaf of a hypersurface
Let $X\subset\mathbb{P}^n$ be an irreducible and reduced hypersurface of degree $d$. How can one explicitly compute the dimension of the vector spaces $H^0(X,T_X),H^1(X,T_X),H^2(X,T_X)$? Here $T_X$ is ...
- 39.3k
5
votes
0
answers
375
views
What is an example of a cokernel $B/\phi(A)$ in group schemes which does not have $A=\mu_d$ and requires the fppf topology to be a sheaf?
Let $S$ be affine. A bit of background: Let us think of $S$-group schemes as abelian sheaves over a given site (etale, Zariski, fppf, etc). When we take a cokernel of a morphism $\phi$ this category: $...
- 3,539
1
vote
0
answers
654
views
Sheafification map is surjective
This is not a research level problem for sure. But, similar question was asked by some one else $2$ years back on Stack exchange has not received any attention. So, I thought it does not suit there....
- 6,023
1
vote
1
answer
187
views
Subbundle generated by linearly dependent sections
On $\mathbb{P}^1$ consider the trivial bundle $\mathcal{O}\oplus \mathcal{O}$, and the subbundle $\mathcal{L}_{a,b}\subset\mathcal{O}\oplus \mathcal{O}$ that on an open subset $U$ of $\mathbb{P}^1$ is ...
- 39.3k
7
votes
1
answer
537
views
Is the quotient presheaf $\mathbb{G}_m/\mu_p$ an étale sheaf?
I apologize if the question is bit trivial for mathoverflow, but I asked on stack exchange a while ago and haven't got any answer.
Let $k$ be a field of characteristic $p > 0$. Consider the ...
- 148k
1
vote
0
answers
309
views
Restriction of the sheaf of relative differentials
Let $f:X\rightarrow C$ be a morphism, where $C$ is a smooth curve, and let $\Omega_f$ be the sheaf of relative differentials.
For $t\in C$ let $i_t:X_t = f^{-1}(t)\rightarrow X$ be the inclusion of ...
CommunityBot
- 1
14
votes
3
answers
1k
views
Counterexamples to gluing complexes of sheaves
Note: I asked the question below last week on MathSE but received no answer.
Background:
I have read the claim that perverse sheaves behave more like sheaves than like complexes of sheaves. This ...
- 9,019
5
votes
0
answers
859
views
How to construct the espace étalé (space of sections) for an arbitrary category?
I want to consider the sheaf valued in an arbitrary category (not only of sets, groups, modules and so on) on a topological space, using the language of étalé space.
In all references I am reading (...
- 35.5k
11
votes
1
answer
1k
views
Etalé space construction for presheaves on a Grothendieck site
As it is described for example in [Mac Lane-Moerdijk, Sheaves in Geometry and Logic, II.6.], one can construct the sheafification functor very lucidly by associating to a presheaf a certain bundle (cf....
- 23.3k
1
vote
0
answers
170
views
Espace étalé for derived category
It is known that for a sheaf $\mathcal{F}$ on $X$, we can associate $X_\mathcal{F}$, the étalé space of $\mathcal{F}$ over $X$ such that section of $X_\mathcal{F}$ coincides with section of $\mathcal{...
- 23.3k
2
votes
0
answers
132
views
Compact generation of quasicoherent sheaves on mapping stack
Let $k$ be a field of characteristic $0$, and let $\mathcal{C}= \mathbf{Vect}_k^{\leq 0}$ be the $\infty$-category of vector spaces concentrated in degrees $\leq 0$. Consider the category $\mathbf{Pr}(...
- 3,019
4
votes
0
answers
347
views
How is the restriction of the dualizing sheaf to an irreducible component related to the dualizing sheaf of the component?
$\DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\hom}{\mathcal{Hom}} \DeclareMathOperator{\ox}{\mathcal{O}_X}$Let $f:X \to Y$ be a proper morphism. In section 6.4. of Liu's book he introduces ...
- 435
1
vote
0
answers
104
views
A sheaf for factorization
Let $R$ be a commutative ring with $1$ and let $X$ be the space of connected componens of $Spec (R) $ with Zariski topology ( The boolean spectrum of $R $ )and let for each $x\in X$ there exists a ...
- 11
6
votes
1
answer
239
views
What does an ideal correspond to in the internal language of sheaves?
Suppose I have a sheaf $\mathcal F$ in some topos $\mathrm{Sh}(\mathcal C)$. Then this becomes the sheaf of rings from algebraic geometry when described as a ring in the internal language of the topos....
- 3,312
1
vote
1
answer
156
views
An ideal and its J-radical
Let $R$ be a commutative ring with $1$ and $I$ be an ideal of $R$. Now let $J=\cap_{I\subseteq m\in Max(R)}m$. Set $A:=\{p\in Spec(R): I\subseteq p\}$ and $B:=\{p\in Spec(R): J\subseteq p\}$, where $...
- 63.9k