All Questions
Tagged with ag.algebraic-geometry sheaf-theory
493 questions
3
votes
0
answers
551
views
Defining Inertia Stack
Let $X$ be a topos and $F: \zeta \rightarrow X$ a stack on $X$. Now in the paper http://arxiv.org/pdf/math/0411337v2.pdf ( Definitions 2.1.1.5 and 2.1.1.1) of Lieblich, he describes the inertia stack ...
- 31
4
votes
1
answer
409
views
Does the nearby cycle functor commute with the Verdier duality?
I would be interested to know the answer to the above question for the constructible bounded derived category on complex analytic or complex algebraic manifolds (or some other context). A reference ...
- 21.8k
6
votes
1
answer
2k
views
Information and intuition packed in the Chern character for coherent sheaves
even after quite some time learning it, I still get somehow puzzled by the Chern character. Let me recall some stuff to get notation and setting.
Let us consider a smooth projective algebraic variety ...
- 155
6
votes
1
answer
752
views
Sections of the conormal bundle
Let $X\subset\mathbb{P}^N$ be a quadratic manifold. That is $I(X)$ is generated by quadratic polynomials $Q_1,...,Q_m$.
Let $\mathcal{I}_X$ be the ideal sheaf of $X$ and $\mathcal{I}_X/\mathcal{I}_X^...
user82886
3
votes
2
answers
488
views
Application of sheaves theory in ring theory
Is there any text that gives some applications of sheaves theory in commutative ring theory? In the other word, is any results in commutative ring theory that be verified by sheaves method?
- 49
1
vote
2
answers
276
views
Sections of a sheaf of differentials on a weighted complete intersection
Let $X\subset\mathbb{P}(a_0,...,a_N)$ be a smooth $n$-dimensional weighted complete intersection in a weighted projective space $\mathbb{P}(a_0,...,a_N)$.
Is it true that if $q\geq 1$ then $H^0(X,\...
user75933
3
votes
0
answers
334
views
Which sheaves on a projective bundle are flat over the base scheme?
Assume $X$ is a noetherian scheme over $\mathbb{C}$ and $E$ a locally free sheaf of finite rank on $X$. Denote the the associated projective bundle by $f: \mathbb{P}(E)\rightarrow X$.
Are there any ...
- 1,025
3
votes
1
answer
515
views
For what kind of sheaves can we always extend a sheaf map from a closed subset to the whole space?
Let $X$ be a topological space. We know that a sheaf on $X$ is call soft if for any closed subset $Z$ of $X$, a section on $Z$ can be always extend to a section on $X$.
Now we consider a similar ...
- 9,019
3
votes
0
answers
102
views
Can we extend a homotopy invertible chain morphisms between complexes of sheaves from a closed subspace to the whole space?
Let $X$ be a (say, topological) space and $i: Z\hookrightarrow X$ be a closed subspace. Let $Sh(X)$ and $Sh(Z)$ denote the categories of sheaves of abelian groups on $X$ and $Z$ respectively. ...
- 9,019
3
votes
0
answers
155
views
Elementary examples on sheaf extension
Let $V\subset\mathbb{P}^n$ be a projective variety and $C_V$ its conormal subvariety in $T^\ast\mathbb{P}^n$. Denote by $\mathscr{O}_{C_V}$ its structure sheaf, then when will the condition
$\mathit{...
- 3,187
15
votes
1
answer
1k
views
Grothendieck - sheaves as meter sticks
I'm trying to read parts of McLarty's Grothendieck on Simplicity and Generality. In the article, I read Grothendieck thought of sheaves over some topological space as meter sticks measuring it.
...
- 10.5k
2
votes
0
answers
272
views
local universal sheaf (moduli of stable sheaves)
I do not know much about moduli of sheaves and I wanted to shows that for a smooth (projective) family over a discrete valuation ring of mixed characteristics (relative dimension 3), the locally free ...
- 101
13
votes
1
answer
583
views
Which nice/deep elaborations on the (operators <-> sheaves) / (endomorphisms <-> objects) theme are there?
A linear operator $T:V\to V$ on a (say) vector space over a field $k$ is just a $k[T]$-module, and may be viewed as the sheaf $\mathscr F_T$ over $\mathbb A^1_k$, with fibre over $\lambda\in k$ equal ...
- 18.4k
2
votes
0
answers
231
views
Surjectivity locus of a morphism of families of sheaves
Let $X$ and $T$ be schemes and assume we have two coherent sheaves $\mathcal{F}$ and $\mathcal{G}$ on $X\times T$ which are flat over $T$, that is these are families of sheaves parametrized by $T$.
...
- 1,025
2
votes
2
answers
607
views
Canonical (tautological) section of a family of sheaves
A couple of months ago, i saw a construction, that somehow looks like the construction of the tautological section of the pullback of a vector bundle to its total space, i am trying to piece it ...
- 1,025
20
votes
1
answer
2k
views
Do I know what "coherent sheaf" means if I know what it means on locally Noetherian schemes?
I've been trying to convince myself that "coherent sheaf" is a natural definition. One way I might be satisfied is the following: for modules over a Noetherian ring $A$, coherent and finitely ...
- 118k
3
votes
0
answers
293
views
Is this diagram of sheaves actually Cartesian as claimed?
The question is about Corollary 1.6.2 (b) in the book by Laumon and Moret-Bailly on algebraic stacks.
There we have a scheme $S$ and morphisms $X \xrightarrow{f} Y \xrightarrow{g} Z$ of sheaves on a (...
- 825
3
votes
3
answers
600
views
A question on flasque sheaf
Let $0\to \mathscr{F}'\to\mathscr{F}\to\mathscr{F}''\to 0$ be an exact sequene of sheaves. It is well known that $\mathscr{F}$ flasque iff $\mathscr{F}''$ flasque provided $\mathscr{F}'$ is flasque. ...
- 153
1
vote
0
answers
135
views
Isomorphism of sheaves
Given a smooth projective variety $X$ and a semiample and big $\mathbb{Q}$-divisor $D$.
We denote by $R:=\sum_{n\in \mathbb{Z}_{\geq 0}} H^0(X,\mathcal{O}_X(nD))$.
Denote by $\tilde R(n)$ the quasi-...
- 1,595
2
votes
0
answers
266
views
Relationship between coherent toposes/coherent logic and coherent sheaves
I've heard it claimed that the adjective "coherent" in logic/topos theory (i.e. coherent logic, coherent toposes, coherent categories) was adopted to fit in with the terminology of coherent sheaves in ...
- 3,058
5
votes
1
answer
529
views
Evaluation maps for moduli of stable maps
Let $\overline{M}_{0,n}(\mathbb{P}^N,d)$ be the moduli space of stable maps of degree $d$ from curves of genus zero with $n$-marked points to $\mathbb{P}^N$.
Consider the product of the evaluation ...
user58604
3
votes
0
answers
422
views
What kind of ringed space $X$ has the property that a locally free sheaf is projective in Qcoh$(X)$?
It is well known that for an affine scheme $X$, every finitely generated locally free sheaf $\mathcal{E}$ is projective in the category Qcoh$(X)$. i.e. the functor $\text{Hom}_{\text{Qcoh}(X)}(\...
- 9,019
2
votes
2
answers
2k
views
Serre's Theorem for Coherent Sheaves
I recently heard a discussion about a certain of Serre which reconstructs the category of coherent sheaves of a variety $V$ as the category of modules over the homogeneous space of $V$ modulo modules ...
4
votes
0
answers
367
views
Is a locally finitely generated sheaf of modules finitely generated on sections?
"Let $(X, \mathcal{O})$ be a ringed space. A sheaf of modules $\mathcal{F}$ on $X$ is finitely generated if for all $a \in X$ there exists a neighbourhood $U$ of $a$, an integer $n$ and a surjective ...
- 191
3
votes
0
answers
217
views
Coherence of $\mathcal O_X[T]$
Let $X$ a complex manifold, and $\mathcal O_X$ the sheaf of holomorphic functions. Oka Coherence Theorem states that $\mathcal O_X$ is coherent (as $\mathcal O_X$-module).
How to prove that also the ...
- 31
5
votes
2
answers
331
views
Sheaf cohomology on non paracompact topological spaces
I have some confusion on the subject of sheaf cohomology on non-paracompact topological spaces, i hope you can help me.
My reference is Godement's book "Topologie algebrique et theorie dex faisceaux"....
- 609
0
votes
1
answer
519
views
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
4
votes
2
answers
809
views
Two basic questions on derived categories
Let $\mathcal{A}, \mathcal{B}$ be two abelian categories with sufficiently many injective objects (in my case these are categories of sheaves of vector spaces on a manifold).
Let $f_*\colon \mathcal{A}...
- 21.8k
3
votes
1
answer
782
views
Is the cokernel of a map of sheaves a seperated presheaf?
The cokernel of a map of sheaves is not necessarily a sheaf until you sheafify. In every example I have seen of the cokernel failing to be a sheaf it is the glueability axiom that fails while the ...
- 31
5
votes
1
answer
2k
views
Stalks of étale sheaves
I want to prove that $0 \to F\to G\to H \to 0$ is an exact sequence of étale sheaves. I understand that it is enough to show that $0\to F_{\bar{x}}\to G_{\bar{x}}\to H_{\bar{x}}\to 0$ is exact at ...
- 1,482
12
votes
1
answer
2k
views
Reference request: Book of topology from "Topos" point of view
Question: Is there any book of topology in the modern language of topos theory?
Motivation:
In "Sheaves in Geometry and Logic" Mac Lane and Moerdijk say: "For Grothendieck, topology became the ...
- 545
4
votes
1
answer
511
views
Nearby cycles and specialisation - properties
I am looking for reference for properties of nearby cycles - specifically, commutation with non-characteristic pull-back (good enough - commutation with pull-back to closed subvariety which is ...
- 5,562
1
vote
1
answer
404
views
Open subset of the moduli space of stable sheaves on a noetherian scheme
This is my question:
Given a projective noetherian scheme $X$, the structural sheaf $\mathcal{O}_X$ is a coherent sheaf, so every locally free sheaf is coherent. This means that the family of stable ...
- 401
3
votes
0
answers
716
views
Two functorial definitions of schemes
I have been reading a bit about the "functor of points" theory for schemes. There seem to be two ways of going about defining schemes this way:
Equip the category $\textbf {Psh}=\operatorname{Fun}(\...
- 2,011
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 $$\mathcal{H}^p(X,F)$...
- 401
1
vote
1
answer
356
views
Examples of nontrivial local systems in Decomposition Theorem
There is a proper map $f: X \rightarrow Y$ of projective varieties. The Decomposition Theorem of Beilinson–Bernstein–Deligne-Gabber states that
$$Rf∗IC_X \cong \oplus_a IC_{\bar{Y_a}}(L_a)[shifts]$...
- 1,719
16
votes
2
answers
9k
views
Canonical Sheaf of Projective Space
I am stuck on one step that occurs without explanation in several Algebraic geometry books.
Starting from the exact sequence
$$0\rightarrow \Omega_{\mathbb{P}^n}\rightarrow \mathcal{O}_{\mathbb{P}^...
- 953
4
votes
1
answer
504
views
About the construction of the Universal Enveloping Lie Algebroid
Let $X$ be a reasonable smooth scheme over some base $S$. The tangent sheaf $T_X$ is a Lie algebroid, locally free as a $\mathcal{O}_X$ module, and its Universal Enveloping Lie Algebroid $\mathfrak{U}(...
- 63
1
vote
3
answers
845
views
Higher cohomology of sheaves on a projective space
Let $S\subset\mathbb{P}^n$ be a finite set of $s$ reduced points. Let $\mathcal{I}$ be the ideal sheaf of $S$ in $\mathbb{P}^n$. We consider the sheaf
$$\mathcal{F}_k:=\mathcal{O}_{\mathbb{P}^n}(kd)\...
user49214
2
votes
1
answer
593
views
Example for pullback of stable sheaf not stable
Suppose $C$ is a complete algebraic curve.
Define a coherent locally free sheaf $\mathcal{F}$ over $C$ to be stable if $\mu(\mathcal{E})<\mu(\mathcal{F})$ for any subsheaf $\mathcal{E}$, where $\...
user39380
1
vote
1
answer
705
views
Unique decomposition of locally free sheaf
Below let's work over coherent sheaves on a smooth projective algebraic curve.
We call a subsheaf $\mathcal{F'}$ of $\mathcal{F}$ saturated if it $\mathcal{F/F'}$ is locally free.
We call a locally ...
user39380
4
votes
1
answer
373
views
flat descent for perverse sheaves
Let $E \in D^{b}_{c}(X,\overline{\mathbb{Q}}_{l})$ where $X$ is a $k$ scheme of finite type for a field $k$.
Let $Y\rightarrow X$ a finite flat surjective morphism such that $f^{*}E$ is perverse and ...
- 3,472
0
votes
1
answer
788
views
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
4
votes
2
answers
491
views
on the Zariski sheafification of Quillen's K-theory
Let $X$ be a smooth k-variety and denote by $K_n$ Quillen's K-theory sheaf, that is: the Zariski sheaf on $X$ associated to the presheaf $U \mapsto K_n(U)$.
The Bloch-Quillen formula says that $CH^n(...
- 43
4
votes
1
answer
263
views
Relating deformations of a scheme to deformations of its singular locus
Let $X$ be a normal scheme with quotient singularities and $Y\subset X$ its singular locus. The first order deformations of $X$ are parametrized by $\mathcal{E}xt^{1}(\Omega_{X},\mathcal{O}_{X})$. ...
- 8,998
0
votes
1
answer
351
views
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
4
votes
1
answer
601
views
Explicit examples presheaves associated to higher direct images which fail to be sheaves
So I would like to have a few simple examples where the presheaf associated to higher direct image of sheaf fails to be sheaf. So I'm looking for two (natural and simple) topological spaces $X$ and $Y$...
- 7,609
1
vote
2
answers
3k
views
Pullback of a constant sheaf
Let $\varphi:X\to Y$ be a surjective morphism of schemes which are connected and of finite type.
Let $A$ be an abelian group, $\mathscr{F}$ be the constant sheaf on $X$ with fibers $A$ and $\mathscr{...
- 251
3
votes
0
answers
160
views
Monodromy along strata of a pushforward
Work with complex varieties and constructible sheaves on the complex analytic site. All functors will be tacitly derived.
Let $X$ be a variety acted upon by a connected linear algebraic group. Let $X ...
- 1,595
4
votes
1
answer
553
views
Representability of a certain group scheme quotient
Let $k$ be a field. Suppose we have an exact sequence of $k$-group schemes (not finite-type)
$$
1\to H\to G\to K\to 1
$$
In other words, the sheaf quotient $G/H$ is representable by a $k$-group ...
- 2,027