All Questions
Tagged with ag.algebraic-geometry sheaf-theory
493 questions
1
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0
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258
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example of rank 2 torsion free sheaf with no global sections that is not stable
Ìn the book "Vector bundles on complex projective spaces" the authors prove in Chapter 2 Lemma 1.2.5 that, if $E$ is a rank $2$ reflexive sheaf on $\mathbb{P}^n$, then $E$ is stable if, and only if, $...
- 556
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0
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139
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Defineing a Sheaf of rings over a topological space
Let $X$ be a topological space and let $R$ be a commutative ring with $1$ such that for each $x\in X$ there exists a multiplicatively closed subset $S_x$ of $R$ such that for each $a\in R$ if $\frac{a}...
- 31
6
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1
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281
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Cokernel of map of étale sheaves
Let $p:\mathbb{G}_m\to \operatorname{Spec} k$ be the structure map, and let $T$ be an algebraic $k$-torus viewed as an étale sheaf over $k$. Why is the cokernel of the canonical map $T\to p_*p^*T$ ...
- 28.7k
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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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0
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85
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A special family of prime ideals
I am looking for a commutative ring $R$ with identity which has a family $F:=\{p_i\}_{i\in A}$ of prime ideals such that for each $n\in \mathbb{N}$ there exists $m\in \mathbb{N}$ with $n\leqslant m$ ...
- 11
1
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0
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180
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Weaker version of smooth base change for étale sheaves
Consider the cartesian square of schemes
$$ \require{AMScd}
\begin{CD}
X' @>{g'}>> X \\
@V{f'}VV @VV{f}V \\
S' @>>{g}> S
\end{CD}
$$
and the base change map
$$ \eta : ...
5
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1
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331
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Is the sheaf associated to a differential structure of a specific type?
On a set $X$, let us define a set $\mathcal{D}$ of functions from $X$ to $\mathbb{R}$. Consider first the initial topology $\tau_\mathcal{D}$ on $X$ with respect to $\mathcal{D}$, i.e. the coarsest ...
9
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1
answer
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Kozsul resolution of $\mathcal{O}_X$
Let $i: X \hookrightarrow Y$ be a closed embedding of smooth algebraic varieties. In the book D-modules, perverse sheaves and representation theory the authors say that there exists a locally free ...
- 3,079
3
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0
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308
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Quotient of a sheaf by group action and representabillity
Let $X$ be a scheme and $S$ be a sheaf of sets over the fppf topology of $X$. Let $G$ be a group scheme over $X$ and there is an action of $G$ on $S$. Now, I want to look at the quotient $G \setminus ...
- 1,277
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151
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Global sections of twisted dualizing sheaf of Hirzebruch surface
Let consider a Hirzebruch surface $S= \mathbb{P}(\mathcal{E})$ over $\mathbb{P}^1$ with invariant $e \ge 0$ where $\mathcal{E}= \mathcal{O}_{\mathbb{P}^1}(e) \oplus \mathcal{O}_{\mathbb{P}^1}$.
Let $\...
- 6,038
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0
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193
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Usage of Leray spectral sequence
Maybe this is an elementary stuff for experts, I could not figure it out by myself. Let $\pi:G\to S$ be an elliptic curve with zero section $e:S\to G$. Take $\mathcal{H}=(R^1\pi_*\mathbb{Q}_\ell)^\vee$...
- 1,428
6
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1
answer
728
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Sheaf cohomology with support vanishes
I am trying to solve the exercise 2.4 chapter III in Hartshorne's "Algebraic Geometry". For this I would like to prove for a sheaf $F$ of Abelian groups on a topological space $X$ and $U$ open subset ...
- 28.7k
3
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1
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203
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Sheaves with $\mathbb{R}$-constructible proper direct image closed under dualizing?
I learned about $\mathbb{R}$-constructible sheaves very recently, so I hope this isn't a silly question:
Let $M$ be a real analytic manifold, and $U\subset M$ an open subanalytic subspace. Denote the ...
CommunityBot
- 1
62
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8
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14k
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Sheaf cohomology and injective resolutions
In defining sheaf cohomology (say in Hartshorne), a common approach seems to be defining the cohomology functors as derived functors. Is there any conceptual reason for injective resolution to come ...
- 6,025
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0
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363
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Singularities of reflexive sheaves
I am studying reflexive sheaves (on $\mathbb{P}^3$) by the Hartshorne's paper ''Stable reflexive sheaves''. As far I understood, reflexive sheaves fail to be locally free at a finite number of points (...
- 556
2
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0
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361
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epimorphism of fppf sheaves is an fppf morphism
I asked this question on math.stackexchange (https://math.stackexchange.com/questions/2693471/epimorphism-of-fppf-sheaves-is-an-fppf-morphism) but didn't get an answer. Maybe someone here can help.
...
- 21
3
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0
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307
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Locality in Grothendieck Topologies
Let $\mathcal{C}$ be a category and $\mathcal{J}$ be a Grothendieck topology on it (i.e., $(\mathcal{C},\mathcal{J})$ is a site). Then what is a good notion of locality in it?
I came up with the ...
- 438
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0
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72
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Support of étale sheaves
Let $X$ be a scheme, $i: Z\to X$ a closed subscheme, $j: U\to X$ its complement in $X$. Assume the codimension of $Z$ in $X$ is large (at least $2$).
Let $A$ be an étale sheaf on $U$, $B$ an étale ...
CommunityBot
- 1
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1
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319
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How are the left and the right group of a bitorsor related?
This question arose from my answer to To what extent does a torsor determine a group: it turns out that I do not know one thing about it.
Let $G$, $G'$ be groups in some nice enough category (you may ...
0
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0
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144
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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
4
votes
0
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369
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Weierstrass model of an elliptic curve: a line bundle over the base
Let $S$ be a Weierstrass model of an elliptic surface (for me it works better to understand it as an elliptic fibration), that is a map $\pi : S \to C$ where $C$ is a compact Riemann surface.
...
- 587
48
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8
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8k
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When are there enough projective sheaves on a space X?
This question is being asked on behalf of a colleague of mine.
Let $X$ be a topological space. It is well known that the abelian category of sheaves on $X$ has enough injectives: that is, every ...
- 148k
2
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1
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460
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Cartier Divisor generated by Global Sections
Let $X$ be an integer curve of (arithmetic) genus $g=0$. (the arithmetic genus $g$ is defined by $g:= 1 -\chi_k(\mathcal{O}_X)$ where $\mathcal{O}_X$ is the structure sheaf of $X$ and $\chi_k(\mathcal{...
- 2,923
35
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5
answers
4k
views
Heuristic explanation of why we lose projectives in sheaves.
We know that presheaves of any category have enough projectives and that sheaves do not, why is this, and how does it effect our thinking?
This question was asked(and I found it very helpful) but I ...
- 118k
3
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2
answers
1k
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Equivalence of Definitions of Twisted Sheaf $ \mathcal {O}(1)$
Let $\mathcal {O}(-1)$ be the tautological line bundle $X$ of $ \Bbb CP^1$, where $X=\{(z,l) \in \Bbb C^2 \times \Bbb CP^1 : z \in l \}$ together with canonical projection $X \to \Bbb CP^1$ (line ...
- 5,215
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163
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Lifting a local section to a global section along a homomorphism of quasi-coherent sheaves
If $X$ is a scheme, is it always possible to find a basis $\mathcal{B}$ for the topology of $X$ (for example, the affine open subsets) with the following property?
For every quasi-coherent sheaf $M$...
- 5,482
15
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2
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2k
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Meaning of the determinant of cohomology
The Arakelov intersection number on arithmetic surfaces is defined as an "extension" of the classical intersection number on algebraic surfaces. It was introduced to get a nice intersection theory ...
- 6,259
4
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1
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550
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Dualizing sheaf and determinant of cohomology
Let $\pi:X\to S=\operatorname{Spec } O_K$ be an arithmetic surface in the sense of Arakelov geometry. Here $K$ is a number field $\pi$ is a flat map and $X$ is a projective surface. For any coherent ...
- 6,259
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4
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5k
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When does the sheaf direct image functor f_* have a right adjoint?
Say f: X → Y is a morphism of schemes. The sheaf direct image functor f★ always has a left adjoint, namely the sheaf inverse image functor f★ (with tensoring).
Under what (...
- 1,161
6
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1
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1k
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Base change for Borel-Moore homology
For a seperated scheme of finite type $X$ over $\mathbf{C}$, let $H_*(X)$ denote its Borel-Moore homology, which is defined by
$$
H_k(X) = R^{-k}\Gamma(X, \omega_X)
$$
where $\omega\in D_c(X, \mathbf{...
- 1,121
3
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1
answer
818
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Serre duality for sheaves of logarithmic differentials
This question is motivated by a comment of Donu Arapura here dimension of compact support cohomology
Let $D$ be a divisor with normal crossings on some smooth projective (complex algebraic) variety $...
- 38k
6
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0
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187
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Algebraic model for the abelian category of descent data for modules in the non-affine case
Let $f: X \to Y$ be a morphism of schemes. I'd like to have a completely algebraic description of the belian category of descent data for modules along $f$. Here's my attempt:
The category of quasi-...
- 7,789
4
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0
answers
432
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Reference request: sheaf-theoretic operations in the classical topology?
Like many graduate students before trying to learn something about étale cohomology and Deligne's proof(s) of the Riemann hypothesis part of the Weil conjectures, I am hunting for references detailing ...
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1
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421
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An example in Mumford's “Picard Groups of Moduli Problems”
I tried asking this at math.stackexchange but I didn't get any responses, so hopefully it's ok to try here.
I'm reading Mumford's paper "Picard Groups of Moduli Problems" and am confused about an ...
- 52.6k
8
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0
answers
470
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Sheaf whose singular support is not Lagrangian
For constructible sheaves $\mathcal F$ on real analytic manifolds $X$, there is a notion of the singular support $SS(\mathcal F)$ which is a radially invariant singular Lagrangian subset of the ...
- 18.7k
2
votes
0
answers
126
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Local cohomology with supports in a constructible set
Let $X$ be a topological space (I'm interested in the case of $X$ being a complex algebraic variety with the Zariski topology) and $Z$ a constructible subset (i.e. a finite union of locally closed ...
- 3,079
2
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0
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101
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Sheaves of functions on open semi-algebraic sets
Let $U\subsetneq\mathbb{R}^n$ be an open semi-algebraic set which is not Zariski open (the case I have in mind is that of an open ball $B(0,1)$). A function $f:U\rightarrow \mathbb{R}$ is called
(1) ...
- 953
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1
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2k
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Functorial characterization of open subschemes?
Given a morphism of schemes f: U → X, can one determine when f is an isomorphism of U onto an open subscheme of X in terms of some induced functors between the categories of quasicoherent modules ...
- 3,079
2
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0
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35
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If the set of non-0 stalks of F is relatively open, is the same true of its Verdier dual?
Let $X$ be a complex manifold, $F$ a bounded complex of $\Bbb C_X$-modules with constructible cohomology. If the set $\{x: F_x\neq0\}$ is relatively open (i.e. open in its closure), is the same true ...
- 3,079
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1
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177
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Is the restriction of a simple sheaf of modules simple?
Let $X$ be a topological space, $A$ a sheaf of (unital and associative but not necessarily commutative) rings on $X$. Suppose $M$ is a simple quasicoherent $A$-module and $U$ an open subset of $X$. Is ...
- 44.7k
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574
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What is the geometric intuition for the sheaf-theoretic terms "soft", "fine", and "flabby"?
The sheaf-theoretic terms "soft", "flabby", and "fine" are of an obviously geometric character, and suggest opposition with "hard", "rigid", and "coarse" sheaves (I'm just inventing these terms here).
...
- 6,025
7
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2
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839
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What is the need for torsion in the definition of lisse sheaves?
I am studying the basics of constructible and lisse sheaves, and am trying to understand SGA 4, IX. As Grothendieck himself observes at the beginning of the chapter, one is forced to work with torsion ...
- 148k
13
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1
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How is a Stack the generalisation of a sheaf from a 2-category point of view?
A stack is usually given in terms of:
-A category $F$ fibered over another $C$ such that the functor $Hom(x,y), x,y \in F(\alpha), \alpha \in C$ is a sheaf
-The descent data are effective.
There ...
- 2,607
9
votes
1
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426
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How to view $\textbf{Sh}(\textbf{CartSp})/X$ as "space" in its own right, étale machinery from abstract nonsense perspective for smooth manifolds
Let $\textbf{CartSp}$ be the category of spaces of the from $\mathbb{R}^n$ with smooth maps between them. This is a site with respect to (differentially) good open covers, so consider the Grothendieck ...
CommunityBot
- 1
2
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0
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258
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Proj construction and pushforward of line bundles
Let $X$ be a variety of dimension $d \geq 2$ (over a field), consisting of two irreducible components meeting transversely in a divisor $D$. (We can assume these components and $D$ are as nice as we ...
- 21
4
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0
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536
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When is a coherent subsheaf determined by its global sections
I am reading an article in which a proof is based on defining a subsheaf by only giving its global sections.
The exact setting is that, one has a surjective finite morphism $f:Y\to X$ between ...
- 333
2
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0
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169
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Restriction of scalars from an Azumaya algebra and preservation of perfect/compact objects of the derived categories
An Azumaya variety over a field is by definition a pair $(X,\mathcal A_X)$, where $X$ is an algebraic variety of finite type over that field and $\mathcal A_X$ is a sheaf of Azumaya algebras, namely ...
- 2,079
25
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3
answers
5k
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Stacks and sheaves
I'm a bit confused by the double role which sheaves play in the theory of stacks.
On the one hand, sheaves on a site are the obvious generalization of a sheaf on a topological space. On the other ...
CommunityBot
- 1
2
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0
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397
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Terminology for "global sections" when sheaf is valued in general category
Let $\mathcal F$ be a sheaf (say on a topological space $X$) valued in some category $\mathcal C$.
What do we call $\mathcal F(X)$?
When $\mathcal C$ is some vaguely linear category (e.g. the ...
- 18.7k
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0
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182
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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, ...
