All Questions
Tagged with sheaves or sheaf-theory
979 questions
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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.
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- 21
2
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264
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Pullback of a constant sheaf over arbitrary sites
Given a geometric morphism between arbitrary Grothendieck topoi, $f:\mathcal{Sh(D)}\to\mathcal{Sh(C)}$, does the pullback $f^{-1}$ (i.e, the left adjoint) take constant sheafs to constant sheafs?
- 187
2
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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 ...
user120812
2
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302
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How to "intersect" or "refine" a pair of abstract simplicial complexes
Let $S,T$ be abstract simplicial complexes.
Is there a (unique) abstract simplicial complex that gives me the most of what is in common with $S$ and $T$?
I'm thinking of this as an "intersection," ...
- 355
2
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0
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143
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Computing derived functor of a complex with non-acyclic terms
Let $A^\bullet =(\dots\to A^i\to A^{i+1}\to\dots)$ be a bounded below complex in an abelian category $\mathcal{A}$ with sufficiently many injectives. Let $F\colon \mathcal{A}\to \mathcal{B}$ be an ...
- 21.8k
2
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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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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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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
2
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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
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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
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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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Chern Classes: two approaches
The following question is closely related to this one.
Let $X$ a non singular projective variety over $\mathbb C$, and let $\mathscr E$ a locally free sheaf of rank $r$ (an algebraic vector bundle), ...
- 1,237
2
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83
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Sheaf of R-modules and modules over compactly supported functions
I'm looking for a reference for the following result:
Let $X$ be a locally compact Hausdorff topological space. let $\mathcal{R}$ be the sheaf of continuous functions with values in $\mathbb{R}$ over ...
- 42.4k
2
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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
2
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272
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Double dual of ample sheaf
Let $X$ be a projective manifold. Then we can define ample sheaves on $X$, and many results of ample vector bundles still hold in this more general case (See K. Kubota, Ample sheaves).
Now I was ...
- 147
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1k
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Equivalent definitions for fine sheaves
There are some different definitions for fine sheaves.
Let X is a paracompact Hausdorff space, a sheaf F over X is a fine sheaf, if
a) Hom(F,F) is soft
b) For every two disjoint closed subsets A,B$\...
- 403
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answers
160
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Pre-cosheaf of connected components
Consider a continous map $f:Y \to X$ between topological spaces. The pre-cosheaf $\mathcal{F}: Open(X) \to Set$ of connected components of the inverse image is defined as $\mathcal{F}(U):= \pi_0(f^{-1}...
- 229
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102
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Are there Coherent Cosheaves?
Is there a well-defined notion of coherent cosheaves in a similar sense to coherent sheaves? If so, what properties do they hold?
- 913
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272
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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
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231
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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
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127
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Locally free sheaves of algebras vs. algebra bundles
It is well known that there is a bijective correspondence between locally free sheaves of modules and vector bundles (cf. https://rigtriv.wordpress.com/2008/04/09/locally-free-sheaves-and-vector-...
- 357
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266
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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
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0
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212
views
Sections of inverse image sheaf of sheaf of sections of vector bundle
Let $\eta \colon Y \to Z$ be holomorphic map between complex manifolds, $E$ holomorphic vector bundle over $Z$, $\eta ^\ast E$ its pullback over $Y$, ${\cal O}_E$ sheaf of sections of $E$, and $\eta^{-...
- 1,368
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0
answers
129
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Sheaves, colimits and closure
I am considering the sheaf topos $\mathbf{Sh}(\mathfrak{X})$ on a topological space $\mathfrak{X}$.
Limits in this category are constructed pointwise, as in presheaves. Colimits, however, are not ...
- 408
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0
answers
116
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Cohomology and quotients for the canonical topology
Recall that for any category $\mathcal C$, there is a unique finest topology, the canonical topology on $\mathcal C$ for which all representable functors are sheaves. I am interested in the example $\...
- 5,759
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306
views
Cech cohomology.
Let us consider the scheme $X$ and the coherent sheaf $\cal F$ on it. We consider finitely many affine open covers $U_{\lambda}$ with $\lambda \in \Lambda$. When I calculate Cech cohomology with $U_{\...
- 181
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180
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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
2
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0
answers
253
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Is there something interesting in the uniqueness condition for a sheaf?
After digesting the Presheaf definition by the very first time, one feels (at least I felt) a strange sensation noticing the existence and uniqueness conditions to graduate that Presheaf as a sheaf, ...
- 51
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0
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450
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large cardinal tree properties as properties of sheaves
As follows from this talk Large Properties for Small Cardinals, p.7,p.4 http://www2.dm.unito.it/paginepersonali/viale/SEMINARS-TORINO/Fontanella-Torino-19.1.2012.pdf, the definitions of weakly compact ...
- 468
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175
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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
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0
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524
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What essential property justifies the name "derivative"?
Most, if not all, of the notions of derivative that I have so far seen have the property that they are locally defined -- meaning that the derivative of a map-type object at a point depends on the map ...
- 643
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358
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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
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414
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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
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0
answers
228
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The Mayer-Viertoris exact sequence as a (Zariski) descent spectral sequence.
For certain 'spaces' $U,V$ (they are certain Henselizations of subvarieties) I would like to compute (certain etale) cohomology of $U\cup V$ in terms of the corresponding cohomology of the diagram $U\...
- 16.9k
2
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0
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486
views
Fine and acyclic sheaves on locales
Hey all. Here's the thing, so I have a measure space. According to Johnstone's 'Topos theory' (page 213), let $(X,\Sigma,\mu)$ be a measure space, we can define a Grothendieck pretopology on it (and ...
- 456
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2
answers
3k
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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
1
vote
2
answers
663
views
Methods of sheaf theory for solving Diophantine equations
What are some examples of sheaf theory used to either provide solutions to Diophantine equations, or to state that no such solutions exist?
- 511
1
vote
1
answer
690
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Beilinson-Bernstein localization: $\mathfrak{g}$ action on $G$-equivariant sheaf
I have a few elementary questions related to Beilinson-Bernstein localization.
Let $G$ be a semisimple algebraic group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}$. Consider the setup of ...
- 174
1
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1
answer
443
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Is this Sequences of Complexes of Sheaves Exact?
So in another question of mine there is a sequence of complexes of sheaves which the author asserts is exact.
Let $K^{\bullet} = \underline{\mathbb{C}}^* \ \underrightarrow{d\ log} \ \underline{...
- 1,611
1
vote
1
answer
186
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Dual of stable vector bundle on a Fano threefold
Let $E$ be a rank $2$ stable vector bundle on a prime Fano threefold of genus $8$, with Chern numbers $c_1=1, c_2=6, c_3=0$.
Question. Is it true that $E(-1)=E^*$?
What I am able to show is that ...
- 565
1
vote
1
answer
749
views
Computing Ext sheaves over complex projective plane
Let $X:=\mathbb{P}^2_K$ with $K$ algebraically closed field. Take $p\in X$ a point and $\mathcal{I}_p$ its ideal sheaf. One can prove (using Serre Duality and the exact sequence defining $\mathcal{I}...
- 395
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2
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431
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Action on group $\operatorname{Ext}^i(\mathcal{L}, \mathcal{M})$ by scalar multiplication
Let $X$ be a proper scheme over field $k$ and $\mathcal{L}, \mathcal{M}$ two invertible $\mathcal{O}_X$-modules. Then $Hom_{\mathcal{O}_X}(\mathcal{L}, \mathcal{M}) \cong Hom_{\mathcal{O}_X}(\mathcal{...
- 6,028
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2
answers
1k
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Hypercohomology of a complex of sheaves that might be acyclic (or might not)
Back again, check this out, let $X$ be a topological space and let $F^{\bullet}$ be a cochain complex of sheaves, I'm trying to compute the cohomology of the complex of global sections of the sheaves
...
- 360
1
vote
2
answers
398
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Cohomology of a cochain complex of acyclic sheaves
Ok, sort of as a follow up to my previous question, let's recall the de Rham-Weil theorem:
Let $F$ be a sheaf on a topological space $X$ and let $\mathcal{L}^{\bullet}$ be an acyclic resolution of $F$....
- 360
1
vote
1
answer
146
views
Is the free R-module on a sheaf of sets still a sheaf?
Let $L$ be a sheaf of sets on some site $S$. Let $F$ be the presheaf obtained by composing $L$ with the free R-module functor, i.e. for any object $U$, we define $F(U)$ to be the free $R$-module on ...
- 5,637
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
1
vote
1
answer
127
views
Vanishing of higher morphisms for pair moduli
Consider a moduli scheme $P$ of sheaves $F$ on a Calabi-Yau n-fold $X$ with a section $s\in H^0(F)$. Alternatively, such objects can be described as maps $\mathcal{O}_X \xrightarrow{s} F$ called pairs....
- 988
1
vote
1
answer
226
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flatness of restriction of structure sheaf over ring of global sections
Let $X$ be an affine scheme. $U \subseteq X$ open. Then I want to show that $\mathcal{O}_X(U)$ is flat over $\mathcal{O}_X(X)$.
But I want to prove it only by knowing the definition of structure sheaf ...
1
vote
1
answer
877
views
Direct image of reflexive sheaf via finite, flat map
Suppose $f: X \rightarrow Y$ is a finite, flat (hence locally free) morphism of curves (i.e. schemes of dimension 1, not smooth or even reduced). Suppose $L$ is a reflexive sheaf on $X$, locally free ...
- 479
1
vote
1
answer
310
views
When is a ring or algebra a ring/algebra of functions?
Note: For the record, exterior algebras and derivations are irrelevant to my question. However, I have a hard time assessing what I want to ask and I find it is the easiest to do so using a direct ...
- 3,292