All Questions
Tagged with sheaves or sheaf-theory
979 questions
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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
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391
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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
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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 ...
- 6,028
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185
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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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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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116
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cohomology of curves
Let $X$ be a smooth projective complex curve. Consider the diagonal $\Delta$
in $X \times X$, and $\mathcal{O}(\Delta)$ the associated line bundle.
If $j$ is the inclusion of $\Delta$ in $X \times X$ ...
- 237
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180
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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 ...
- 6,028
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657
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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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635
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A coherent sheaf is a vector bundle over subvariety?
Over a complex manifold, can every coherent sheaf be seen as a holomorphic vector bundle over an analytic subset?
Thanks in advance.
- 789
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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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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 ...
- 6,028
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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, ...
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193
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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
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303
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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
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265
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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
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191
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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.$$H^1(\mathscr{O}_D, \mathbb{...
user76513
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239
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Cohomology group vs sheaf of cohomology group
Suppose $F$ is a coherent sheaf on a smooth (algebraic or complex) variety $X$. Then we can consider the cohomology groups $$H^p(X,F)$$ for all $i$. Now, let we consider the sheaf $$\mathcal{H}^p(X,F)$...
- 401
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85
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$\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
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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
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421
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Split and pure exact sequence of sheaves
Let $X$ be a topological space and
$$\varepsilon \ :\ 0 \to A \to B \to C \to 0$$
be an exact sequence of sheaves of ${\cal O}_X$-modules. $\varepsilon$ is said to be pure if
for each point $x\in X$,...
- 63
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859
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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
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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
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186
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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
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0
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187
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Are sieves in locally small categories still sets?
In "Sheaves in Geometry and Logic", M&M define a sieve of an object $C$ as a downward-closed set of arrows $S$ with codomain $C$. They go on to say that for a locally small category, a sieve of $C$...
- 1,872
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1
answer
177
views
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
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1
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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 $\mathcal{F}$ ...
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1
answer
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Why sheaves are important and why do we care about them? [closed]
Presheaves are contravariant functors from a category $C$ to the category $Set$, that is functors $P$:
$$P:C^{op}\to Set.$$
For every topology $J$ on $C$ we can generate a reflexive subcategory
$$Sh(...
- 391
-3
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1
answer
218
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Exact sequence of sheaves that generates an exact sequence of Abelian groups [closed]
Let $X$ be a smooth manifold of dimension $n > 1$. Let us denote by $\underline{\mathbb{S}}^{1}$ the sheaf of the smooth functions over circle, $C^{\infty}$ the sheaf of the smooth functions over $\...
- 19
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1
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286
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Does the 1-category construction of a topos of presheaves extend to the 2-Category of Groupoids?
In the case of 1-categories, we know there is a functor category
$PSh(C):=[C^{op},Set]$, where $C$ is a small category,
and this functor category is a topos. I am hoping this will extend to the case ...
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