Questions tagged [sheaf-theory]
For questions about sheaves on a topological space. Usually you think of a sheaf on a space as the data of functions defined on that space, although there is a more general interpretation in terms of category theory. Use this tag with the broader (algebraic-geometry) tag.
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Etale Stalks of the Hom sheaf
I am reading Milne's lectures on etale cohomology. In the note he states:
Proposition 6.16 Assume $X$ is connected, and let $\bar{x}$ be a geometric point of $X$. The map $\mathcal{F} \mapsto \mathcal{...
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Structure sheaf of the projectivization of a bundle
Let $0\to V'\to V\to M \to 0$ be an exact sequence with $V$ a vector bundle over a scheme $B$, $M \in \operatorname{Pic}(B) $. Let $H:=\mathbb{P}(V')$ be the Cartier divisor, let $p: \mathbb{P}(V) \to ...
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(Interlude) Question for fun : What would be simplest way of proving "a presheaf isomorphic to a sheaf is also sheaf"
This is a question for fun.
Literally, let $\mathcal{F}$ on $X$ be a presheaf which is isomorphic to some sheaf $\mathcal{G}$.
Then my question is, among various possible proofs that '$\mathcal{F}$ is ...
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Why is $ \mathfrak{Mod}(A_{Y}/f) $ a thick subcategory of $\mathfrak{Mod}(A_{Y})$?
Let $f: Y \to X$ be a continuous map and $\mathfrak{Mod}(A_{Y}/f)$ be the full subcategory of $\mathfrak{Mod}(A_{Y})$ (categories of $A_{Y}$-modules, with $A$ a fixed ring) whose sheaves $\mathcal{F}$ ...
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The intersection of $\mathcal{O}_X$-submodules is an intersection as the categorical sense?
I have simple question in studying algebraic geometry.
First, next is the definiiton in the Gortz's Algebraic geometry book (p.174)
Let $\mathcal{F}$ be an $\mathcal{O}_X$-module and let $(\mathcal{F}...
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Real-world examples of perverse sheaves
If I were to have asked this question about local systems, it would be extremely easy to give examples from physics/biology, etc. I can give example if people want, but this should be easy to Google.
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monodromy representation associated with pushforward of constant sheaf
I am reading Geordie Williamson's guide
to perverse sheaves and stuck on Example 5.11.
Consider the map $f:\mathbb{C}^* \to \mathbb{C}^*: z \mapsto z^m$. Let
$\underline{k}$ be the constant sheaf on $\...
- 127
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Relation between the sheaf of relative differentials and the canonical divisor
Let $\hspace{0.2cm}f:$ $X\longrightarrow Y \hspace{0.2cm}$ be a finite morphism of curves over $K$.
Consider $\hspace{0.2cm}\Omega_{X/K}\hspace{0.2cm}$ and $\hspace{0.2cm}\Omega_{Y/K}\hspace{0.2cm}$ ...
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Lifting morphism of stalks to sheaf morphisms
I attempted to prove Hartshorne, Proposition 2.2, which is
Proposition. Let $(X,\mathcal{O}_X)$ be a ringed space. Then the category of $\mathcal{O}_X$-modules has enough injectives.
Pick some ...
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Do quasi-coherent sheaves form a reflective subcategory?
Let $X = $ Spec $A$ be an affine scheme.
I know that there is an inclusion of categories from $A$-modules to sheaves of $\mathcal O_X$-modules on $X$, which is exact and fully faithful.
It seems to me ...
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Coherent subsheaf of $\mathcal{O}(-1)$ on $\mathbb{P}^1$
I'd like to ask that if there is a classification of coherent subsheaf of $\mathcal{O}(-1)$ on $\mathbb{P}^1$.
Thanks.
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Cokernel of $\mathcal{O}_m\to\mathcal{O}_n$ on $\mathbb{P}^1$
Consider the Serre twisted sheaf $\mathcal{O}_m$ and $\mathcal{O}_n$ with $m<n$ on $\mathbb{P}^1_k$ where $k$ is a field, as $\mathrm{Hom}(\mathcal{O}_m, \mathcal{O}_n)=\Gamma(\mathcal{O}_{n-m})$, ...
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Constructing injective resolution on big sites.
Let's work over a category of varieties on some field. Let $\mathcal{F}$ be a presheaf on varieties that satisfies Zariski/etale sheaf conditions whenever restricted to the opens of a specific variety....
- 1,354
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Pushforward sheaf of a sheaf on a singleton?
I'm trying to understand the following proof:
Proposition. Let $(X,\mathcal{O}_X)$ be a ringed space. Then the category of $\mathcal{O}_X$-modules has enough injectives.
Proof. Let $\mathcal{F}$ be ...
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Sub-problem in the definition of the pullback of the sheaf of differentials
My question is a sub-problem I came up while trying to solve the question I proposed here (Definition of the pullback of the sheaf of differentials). But I resume it here anyway for completness:
...
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Definition of the pullback of the sheaf of differentials
I premise that I'm very new to the study of algebraic geometry so probably things that can look trivial are not so clear to me.
Let $f : X \rightarrow Y$, $g: Y\rightarrow Z$ be morphisms of schemes.
...
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Sheaves for a Grothendieck topology compatible with a pretopology
$ \newcommand{\cat}[1]{\mathcal{#1}} $ Let $ (\cat C,J) $ be a site. I'm trying to show that if the topology $ J $ comes from a pretopology $ K $ in the sense that for each $ X\in \cat C $ and for ...
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Exact sequences of morphisms in Hartshorne
I want a bit clarity. In Hartshornes book he states
if we have an exact sequence $\cdots\rightarrow \mathcal{F}^{n-1} \rightarrow^{f^{n-1}} \mathcal{F}^{n} \rightarrow^{f^n} \mathcal{F}^{n+1} \...
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Extension of line bundle on regular deformation
Let $R$ be a DVR and $\pi : S \rightarrow \text{Spec}(R)$ a regular smoothing of a nodal curve $C$ (with regular components). Given a line bundle $L$ on the (regular) generic fiber $\pi^{-1}(\eta)=S_\...
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On the definition of sheaf on a site "without matching families"
I'm reading Mac Lane & Moerdijk's Sheaves in Geometry and Logic and I got stuck at the definition of sheaves on a site.
The authors declare that a presheaf $ F $ on a site $ (\mathcal C,J) $ is a ...
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How to prove a sheaf to be a fine sheaf?
My inquiry pertains to the reference [1]. In section 2.3 on page 7 of this paper, the author asserts that if $u\in \Gamma(U,\Omega_{(2)}^{p,q}(X,E))$ and $f\in C^\infty (X)$, then $fu\in \Gamma(U,\...
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The relations between two definitions of Hodge bundles
I recently learnt the notion of "Hodge bundle", primarily for families of abelian varieties. This is usually defined as follows: Let $\pi:X\to S$ be an abelian scheme and $e:S\to X$ be its ...
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Is the quotient of separated presheaves separated? [closed]
Let $X$ be a topological space. If $A \leq B$ are Abelian group-valued sheaves on $X$, then the presheaf quotient $U \mapsto B(U)/A(U)$ is a separated presheaf.
Does this still hold if we require $A$ ...
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Fiber functors and lifting property of coverings
Let $X$ be a sufficiently good topological space and $x \in X$, then it is well-known that there are equivalences of categories
$$\begin{equation*} \begin{split} \left \{\text{coverings} \ p: Y \...
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Definitions of the Atiyah class
Let $E$ be a coherent sheaf on $X$, then its Atiyah class is defined as follows: There is a map $\alpha: \mathcal{O}_{\Delta X}\rightarrow \Delta_*\Omega^1_X$ where $\Delta:X \rightarrow X\times X$ ...
- 1,758
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Kernel of an epimorphism of coherent sheaves on Noetherian schemes
Let $f\colon\mathcal{E}_1\to\mathcal{E}_2$ be an epimorphism of locally free sheaves on a Noetherian scheme $X$. Then also $\ker(f)$ is a locally free sheaf.
Proof. For all $x\in X$ one has a short ...
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Why can $(f)\ge -D_0$ conclude that $f$ gives a global section of $\mathscr L(D_0)$ whose divisor of zeros is $D$?
I'm reading Hartshorne (II) Proposition 7.7.
Proposition 7.7. Let $X$ be a nonsingular projective variety over the algebraically closed field $k$. Let $D_0$ be a divisor on $X$ and let $\mathscr L\...
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What exactly is a ringed space?
The Question:
What is a ringed space? Specifically, how does one think about them?
Context:
Ringed spaces are important for many fields of mathematics, but for me, I use them in the context of ...
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Tensor with the pushforward of the stalk of the structure sheaf equals pushforward of the stalk: true in a more general case?
$\def\sF{\mathcal{F}}
\def\sO{\mathcal{O}}
\def\spec{\operatorname{Spec}}
\def\frp{\mathfrak{p}}
\def\im{\operatorname{Im}}$Consider the following result:
Lemma. Let $\sF$ be a quasi-coherent sheaf ...
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Sheaf on Empty Set in Zariski Topology over Spectrum of Integers [duplicate]
I am confused about how to understand the sheaf on empty set (as an open subset) of the Zariski Topology over the Spec(Z), which is generated by the additive identity zero in Z. The sheaf I have in ...
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When is the pushforward of a sheaf isomorphic to the pullback along a section?
Let $p:X\to S$ be a morphism of schemes, and let $\mathcal{F}$ be a $\mathcal{O}_X$-module on $X$. Suppose $s:S\to X$ is a section of $p$. Then when do we have an isomorphism $s^{\ast}\mathcal{F}\cong ...
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When can we define the stalks of a sheaf?
So i am reading rotman's book on homological algebra, and in it he states in Theorem 5.91, the following:
Let $X$ be a topological space and $\mathcal{A}$ an abelian category and consider the category ...
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How to show $(dx_1)_y,\cdots,(dx_r)_y$ generate a free submodule of rank $r$ of $\mathcal O_{X,y}$-module $(\Omega_{X/k})_y$?
Let $X$ be a nonsingular variety of dim $n $ over an algebraically closed field $k$. Let $Y$ be an irreducible closed subscheme defined by a sheaf of ideals $\mathscr I$.
Then I want to prove that
$ Y ...
- 2,603
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Why is $Ext_{\mathcal{S}}^{j}(H,G) = 0$ if $H$ is a free sheaf and $G$ a flabby sheaf?
So, I am reading Schapira's and Kashiwara's "Sheaves on Manifolds" and in the proof of proposition 2.6.3, it is stated that, for a free $\mathcal{S}$-module $H$ and a flabby $\mathcal{S}$-...
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Questions about étale space construction of sheafification
I'm trying to wrap my head around sheafification via the étale space construction from, say, Hartshorne's or Liu's books on algebraic geometry (I guess this is a construction of Bourbaki?). In these ...
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Definition of Supermanifold
defining a super-domain as a pair $(U\subset\mathbb{R}^n, C^\infty(U)\otimes\bigwedge[\theta^1\cdots\theta^m])$, a supermanifold is usually defined as a topological space $M$ endowed with a sheaf of ...
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Pushforward under Segre Embedding
This question originated from an answer to this post. Take $\mathcal{O}(1,0)$ on $\mathbf{P^1}\times \mathbf{P}^1$ and consider its push-forward $F$ on the quadric $\subset \mathbf{P}^3$ under the ...
- 1,758
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Čech Cohomology on Sheaf of maps for a Contractible space is 0
I have been pondering on this question for half a year and haven't gotten an ideal answer. Hope someone can help!
Conjecture:
Given a topological space $X$ and any dimension $i\in \mathbb{N}$, the ...
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Understanding Daniel Rosiak's intuitive explanation for adjoints
Example 202 Both small-scale and large-scale projects, such as those in research or development, require resources. Resource allocation (through grants, investment funding, contracts, etc.) requires a ...
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Dualizing sheaf - normalization of nodal curve
Given a normalization of a nodal curve $\alpha : \tilde{C} \to C$ over an algebraically closed field. Assume for simplicity we only have one node $p$ with $\alpha^{-1}\left(p\right)=\left(q_1,q_2 \...
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Ideal sheaves of diagonal maps
Let $X\subset A$ be the inclusion of a scheme $X$ into an ambient space with ideal sheaf $J \subset \mathcal{O}_A$ and let $\Delta_A,\Delta_X$ be the images of the diagonal embeddings $A \rightarrow A\...
- 1,758
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Compute the stalk and costalk of a pushforward of the constant sheaf
I am reading Achar's book about perverse sheaves. Now I am trying to solve the exercise 1.10.5 in this book (all varieties are assumed over $\mathbb{C}$ and sheaves are over a field $k$):
Define $$
\...
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Proof that the push-forward of a sheaf is unstable
Let $\pi:X\to C$ be a (proper) elliptic surface with $F$ as a general fiber and H a polarization on $X$.
Let $\mathcal F$ be a bundle on $C$ and let $\mathcal E:=\pi^* \mathcal F$.
Assume $\mathcal E$ ...
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The stalk of a point on a scheme is a localization of ring of affine open?
Let $(X,\mathcal O_X)$ be a Noetherian scheme. For every affine open subset $U$ of $X$, it holds that $U=\text{Spec}(\mathcal O_X(U))$. Let $x \in X$, and let $U$ be an affine open subset of $X$ ...
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Fixed part of the induced representation. ntation.
Suppose the topological groups $h, G$ satisfy that $H < G$ and that $[G \colon H] < \infty$. Let $V$ be a $K$-vector space on which $H$ acts continuously. Then we consider the induced $G$-...
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Is relative spectrum a left adjoint?
Given a scheme $X$ and a quasicoherent sheaf of algebras $\mathscr{R}$ on it. Vakil's FOAG, section 17.1.2, page 470 says that the relative spec $\beta: Spec \mathscr{R} \to X$, representing the ...
- 1,224
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The structure sheaf of a manifold is always (isomorphic to) a sheaf of functions
I'm reading the chapter on manifolds in Wedhorn's Manifolds, Sheaves, and Cohomology. At some point the author (in a somewhat terse manner) gives a characterization of the structure sheaf of a (pre)...
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Direct Image and the induced representation.
Suppose $f \colon \mathrm{Spec}\, L \to \mathrm{Spec}\, K$ be a finite covering. Given a smooth sheaf ${\cal F}_{\rho}$ on $\mathrm{Spec}\,L$ corresponding to the representation $\rho \colon {\pi}_1(\...
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Are separated flabby (flasque) pre-sheaf sheaf?
The question is as in the title and comes from Justin Smith's book «Introduction to Algebraic Geometry» Appendix Proposition B.2.7. It is given without proof :
A flasque-separated presheaf is a sheaf
...
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Finite injective resolution of a constant sheaf
Is it known if the injective resolution of the constant sheaf $\mathbb C_X$ on a smooth manifold $X$ is of finite length? I am asking this because the fine resolution of $\mathbb C_X$ in terms of the ...
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