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Questions tagged [representable-functor]

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2
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1answer
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Not able to make sense of gluing via the functor of points in EGA

I am trying to use a well known result of Grothendieck to show that if $S$ is a scheme, and $\mathcal{B}$ is a quasi coherent sheaf of $\mathcal{O}_{S}$-algebras, then there is a relative affine ...
2
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1answer
107 views

Proof that a functor covered by open representable subfunctors is representable by a scheme

So I'm led to believe this is a fairly standard result. See, for example Lemma 25.15.4 here. I am trying to prove a criterion for the representability of a (contravariant) functor from schemes to sets....
0
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1answer
43 views

Brown representability for model categories

In Jardine's article on the subject: https://ncatlab.org/nlab/files/JardineBrownrep.pdf He shows a version of Browns representability, which asserts the representability of a functor out of a model ...
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172 views
+50

Proving the representability of a functor, that is covered by open subfunctors

I want to proof Theorem 8.9 from Algebraic Geometry I ( U.Görtz, T.Wedhorn), which reads as follows: Let $S$ be a scheme $F: Sch/S°\rightarrow Set $ a functor such that: F is a sheaf for the ...
3
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0answers
90 views

Using the functor of points approach to show a scheme theoretic construction exists

T am trying to refram what I have learned in algebraic geometry in the context of the functor of points approach. In particular, I want to practice proving the existence of a scheme satisfying a ...
2
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0answers
97 views

Representability criterion for Zariski sheaf in terms of open subfunctors

I've been trying to prove a fairly classical, well-known result, but am running into a lot of trouble following any of the proofs I have found. At the moment I am following EGA I 0.4.5.4. Let $F: \...
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0answers
12 views

Quot-like scheme for torsion sheaves

I am wondering if, as the Quot schemes parametrizes flat (quotients of) sheaves over schemes, there is anything similar for torsion sheaves. In first approximation, if $I$ is a sheaf of ideals over a ...
1
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0answers
49 views

Ulrich, Torsten Proposition 8.4.2 (Closed Subscheme)

Let $S$ be a scheme and let $v : \mathcal E \longrightarrow \mathcal F$ be a morphism of quasi-coherent $\mathcal O_{S}$-modules. Let $\mathcal F$ be finite locally free. Then the locus $v=0$ is ...
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0answers
29 views

Irreducibility of the Jacobians of a curves.

I'm studying Jacobian varieties.I assume that the existence of the Jacobian variety for a curve and attempt to show irreducibility of the Jacobian for a curve according to Remark:IV.4.10.9 of ...
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0answers
30 views

How to arrive at unique factorization through the limit given naturality compatibility conditions?

If $\alpha : I \to C$ from a small category to any category $C$. Define a functor $\lim\limits_{\rightarrow} \alpha : X \mapsto \lim\limits_{\leftarrow} \text{Hom}_C(\alpha, X)$ from $C^{op}$ to $\...
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0answers
87 views

Nearly locally presentable categories

Here1, in the remark $2.3 (1)$ how from the fact that ${\cal K}(A,-)$ does not preserve coproducts it follows that ${\cal K}(A,-)$ sends special $\lambda$-directed colimits to $\lambda$-directed ...
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0answers
39 views

Explicit description of the scheme obtained by relative gluing data over a base scheme

I have recently been trying to get a better understanding of the projective space bundle of a quasi-coherent sheaf of graded algebras over a scheme $X$. The key idea is the following construction ...
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0answers
85 views

Expressing Representation of a Colimit as a Limit

$\newcommand{\C}{\mathcal{C}} \newcommand{\I}{\mathcal{I}} \newcommand{\L}{\mathcal{L}} \newcommand{\Hom}{\mathrm{Hom}_\C} \newcommand{\op}{\mathrm{op}} \newcommand{\colim}{\mathrm{colim}}$ Let $\C$ ...
1
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0answers
10 views

Reference request: Representability of multiplicative equivariant cohomology theories

Let $G$ be a topological group, say a compact Lie group, and $e^*_G$ a multiplicative $\mathbb Z$-graded $G$-equivariant cohomology theory defined on $G$–CW complexes. Is there some analogue result to ...
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0answers
67 views

Representability of zariski functors on schemes over a base by gluing

Suppose $F$ is a Zariski functor $(Sch/S)^{opp}\rightarrow Set$ where $S$ is a scheme. Suppose that I can show that for every affine $j:U\subseteq S$, I can show that the functor $F_U:(Sch/U)^{opp}\...
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0answers
23 views

Induced map between Grassmannian

Definition : Let $S$ be a scheme, $\mathcal E$ a quasi coherent $\mathscr O_S$-module, and $e \geq 0$ an integer. For every $S$- scheme $h : T\longrightarrow S$ denote by $$ Grass^e(\mathcal E)(T) = \{...
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0answers
171 views

EGA I (Springer), Proposition 0.4.5.4.

I do not understand one argument in the proof of Proposition 0.4.5.4. in the new version by Springer of EGA I. When proving that the functor $F$ is representable by $(X, \xi)$, where we obtained $X$ ...
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0answers
39 views

Showing the Affine functor $\underline{\mathbb{A}}^r$ is representable by Affine Space $\mathbb{A}^r := Spec(\mathbb{Z}[X_1, \dots, X_r])$

Let $\underline{\mathbb{A}}^r$ be the functor from $\textbf{Schemes}$ to $\textbf{Sets}$ which associates to each scheme $S$ the set of morphisms $\bigoplus_{k=1}^rO_S \to O_S$ ($O_S$ is the structure ...
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37 views

Are the Following Endomorphism and General Linear Group Functors Representable?

Let $k$ be a unital commutative ring. Let $k$-Alg denote the category of commutative and unital $k$-algebras. Let Set denote the category of Sets Fix an arbitrary $k$-module $V.$ Consider the ...
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35 views

Brown representability and based homotopy classes

The following statement is the version of Brown's representability theorem I learned: Let $\mathbf{CW}$ be the category of based, connected CW complexes together with based homotopy classes of ...
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62 views

The isomorphism $F \simeq h_C(X)$ determines $X$ how?

I've asked something similar before, here. But I didn't quite understand their reasoning. So I'm breaking the problem down. First of all, how is $X$ determined? By yoneda $\text{Hom}_{C^{\wedge}}(...