# Tagged Questions

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### representing base changes of the unit section

Let $S$ be a scheme and $G$ be a sheaf in groups on the big étale site over $S$. Let $e:S\rightarrow G$ be the unit section. Is it true that given an algebraic space in groups $H$, étale over $S$, and ...
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### History of the functor of points

Until now, I thought the functor of points approach was introduced by Grothendieck at the 1973 Buffalo seminar. However, in this note by Lawvere the author writes: "I myself had learned the ...
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### On functors which are generically representable

Let $F$ be a set-valued (contravariant) functor on the category of schemes. Let $F_{\mathbb Q}$ be the associated functor on the category of schemes over $\mathbb Q$. Suppose that $F_{\mathbb Q}$ is ...
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### What if the base change of an algebraic space is representable

Let $k\subset L$ be an extension of fields of characteristic zero. Suppose that $X/k$ is an algebraic space such that $X\otimes_k L$ is representable by a finite type $L$-scheme. I am sure there are ...
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### How to determine a functor (natrually arising from geometry or homological algebra) to be locally of finite presentation?

How to determine a functor (natrually arising from geometry or homological algebra) to be locally of finite presentation? Is there any reference for such staff? My example of functors underlying this ...
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### Are Brown representable functors determined by restriction to finite complexes?

Assume two $CW$ complexes $X,Y$ give two functors $h_X=[-,X], h_Y=[-,Y]$ on the homotopy category of $CW$ complexes whose restrictions to the full subcategory of finite $CW$ complexes are naturally ...
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### Is Mumford's statement about the representability of some functor wrong?

I am having trouble proving a result in Mumfords book 'Lectures on Curves on an Algebraic surface. It is a statement about the representability of some functor. It is stated on page 108 and says the ...
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### Algebraic objects and lifts of their represented functors

I've seen the following theorem around in various forms: To give an object $A \in \mathcal{C}$ the structure of a $\Omega$-algebra object in $\mathcal{C}$ is equivalent to giving a lift of the ...
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### Representability of sheaf of Ext^1 of a Néron model by $\mathbb{G}_m$

Let's work over a trait $S=\mathrm{Spec}R$, where $R$ is a dvr with fraction field $K$, residue field $k$. Given an abelian variety $A_K$ with semi-stable reduction, let $A$ over $S$ be its Néron ...
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### Schemes associated to vector spaces

Let $k$ be a field. Let $F$ be a covariant functor on the category of $k$-algebras to the category of sets. Assume that the opposite functor $F^{op}$ on the category of affine $k$-schemes is a sheaf. (...
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### Extending smooth irreducible representations

Hi, Let $G_1, G_2$ be topological groups with $G_1 \subset G_2$ is closed. Let $\rho:G_1 \to Aut(V)$ be a smooth irreducible representation. Can anyone tell me if there is a criterion/example/idea ...
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### Where is the representability of the moduli of curves with framed points proved?

There is a variant of the Knudsen-Mumford moduli problem $\mathcal{M}_{g,n}$ of pointed curves, where one endows the $n$ marked points with non-zero tangent vectors. It shows up in the theory of ...
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### construction of the Jacobian of a curve

I am trying to understand the construction of the Jacobian of a curve following the notes of J. S. Milne The question is going to be about a particular step in the proof of Proposition 4.2b in ...
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### Do coarse moduli spaces respect Galois actions?

To explain, I will use the following concrete example: Let $\mathcal{M}_g$ be the functor for the moduli problem of classifying genus $g$ smooth projective curves (taking a scheme $S$ to the set of ...
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### When is a sheaf on a scheme extendable to a representable functor?

I'll start with example: Let $X$ be a scheme, and $O_X$ be its structure sheaf. It is defined at the moment on open sets of $X$, and it takes them to $Sets$. However, it is extendable to a sheaf on ...
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### Relationship between Hilbert schemes and deformation spaces

Hi, I'm just starting to learn about deformation theory (via Hartshorne's Deformation theory, as well as Fantechi's section of FGA explained), and I feel like I'm confused about fundamental concepts. ...
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In any presheaf topos, there exists an object called Lawvere's segment, which can be described as the presheaf $L:A^{op}\to Set$ such that for each object $a\in A$, $L(a)=\{x\hookrightarrow\ h_a: x\in ... 1answer 370 views ### Comparing colimits in schemes with colimits in sheaves of sets Suppose I have a diagram of schemes, and I know that the colimit exists in the category of schemes. How does this colimit compare with the colimit of the corresponding sheaves (I'm being nonspecific ... 2answers 451 views ### Why is Maps(X,Y) an open subfunctor of Hilb(X x Y)? Let$X$and$Y$be projective schemes. Then we can define the mapping scheme between them,$\rm{Maps}(X,Y)$as follows: To any map$f:X\rightarrow Y$we consider the graph$\Gamma_f$as a closed ... 1answer 1k views ### Picard groups of non-projective varieties As far as I know, the main representability result for the relative Picard functor$Pic_{X/k}$, for a noeth. sep. scheme of finite type over a field$k$is: If$X$is proper then$Pic_{X/k}$is ... 1answer 573 views ### Representing cohomology of a sheaf à la Eilenberg-Maclane Suppose that we are given a nice space$X$and a sheaf of abelian groups$F$on$X$. Fix an integer$n$. Then We have a contravariant functor from nice spaces over$X$to abelian groups; Namely, to a ... 1answer 627 views ### When is a stack (NOT) geometric? Following the terminology of$n$-Lab, a geometric stack$\mathcal{X}$on a site$\mathcal{(C,J)}$is a stack for which there exists a representable epimorphism$X \to \mathcal{X}$from an object$X$... 1answer 1k views ### When does a “representable functor” into a category other than Set preserve limits? This might be a dumb question. If$C$is an ordinary category, then for any$c \in C$the covariant representable functor$\text{Hom}(c, -) : C \to \text{Set}$preserves limits. However, it can ... 1answer 399 views ### Sheaf condition and representability in the category Top This is a rather nice question I got from this user via private communication. Let$\mathcal{C} = Top$the category of topological spaces. Let$\mathcal{C}^\prime$be the category$Funct(\mathcal{C}^{...
Consider the following: Let $A$ be a commutative ring, let $M$ be an $A$-module. When is the functor from $A$-algebras to Sets given by $R \mapsto R \otimes M$ representable by an $A$-scheme? Unless ...