1
vote
1answer
517 views

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 ...
2
votes
0answers
147 views

Two functorial definitions of schemes

I have been reading a bit about the "functor of points" theory for schemes. There seem to be two ways of going about defining schemes this way: Equip the category $\textbf ...
4
votes
1answer
237 views

Is a group scheme determined by its category of representations?

More precisely, let $G$ be an affine group scheme over a field $k$, $Rep_k(G)$ be the category of finite dimensional representations of G, and $\omega_0$ be the forgetful functor from $Rep_k(G)$ to ...
6
votes
2answers
423 views

Is an algebraic space over a DVR, whose special fibre and generic fibre are schemes, actually a scheme?

Is an algebraic space over a DVR, whose special fibre (and all its infinitesimal neighborhood) and generic fibre are schemes, actually a scheme?
0
votes
1answer
117 views

Prorepresentable functors repres. by alg. spaces? Covering spaces by alg. spaces.

Let $X$ be a (reasonable) scheme. I'm curious about constructing the constructing the covering space of a scheme algebraically. The covering space functor $F$ (below) can be represented by a ...
3
votes
2answers
287 views

Representability of sheaves of groups

There are lots of natural functors (that define sheaves in the fppf topology) that are not representable by schemes. For example, hilbert schemes of proper non-projective schemes in general need ...
0
votes
0answers
151 views

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 ...
6
votes
1answer
213 views

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 NĂ©ron ...
1
vote
1answer
351 views

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. ...
9
votes
0answers
324 views

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 ...
6
votes
1answer
994 views

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 ...
10
votes
2answers
459 views

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 ...
3
votes
1answer
424 views

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 ...
6
votes
2answers
746 views

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. ...
8
votes
1answer
345 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 ...
10
votes
2answers
432 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 ...
12
votes
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 ...
4
votes
2answers
471 views

Is the functor of open subschemes representable?

First some simple observations in order to motivate the question: The functor $Set^{op} \to Set, X \to \{\text{subsets of }X\}, f \to (U \to f^{-1}(U))$ is representable. The representing object is ...
3
votes
1answer
370 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 ...
5
votes
4answers
347 views

Is tensoring with a module representable iff it is locally free of finite rank?

Motivation: It's nice when you can think of the elements of an $A$-module $M$ as sections some $A$-scheme $Y\to Spec(A)$. That is, maps $Spec(A)\to Y$ such that $Spec(A)\to Y \to Spec(A)$ is the ...
9
votes
2answers
460 views

When is tensoring with a module representable by a scheme?

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 ...
7
votes
1answer
631 views

What functor does a Schubert variety represent?

I'm inspired by Yuhao's question. The functor that takes a scheme S to the set of k-dimensional vector subbundles of C^n x S (understanding "subbundle" to mean that the quotient by it is another ...
9
votes
3answers
1k views

What functor does Grassmannian represent?

As we know, the Projective space P^n represent the functor sending X to the set of line bundles L on X together with a surjection from the trivial vector bundle to L. My question is, what functor ...
6
votes
2answers
557 views

Do quotients of representable sheaves represent quotients?

Here's the context for the question: Proposition 4.6 of Freitag and Kiehl's book on etale cohomology shows that a sheaf (of sets) $\mathcal{F}$ (on the site Et(X)) is constructible if and only if it ...