# Questions tagged [representable-functors]

The representable-functors tag has no usage guidance.

73
questions

**2**

votes

**0**answers

133 views

### Algebraic groups as functors of points vs maximal points over algebraically closed field

So I only started learning about group schemes this summer, and I found two approaches. For the record I am interested in affine group schemes $G$ of finite type over a field $k$ (algebraic groups).
...

**2**

votes

**0**answers

89 views

### Representability result

Let $X$ and $S$ be schemes over a field $k$.
Reading this paper, there is a result on the representability of a morphism (proposition 3.1, page 4).
Which result or reference on representability is ...

**2**

votes

**0**answers

63 views

### Grassmannian of vector bundles with determinant being a square

I want to find a classfying space of vector bundles of rank $n$ generated by its global sections with determinant being a square of a line bundle. Stating in a formal way, suppose $X$ is a scheme, ...

**5**

votes

**1**answer

227 views

### Any exact faithful functor is represented by a unique projective generator

In the book 'Tensor Categories' by Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik on page 10 it says:
'Conversely, it is well known (and easy to show) that any exact faithful functor ...

**1**

vote

**0**answers

47 views

### Representable functors and symplectic co-tangent bundles

I've been banging my head against something that I feel should follow from abstract non-sense, and I hope someone here can set me straight.
Let $\mathcal{M}$ be the category of smooth manifolds, with ...

**1**

vote

**0**answers

261 views

### Constructions that can be seen as objects representing a functor

Some constructions can be seen as objects representing a functor.
For example,
Consider a topological group $G$ and a functor $\mathcal{F}:\text{Top}\rightarrow \text{Gpd}$ defined as $M\mapsto \...

**6**

votes

**1**answer

228 views

### Why does every chain complex have a map into its cone?

In Weibel's An introduction to homological algebra he defines a cone as an explicit chain complex associated to the given one -i.e. for a chain $C=(C_i, d)$ he defines $Cone(C)=\left(C_{i-1} \oplus ...

**5**

votes

**1**answer

167 views

### Does every functor between Grothendieck categories have adjoints?

Let $F:\mathcal C\longrightarrow \mathcal D$ be an additive functor that preserves colimits.
Suppose that $\mathcal C$ and $\mathcal D$ are Grothendieck categories.
Does $F$ have a right adjoint? ...

**4**

votes

**0**answers

63 views

### Finding a cover of the Hilbert functor, that corresponds to a cover of the Grassmannian

Let $\mathrm{Grass}_{k+1,n+1}:\mathrm{Sch}^{\circ}\rightarrow\mathrm{Set}$ be the Grassmann functor, which maps a scheme $S$ to the set:
$$\left\{\mathscr{U}\subseteq\mathscr{O}_S^{n+1}:\,\,\mathscr{...

**6**

votes

**0**answers

105 views

### Open subfunctor of Quot Functor induced by open immersion

Let $f: X \rightarrow S$ be a morphism of noetherian schemes and $\mathfrak{Q}uot_{\mathcal{E}/X/S}$ be the functor parametrizing families of quotients of $\mathcal{E}$ in the category of locally ...

**2**

votes

**1**answer

325 views

### Representability of Grassmannian functor by a scheme

I am having some trouble following a proof that the Grassmannian functor is representable by a scheme. I am following the proof in EGA 9.7.4. It is only a small step that I am stuck on. For reference, ...

**1**

vote

**1**answer

284 views

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

I already posted this on math.stackexchange some while ago, but haven't received any answers yet. (https://math.stackexchange.com/questions/3221006/proving-the-representability-of-a-functor-that-is-...

**0**

votes

**0**answers

106 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 $$ \operatorname{Grass}^e(\...

**4**

votes

**0**answers

242 views

### EGA I (Springer), Proposition 0.4.5.4 [closed]

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 $X$ ...

**3**

votes

**0**answers

109 views

### is the functor of finite order elements in grp representable

My question is whether the functor F:Grp->Set that sends a Group to its Set of elements with finite order is representable.
I have the sense that it shouldn't be but I've so far failed to prove it in ...

**1**

vote

**0**answers

90 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 ...

**6**

votes

**0**answers

165 views

### representability of $\mathrm{R}^1f_{*,\mathrm{fppf}}\mathscr{A}$

Let $f: X' \to X$ be a finite flat morphism of (nice) schemes and $\mathscr{A}/X'$ be a smooth commutative group scheme such that the Weil restriction $f_*\mathscr{A}/X$ is representable by a ...

**2**

votes

**0**answers

175 views

### Quotient of a sheaf by group action and representabillity

Let $X$ be a scheme and $S$ be a sheaf of sets over the fppf topology of $X$. Let $G$ be a group scheme over $X$ and there is an action of $G$ on $S$. Now, I want to look at the quotient $G \setminus ...

**2**

votes

**0**answers

130 views

### Representability of Flattening stratification functor

Let $f:X \to Y$ be a projective morphism between noetherian scheme. Is there any know condition under which there exists a functorial stratification of $Y$ i.e., there exists a filtration by locally ...

**1**

vote

**0**answers

133 views

### Internal Mor of schemes

Let $S$ be a Noetherian scheme and $X,Y$ be $S$-schemes of finite type. Consider the functor $X^Y$ given by $T \mapsto Mor_S(Y \times_S T,X)$. When is this functor representable by an $S$-scheme of ...

**4**

votes

**0**answers

193 views

### Representable $\text{Hom}$ functors

Let $X, Y, S$ be noetherian schemes, $X$ flat and quasi-projective over $S$, $Y$ projective over $S$.
Is the hom-functor $T\mapsto\text{Hom}_T(X_T, Y_T)$ representable?
If $X$ is flat and projective,...

**4**

votes

**0**answers

198 views

### Is the relative moduli space of semi-stable sheaves on families of curves fine

Let $\pi:X \to B$ be a family of smooth, projective curves. Fix coprime integers $r,d$. Denote by $\mathcal{M}(r,d)$ the relative moduli functor corresponding to rank $r$, degree $d$, semi-stable ...

**7**

votes

**1**answer

655 views

### Algebraic cycles, Chow spaces and Hilbert-Chow morphisms

In the sequel, let $S$ be a scheme, and $X$ a locally of finite type algebraic space over $S$.
In his thesis ([R1-R4]), David Rydh introduces, among several others, the notion of relative cycles on $...

**4**

votes

**3**answers

287 views

### About the Yoneda objects of a locally presentable category

This question is a follow-up of Extending functors defined on dense subcategories.
Let $\mathcal{K}$ be a locally presentable category. An object $X$ of
$\mathcal{K}$ is called a Yoneda object if ...

**7**

votes

**0**answers

202 views

### When is the character group scheme of a group scheme representable? (Affine Case)

While reading Tate's article on Finite Flat Group Schemes in "Modular Forms and Fermat's Last Theorem" I was lead to this question. Let $S$ be a scheme, $G$ a group scheme over $S$, and $T$ an $S$-...

**3**

votes

**1**answer

91 views

### Why does the variety of ideals in this quaternion type algebra have a non-reduced structure?

Let $A$ be the $\mathbb{C}$-algebra generated by elements $i,j$ with relations $i^2=j^2=0$ and $ij=-ji$, i.e. we have $A=\mathbb{C}\oplus\mathbb{C}i\oplus\mathbb{C}j\oplus\mathbb{C}ij$.
Let $\mathcal{...

**2**

votes

**0**answers

99 views

### How to describe the closure of a sub-group functor of a group scheme

Let $G$ be a group scheme over a field $k$, assume it has all the good conditions one may need (smooth, affine, algebraic, you name it).
Let now $H:Aff/k\to Gp$ be a group functor and $i:H\to G$ be a ...

**2**

votes

**0**answers

159 views

### Infinite iterates of the contravariant hom endofunctors on sets

My recent answer to Is it possible to define higher cardinal arithmetics (about defining infinite tetrations) requires something I don't know. Here is the simplest case.
Take a set $S$ and consider
$$...

**0**

votes

**1**answer

81 views

### 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 ...

**8**

votes

**0**answers

181 views

### Yoneda embedding and Horn sentences

The following is taken from Borceux and Bourn's Mal'cev, Protomodular, Homological, and Semi-Abelian Categories.
Metatheorem 0.1.3. Let $\mathcal P$ be a statement of the form $\varphi\implies \psi$, ...

**1**

vote

**0**answers

72 views

### A question on uniformly corepresented functor

Let $\mathcal{F}$ be a functor from the category of $k$-schemes to sets, uniformly corepresented by $M$. Suppose $U$ is an open subscheme of $M$. I could not find a good reference for uniformly ...

**3**

votes

**1**answer

512 views

### Is a left-exact limit-preserving functor $Ab \to Ab$ necessarily representable?

Let $Ab$ be the category of abelian groups, and let $F: Ab \to Ab$ be a covariant functor which is left-exact and limit-preserving. Is $F$ necessarily naturally equivalent to a functor of the form $\...

**18**

votes

**0**answers

1k views

### 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 ...

**0**

votes

**1**answer

583 views

### Representable functors and direct limits

Let $\mathcal{F}:\mathrm{Sch}/S \to \{\mathrm{Sets}\}$ be a representable functor. Denote by $X$ the scheme representing $\mathcal{F}$. The question is whether the natural tranformation $\mathcal{F}(-)...

**1**

vote

**1**answer

207 views

### 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 ...

**2**

votes

**1**answer

384 views

### When is the Hom-scheme connected?

Suppose that $A$ and $B$ are two algebras finite over a field $K$ (which may be assumed to be separably closed, if that helps), then we know that the functor $\mathrm{Hom}_K(\mathrm{Spec}(A),\mathrm{...

**3**

votes

**1**answer

350 views

### 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 ...

**2**

votes

**0**answers

166 views

### 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 ...

**9**

votes

**1**answer

251 views

### 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 ...

**1**

vote

**1**answer

659 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

**1**answer

150 views

### 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 ...

**3**

votes

**0**answers

355 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 {Psh}=\operatorname{Fun}(\...

**7**

votes

**1**answer

793 views

### formally smooth functor

Let $p$ be a prime number, $\mathcal{O}$ the integers of a finite extension of $\mathbb{Q}_p$ with residue field $k$. Let $\mathcal{C}$ be the category of complete, local, noetherian $\mathcal{O}$-...

**5**

votes

**2**answers

597 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 ...

**0**

votes

**1**answer

428 views

### Terminal object of category of elements of a representable functor

In Awodey's Category Theory (2nd edition), page 229, I read:
the category of elements $J$ of a representable $yC$ has a terminal
object, namely the element $1_C \in Hom_{\mathbf{C}}(C,C)$
...

**7**

votes

**2**answers

654 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

**1**answer

212 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 ...

**4**

votes

**2**answers

760 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 ...

**3**

votes

**0**answers

228 views

### Examples of Sheafification via Hypercovers

For a presheaf $F$ on a category equipped with a pretopology, one has the sheafification $F^{\sharp}$ of $F$.
I know well the plus-construction of sheafification, which is presented in Artin's paper "...

**0**

votes

**0**answers

551 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 understanding ...