3
votes
0answers
66 views

Weil restriction - from abstract nonsense to a practical procedure

Let $L/K$ be a finite field extension. Let $X$ be a $L$-scheme, that is $\exists$ a morphism of schemes $\pi : X \to \operatorname{Spec} L$. The Weil restriction of $X$ is a $K$-scheme $W_{L/K}X$ ...
1
vote
0answers
41 views

What is the opposite of category of schemes

The category of schemes sits between $Aff=CRing^\text{op}$ and its free cocompletion $\hom(Aff^\text{op},Sets)$, as the locally representable functors. Dualizing, we have $CRing\subseteq ...
7
votes
2answers
257 views

Functor of points and category theory

I am trying to read the section on Functor of points from Eisenbud - Harris (and I also referred to Mumford's book). They all motivate functor of points this way : In general, for any object $Z$ of a ...
2
votes
0answers
33 views

Unipotent Group Scheme

Can someone give me a reference of why this is true? Let A an associative k-algebra in wich every element is nilpotent, (non unital) and free of finite rank k-module, then for every commutative ...
0
votes
1answer
58 views

Representability of transformations by morphisms

This is a very naive question. Suppose I'm given two functors $F,G\colon(Schemes/S)^{op}\to (Sets)$, over some base scheme $S$, represented by $S$-Schemes $X_F$ and $X_G$ respectively. There are ...
2
votes
1answer
63 views

Retrieve affine schemes by adjunction

In an introductory course about schemes, I've seen the adjunction $$ {\mathbf{LocRngSpace}}^{\mathrm{op}} \overset{\mathrm{Spec}}{\underset{\Gamma}{\leftrightarrows}} \mathbf{Ring}, $$ where ...
4
votes
1answer
63 views

Exact sequences in the Mumford example

This question concerns the proof for the Mumford example of an Hilbert Scheme that has a non-reduced component. I am studying the proof given on R. Hartshone, "Deformation Theory", pp. 91-94. I do not ...
0
votes
1answer
81 views

Are the hom sets in the category of varieties abelian groups?

This is supposedly (though I know of the proof bud haven't read it) for the Hom sets of noetherian schemes. Since every variety can be thought of as a noetherian scheme then it seems right... when ...
5
votes
1answer
212 views

Why is the category of coherent sheaves not grothendieck?

Let $(X, \mathscr{O}_X)$ be a ringed space. It is well-known that the category $\mathbf{Mod}(\mathscr{O}_X)$ of $\mathscr{O}_X$-modules is a grothendieck abelian category (see e.g. Grothendieck's ...
3
votes
1answer
100 views

Does the direct image functor on sheaves reflect epimorphisms?

Let $f:X\to Y$ be a morphism of schemes, and let $f_*:\mathbf{Sh}(X)\to\mathbf{Sh}(Y)$ be the direct image functor. Does $f_*$ reflect epimorphisms? That is, suppose ...
3
votes
3answers
122 views

Does the inclusion from affine schemes into schemes preserve pushouts?

Let $K$ be a field. What is an example of two $K$-algebra morphisms $R\to T$ and $S\to T$ such that $\operatorname{Spec}(R\times_T S)$ is not the pushout of the diagram $$ ...
3
votes
1answer
78 views

Property kept under base change and composition is preserved by products

The following is true? Why? Let $P$ be a property of morphisms preserved under base change and composition. Let $X\to Y$ and $X'\to Y'$ be morphisms of $S$-schemes with property $P$. Then the unique ...
4
votes
2answers
209 views

Basic definition of Inverse Limit in sheaf theory / schemes

I read the book "Algebraic Geometry" by U. Görtz and whenever limits are involved I struggle for an understanding. The application of limits is mostly very basic, though; but I'm new to the concept of ...
3
votes
1answer
157 views

Does the functor of points commutes with inverse limits?

A scheme $X$ defines a (covariant) functor from commutative rings to sets by $A\mapsto X(A)=Mor(SpecA,X)$ Does this functor commutes with inverse limits? It does when $X$ is affine, but I can't see ...
3
votes
1answer
179 views

For which categories is the Hom functor left exact?

Sorry for this silly question but I can't find a reference. Let $\mathcal C$ be a preadditive category and $A\in \mathcal C$ an object. Under what conditions (on $\mathcal C$) can one say that the ...
8
votes
1answer
360 views

On limits, schemes and Spec functor

I have several related questions: Do there exist colimits in the category of schemes? If not, do there exist just direct limits? Do there exist limits? If not, do there exist just inverse limits? ...