5
votes
1answer
45 views

quasicompact scheme are finite union of affine scheme

This is a problem from Liu`s book. Show that a scheme $X$ is quasi-compact if and only if it is a finite union of affine schemes. If the scheme is quasicompact then it is obviously a finite union of ...
1
vote
1answer
53 views

Connected scheme but not quasicompact

I am searching for a connected scheme which is not quasi compact. My try: Suppose $A_i$=Spec$k[x]$, where $k$ is algebraically closed field are affine schmes . I am planning to glue $0$ of the ...
0
votes
0answers
34 views

A doubt in affine schemes

Consider the affine scheme (Spec($A$),$O_{Spec(A)}$). where $A$ is a commutative ring. Then I know (from hartshrone page 71) that $\Gamma (Spec(A),$ $O_{Spec(A)}$) is isomorphic to $A$ . I also know ...
5
votes
1answer
84 views

Why is Weil restriction right adjoint to base change?

Let $k'/k$ be a finite field extension, and let $X'$ be an affine group scheme over $k'$. We can define the Weil restriction of $X'$ to be the affine group scheme $\mathrm{Res}_{k'/k}(X')$ over $k$ ...
4
votes
0answers
108 views

Orbits of affine group scheme actions

Motivation Let $F$ and $K$ be distinct algebraically closed fields. Then $GL_n(F)$ acts on $M_n(F)$ by conjugation, and the nilpotent orbits of this action can be characterized by the Jordan normal ...
6
votes
1answer
225 views

Toric Varieties: gluing of affine varieties (blow-up example)

Let $\Delta$ be a fan, consisting of cones $\sigma_0=conv(e_1,e_1+e_2)$ and $\sigma_2=conv(e_1+e_2,e_2)$ and $\tau=\sigma_0\cap\sigma_1=conv(e_1+e_2)$. The dual cones are $\sigma_0^\vee= ...
2
votes
1answer
66 views

Limits of subrings and surjectivity

Let $A$ be a ring and let $\mathcal{F}$ be the inductive system of subrings of $A$ which are of finite type over $\mathbb{Z}$: $$ \mathcal{F} = \{ \mathbb{Z}[a_1,\dots,a_n] \subseteq A \mid n \geq 0, ...
1
vote
1answer
106 views

Functional sheaf (Hartshorne, Cartier divisors)

In Hartshorne there is the following description of the sheaf $K$ on the scheme. For each open $U = Spec \, A$ we define $K(U) = S^{-1} A$, where $S$ is the set of non-zero-divisors. Why is it a ...
1
vote
1answer
85 views

Torsion free monoids and different definitions

I just read the following question Groupification and perfection of a commutative monoid and being also interested in the same topic, I found out the definition of torsion-free monoid I am using is ...
2
votes
0answers
96 views

Is there a $\mathbb{C}_g$, where $g$ is composite?

Analogous to the $p$-adic ring $\mathbb{Z}_p$, you can (at least formally), define the $g$-adic ring $\mathbb{Z}_g$, where $g$ is composite. Of course when completing to a field, you get in trouble ...