Points on Schemes

Posts: 866
Joined: Jan 12, 2011

by iarnab_kundu, Dec 17, 2018, 11:28 AM

Definition- Let $S$ be a scheme, and $K$ be a field. Then a $K$ -point on $S$ is a morphism $x:\text{Spec}(K)\to S$ . The set of $K$ -points on $S$ shall be denoted by $S(K)$ .

Given a point $p\in S$ we denote the local ring of $S$ at $p$ to be $\mathcal{O}_{S,p}$ . The maximal ideal of the this local ring is denoted to be $p$ , excusing the abuse of notation. The residue field of $S$ at $p$ is denoted to be $\kappa(p)\coloneqq \mathcal{O}_{S,p}/p$ .

Proposition- There is a natural bijection between the set of morphisms $\text{Spec}(K)\to S$ with the image as $p\in S$ and the set of embeddings $\kappa(p)\hookrightarrow K$ .
Proof- Suppose $x:\text{Spec}(K)\to S$ is a morphism with image $p$ . Then we get a map induces on local rings $\mathcal{O}_{S,p}\to\mathcal{O}_{\text{Spec}(K),q}$ , where $q$ is the unique point in $\text{Spec}(K)$ . But $\{q\}$ is open and therefore $\mathcal{O}_{\text{Spec}(K),q}=K$ . The morphism $\mathcal{O}_{S,p}\to K$ , induces a unique morphism $\kappa(p)\hookrightarrow K$ .

Now let $\kappa(p)\hookrightarrow K$ be an embedding. Suppose $p\in U=\text{Spec}(A)$ be an open affine sub-scheme of $S$ . Thus we have a morphism $A_p/p\hookrightarrow K$ which gives us a unique morphism $A\to K$ and this gives us a unique morphism $\text{Spec}(K)\to U=\text{Spec}(A)$ . We can compose with the open immersion $U\hookrightarrow S$ to get a morphism $\text{Spec}(K)\to S$ with image $p$ .

This post has been edited 1 time. Last edited by iarnab_kundu, Dec 17, 2018, 11:50 AM

algebraic geometry

Comment

Cracking the Nut Open

Points on Schemes

by iarnab_kundu, Dec 17, 2018, 11:28 AM

0 Comments