let $K$ be a field and $X$ a scheme. I'd like to understand the bijection

$Hom_{Sch}(Spec(K), X) \cong \{x \in X | \exists \kappa(x) \to K \}$

That map is given by sending a morphism $f: Spec(K) \to X$ to the value $f((0))$ of the only prime ideal in $K$. Why is this map injective? I don't understand why $f^\#$ is determined by $f((0))$.

  • $\begingroup$ As you note below, the RHS needs to be modified a little. $\endgroup$ – Hoot Dec 29 '14 at 17:15

Let $f :$ Spec$K \rightarrow X$ be given and let $f((0)) = x \in X$. Then it will induce a map at the local ring level: $\mathcal{O}_{X,x} \rightarrow \mathcal{O}_{K, (0)}=K.$ So we have an induced map $k(x) \rightarrow K.$

On the other hand, suppose a map $k(x) \rightarrow K$ is given, for some $x \in X.$ Define $f:$ Spec$K \rightarrow X$ by $(0) \mapsto x.$ Let $U \subseteq X$ be open. Now define $f^\#: \mathcal{O}_X(U) \rightarrow \mathcal{O}_{Spec K}(f^{-1}(U))$ as follows: if $x \in U,$ then $\mathcal{O}_X(U) \rightarrow \mathcal{O}_{X,x} \rightarrow k(x) \rightarrow K = \mathcal{O}_{Spec K}(f^{-1}(U)),$ and if $x \notin U,$ then $\mathcal{O}_X(U) \rightarrow 0 \cong \mathcal{O}_{Spec K}(f^{-1}(U)).$ This is a morphism between two sheaves: $\mathcal{O}_X$ and $ f_* \mathcal{O}_{SpecK}$ (why?). The only thing that remains to check is that it's a morphism between between local rings. For that use the homomorphism $k(x) \rightarrow K.$

  • $\begingroup$ In the end you mean that I have to check that it's a morphism between locally ringed spaces, right? I would do it like that: $\endgroup$ – Leon Lang Dec 29 '14 at 16:32
  • $\begingroup$ yeah!! it's a morphism between locally ringed spaces. $\endgroup$ – Krish Dec 29 '14 at 16:40
  • $\begingroup$ $(f^\#_{(0)})^{-1}((0)) = \{[s] | s \in \mathcal{O}_X(U), f^\#_U(s) = 0\} = \{s | (\mathcal{O}_{X,x} \to \kappa(x))([s]) = 0\} = \mathcal{m}_x$. That follows from the fact that $\kappa(x) \to K$ is injective. $\endgroup$ – Leon Lang Dec 29 '14 at 16:50
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    $\begingroup$ The main problem was that I did not know that on the right hand side the morphism $\kappa(x) \to K$ is also important. So on the right side we have $\{ (x, f) | x \in X, f: \kappa(x) \to K \text{ a homomorphism } \}$. Thank you! $\endgroup$ – Leon Lang Dec 29 '14 at 17:01
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    $\begingroup$ Thanks again: To construct this bijection was one task in the exam, and of course I could answer it ;-) I'm wondering now: Shouldn't this construction work even if X is just a locally ringed space? I don't think that there is used that it locally looks like an affine scheme. $\endgroup$ – Leon Lang Feb 4 '15 at 21:36

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