# Morphism from a scheme to the spectra of global section

This is a small inquiry, but I just want to be sure about it.

Let $$X$$ be a scheme. I am trying to understand what it is meant by the cannonical morphism between $$X \rightarrow Spec (\Gamma(X,\mathcal{O}_{X})).$$ Is it the morphism of schemes induced by the identity map of rings $$\Gamma(X,\mathcal{O}_{X}) \rightarrow \Gamma(X,\mathcal{O}_{X})$$ and hence by bijection between morphisms of scheme to affine scheme and morphisms of rings of global sections, the aforementioned morphism of scheme is isomorphism?

I want to respond to one particular statement you made, but first I'll make things a bit more precise.

We have the adjunction $$\newcommand\Hom{\operatorname{Hom}}\newcommand\Spec{\operatorname{Spec}}\newcommand\calO{\mathcal{O}}\Hom(X,\Spec A)\simeq \Hom(A,\Gamma(X,\calO_X)),$$ and in particular this tells us that $$\Hom(X,\Spec \Gamma(X,\calO_X))\simeq \Hom(\Gamma(X,\calO_X),\Gamma(X,\calO_X)).$$

The identity map $$\Gamma(X,\calO_X)\to\Gamma(X,\calO_X)$$ therefore gives us a map of schemes $$X\to \Gamma(X,\calO_X)$$ as you noticed, and this is indeed the canonical map. However, you seem to be under the impression that this map is therefore an isomorphism.

In general this cannot possibly be true, since if the map were an isomorphism, $$X$$ would necessarily have to be affine. However, if $$X$$ is affine, this map is indeed an isomorphism.

Let's be a little more clear how this map works then.

In fact let's be a little more clear how it works in general. Let $$\phi : A\to \Gamma(X,\calO_X)$$ be a ring morphism. Let's try to understand the induced map $$f : X\to \Spec A$$.

Let $$U$$ be an affine open in $$X$$. Then we have the maps $$\newcommand\toby\xrightarrow A\toby{\phi}\calO_X(X)\toby{r_{XU}} \calO_X(U).$$ Taking $$\Spec$$ of this sequence gives $$U\toby{\Spec r_{XU}} \Spec \calO_X(X) \toby{\Spec \phi} \Spec A.$$ Gluing these maps together gives the desired map from $$X$$ to $$\Spec A$$.

Observe then that if $$\phi=\newcommand\id{\operatorname{id}}\id$$, that the map $$X\to \Spec \Gamma(X,\calO_X)$$ is the result of gluing the maps obtained from applying the Spec functor to the restrictions $$r_{XU}:\calO_X(X)\to \calO_X(U)$$.

If $$X$$ is affine, then we can take $$U=X$$, and there's no need to glue, the map $$X\to \Spec\Gamma(X,\calO_X)$$ is $$\Spec \id=\id$$. On the other hand, if $$X$$ is not affine, for example, if $$X$$ is a projective $$k$$-scheme with $$k$$ algebraically closed, then $$\Gamma(X,\calO_X)=k$$, and $$X\to \Spec\Gamma(X,\calO_X)$$ is the $$k$$-scheme structure morphism $$X\to \Spec k$$, which in general is clearly not an isomorphism.