# Morphisms between varieties

I want to verify my understanding.

I am working with Hartshorne definitions, in particular a variety is either affine, quasi-affine, projective, or quasi-projective.

A morphism is one that sends regular functions to regular functions.

Explicitly saying what morphisms are:

1. A map to an affine $$Y$$ is a morphism iff each component is a regular function
2. A map to a quasi-affine $$Y$$ is a morphism iff each component is a regular function
3. A map to a projective $$Y$$ is a morphism iff if on $$U_i \cap Y$$, normalizing the ith coordinate to be $$1$$, the rest are regular
4. A map to a quasi-projective $$Y$$ is a morphism iff if on $$U_i \cap Y$$, normalizing the ith coordinate to be $$1$$, the rest are regular

The proof to all of these is just explicitly saying that composing with a regular function gives again an appropriate regular function (one has to divide to cases based on the preimage being affine or projective).

• It seems correct. Though, Hartshorne Chapter II gives a better definition of morphisms using the notion of \textit{ringed spaces} (and schemes in particular). But for local computation this description can be useful. Commented Feb 11, 2020 at 7:24
• @DominiqueMattei Thanks, if you put this as an answer I can accept it
– Andy
Commented Feb 11, 2020 at 7:56

As I said in comments: If you think of varieties as locally ringed spaces then a morphism $$(f,f^\sharp): (X,\mathcal{O}_X) \to (Y,\mathcal{O}_Y)$$ is the data of both a continuous map $$f : X\to Y$$ and a morphism of sheaf $$f^\sharp : \mathcal{O}_Y \to f_*\mathcal{O}_X$$.
In the case of a (let's say affine) variety $$X=V(P)$$ for $$P\in K[X_1,\dots,X_n]$$, the sheaf $$\mathcal{O}_X$$ is associated to $$K[X_1,\dots,X_n]/(P)$$, that is the regular functions on $$X$$. Hence the component $$f^\sharp$$ is just precomposition by $$f$$, thus we find back the condition "regular functions are sent to regular functions".