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I have a naive question on algebraic geometry.

To fix a context we consider $\phi:X\to Y$ a morphism between two affine varieties over an algebraic closed field $k$.

This give under the anti-equivalence of categories a k-algebra morphism $\phi^*$ between coordinate algebras of $Y$ and $X$.

However, $\phi^*$ injective doesn't imply $\phi$ surjective. Think for example to the projection of the hyperbola $XY=1$ to the $X$-axis.

Shouldn't the anti-equivalence of categories force a correspondence between injective and surjective morphisms? It must be that surjective morphism of affine varieties are not the epimorphisms in the categorical sense, but I don't understand why?

I know that for finite morphisms for instance, we have this correspondence. So I am wondering what are the epimorphisms and monomorphisms in the category of affine varieties?

Thanks in advance!

Regards, Moustik

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    Monomorphisms in the category of rings give rise to epimorphisms of affine varieties, namely dominant maps. This is not surprising since it is the same in topology: morphisms of varieties are continuous and such maps are uniquely determined on a dense subset. – 2017-02-13
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    Thank @Moos.I think I just figured one thing: if $\phi ^*$ is injective, it is a monomorphism. Then $\phi$ is of course an epimorphism. But epimorphisms are not necessarily surjective... That's just what I was missing... – 2017-02-13
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    Anyway the question stands. Is there a description of epimorphisms and monomorphisms in the category of affine varieties? – 2017-02-13
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    For monomorphisms of affine schemes/epimorphisms of commutative rings, see [this MathOverflow question](http://mathoverflow.net/q/109/33088). – 2017-02-14
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    @TakumiMurayama I think that link qualifies as answer and should be posted as such. – 2018-05-14

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I am posting my comment as an answer as requested, with some more concrete references.

We start with monomorphisms:

Proposition [Borceux, Ex. 1.7.7.d]. A homomorphism of unital commutative rings is a monomorphism if and only if it is injective.

Epimorphisms are harder to describe; see this MathOverflow question, and especially David Rydh's answer, which states a characterization of monomorphisms among morphisms of schemes which are locally of finite type.

We state a general characterization of epimorphisms of rings:

Theorem [Roby, Thm. 1]. Let $f \colon R \to S$ be a homomorphism of unital commutative rings. The following are equivalent:

  1. $f$ is an epimorphism;
  2. For every $R$-algebra $T$, there exists an $R$-algebra homomorphism $S \to T$;
  3. For all $s \in S$, the relation $1 \otimes s = s \otimes 1$ holds in $S \otimes_R S$;
  4. The multiplication map $\mu \colon S \otimes_R S \to S$ is injective (in which case it is an isomorphism of $R$-algebras);
  5. The canonical injection $i_1 \colon S \to S \otimes_R S$ (resp. $i_2 \colon S \to S \otimes_R S$) defined by $s \mapsto s \otimes 1$ (resp. $s \mapsto 1 \otimes s$) is surjective (in which case it is an isomorphism of $R$-algebras);
  6. The tensor algebra $T_R(S)$ is commutative;
  7. $S \otimes_R S/R = 0$.

I've skipped some of the conditions; see [Roby] for more. Roby's article is from a seminar directed by Samuel, which is probably the most thorough source on epimorphisms of rings. Finally, see [Lazard] for results on flat epimorphisms, and [Stacks, Tag 04VM] for a modern reference.