In Example 3.3.1 Chapter II Hartshorne, he considers a morphism of schemes $F : X \to Y$, where $X = Spec \ k[x,y,t]/(ty - x^2)$ and $Y = Spec \ k[t]$. And he says $f$ is a surjective morphism. I was wondering what is the definition of surjectivity for morphisms of schemes? Thank you
2026-03-30 23:12:40.1774912360
Definition of a surjective morphism of schemes
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Usually it is just supposed to mean that the underlying map on the topological spaces is surjective.
There are various other possibilities for ”surjective-like“ morphisms (e.g. epi, dominant etc.), but they have other names.
There are non-surjective epis and surjective non-epis in the category of schemes. Thus do not arbitrarily switch between these two.