Properties of Morphisms

Here we lost some properties of morphisms

Definition of Morphism

A morphism of schemes are morphisms of locally ringed spaces

Properties of Morphims

Open Embedding

A morphism $X\rightarrow Y$ is called open embedding if the it’s an ismoprhism from $X$ to an open subset $U$ in $Y$.

Quasicompact

A morphism $X\rightarrow Y$ is called quasicompact if for any affine open subset $U\subset Y$, the preimage of $U$ is quasicompact.

Proposition
Quasicompactness is affine-local on the target.

Quasiseperated

A morphism $X\rightarrow Y$ is called quasiseperated if for any affine open subset $U\subset Y$, the preimage of $U$ is a quasiseperated scheme.

Proposition
Quasiseperatedness is affine-local on the target.

Affine Morphisms

A morphism $X\rightarrow Y$ is called affine if for any affine open subset $U\subset Y$, the preimage of $U$ is affine.

Proposition
The affineness of a morphism is affine-local on the target.

Finiteness

A morphism $X\rightarrow Y$ is called finite if it is affine and for any affine open subset $U=\operatorname{Spec}(B)$, the preimage of $U$ is $\operatorname{Spec}(A)$ and $A$ is a module-finite over $B$.

Proposition
Finite morphisms are closed and of finite fiber.

Integral

A morphism $X\rightarrow Y$ is called integral if it is affine and for any affine open subset $U=\operatorname{Spec}(B)$, the preimage of $U$ is $\operatorname{Spec}(A)$ and $A$ is a integral over $B$.

Proposition
Integral morphisms are closed.

Locally of Finite Type

A morphism $X\rightarrow Y$ is called locally of fintie type if for any affine open subset $U=\operatorname{Spec}(B)$ and any $\operatorname{Spec}(A)$ contained in the preimage of $U$, $A$ is a finitely generated $B$-algebra.

A morphism locally of finite type is called of finite type if in addition it is quasicompact.

Proposition
Finite = Integral + of Finite Type