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