• Which morphisms of schemes $\pi: X \rightarrow Y$ are quotient morphisms, i.e. satisfy the following universal property (*)?

(*) For any morphism $f:X \rightarrow Z$, such that its restriction to the fibers of $\pi$ is constant, there is an $\bar{f}: Y \rightarrow Z$ with $f= \bar{f} \circ \pi$.

[btw: is it the right definition of quotient morphism? Is it ok if we consider only fibers over closed points?]

Categorical quotients by group actions $X \rightarrow X/G$ have this property.

• Do flat surjective morphisms have this property?

• What about the first question, if we only look at varieties (and closed points)?

• Which morphisms of schemes (or varieties, if you prefer) $\pi: X \rightarrow Y$ are quotient morphisms, i.e. satisfy the following universal property (*)?

(*) For any morphism $f:X \rightarrow Z$, such that its restriction to the fibers of $\pi$ is constant, there is an $\bar{f}: Y \rightarrow Z$ with $f= \bar{f} \circ \pi$.

[btw: is it the right definition of quotient morphism? Is it ok if we consider only fibers over closed points?]

Categorical quotients by group actions $X \rightarrow X/G$ have this property.

• Do flat surjective morphisms have this property?

• Which morphisms of schemes $\pi: X \rightarrow Y$ are quotient morphisms, i.e. verify the following universal property (*)?

(*) For any morphism $f:X \rightarrow Z$, such that its restriction to the fibers of $\pi$ is constant, there is an $\bar{f}: Y \rightarrow Z$ with $f= \bar{f} \circ \pi$.

[btw: is it the right definition of quotient morphism? Is it ok if we consider only fibers over closed points?]

Categorical quotients by group actions $X \rightarrow X/G$ have this property.

• Do flat surjective morphisms have this property?

• What about the first question, if we only look at varieties (and closed points)?

• Which morphisms of schemes $\pi: X \rightarrow Y$ are quotient morphisms, i.e. satisfy the following universal property (*)?

(*) For any morphism $f:X \rightarrow Z$, such that its restriction to the fibers of $\pi$ is constant, there is an $\bar{f}: Y \rightarrow Z$ with $f= \bar{f} \circ \pi$.

[btw: is it the right definition of quotient morphism? Is it ok if we consider only fibers over closed points?]

Categorical quotients by group actions $X \rightarrow X/G$ have this property.

• Do flat surjective morphisms have this property?

• What about the first question, if we only look at varieties (and closed points)?