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replaced K by Z
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Andrew Critch
Andrew Critch
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Nice fact: Suppose f:X->Y is a map of schemes and K⊆YZ⊆Y is a subscheme (locally closed immersion) containing the set-image of X. If X and KZ are reduced, then it follows that f factors through KZ. This is nice because it makes "factoring through" purely a consideration of the underlying topological spaces.

So now I'm wondering, to what extent does "reduced" allow us to think only terms of topological spaces? Suppose we weaken the assumption that K→YZ→Y is an inclusion. When can we say f factors through KZ? More precisely:

Suppose X,KZ are reduced schemes, f:X→Y and g:K→YZ→Y are scheme morphisms such that f factors through g in Top. When does f factor through g in Sch?

I know the answer is "not always", for example if X,K,Y are fieldsY is a field and X,KZ are incomparable field extensions of Y (in Ringop). But does anyone know any positive results we can state here?

Nice fact: Suppose f:X->Y is a map of schemes and K⊆Y is a subscheme (locally closed immersion) containing the set-image of X. If X and K are reduced, then it follows that f factors through K. This is nice because it makes "factoring through" purely a consideration of the underlying topological spaces.

So now I'm wondering, to what extent does "reduced" allow us to think only terms of topological spaces? Suppose we weaken the assumption that K→Y is an inclusion. When can we say f factors through K? More precisely:

Suppose X,K are reduced schemes, f:X→Y and g:K→Y are scheme morphisms such that f factors through g in Top. When does f factor through g in Sch?

I know the answer is "not always", for example if X,K,Y are fields and X,K are incomparable field extensions of Y (in Ringop). But does anyone know any positive results we can state here?

Nice fact: Suppose f:X->Y is a map of schemes and Z⊆Y is a subscheme (locally closed immersion) containing the set-image of X. If X and Z are reduced, then it follows that f factors through Z. This is nice because it makes "factoring through" purely a consideration of the underlying topological spaces.

So now I'm wondering, to what extent does "reduced" allow us to think only terms of topological spaces? Suppose we weaken the assumption that Z→Y is an inclusion. When can we say f factors through Z? More precisely:

Suppose X,Z are reduced schemes, f:X→Y and g:Z→Y are scheme morphisms such that f factors through g in Top. When does f factor through g in Sch?

I know the answer is "not always", for example if Y is a field and X,Z are incomparable field extensions of Y (in Ringop). But does anyone know any positive results we can state here?

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Andrew Critch
Andrew Critch
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Comparing maps of reduced schemes

Nice fact: Suppose f:X->Y is a map of schemes and K⊆Y is a subscheme (locally closed immersion) containing the set-image of X. If X and K are reduced, then it follows that f factors through K. This is nice because it makes "factoring through" purely a consideration of the underlying topological spaces.

So now I'm wondering, to what extent does "reduced" allow us to think only terms of topological spaces? Suppose we weaken the assumption that K→Y is an inclusion. When can we say f factors through K? More precisely:

Suppose X,K are reduced schemes, f:X→Y and g:K→Y are scheme morphisms such that f factors through g in Top. When does f factor through g in Sch?

I know the answer is "not always", for example if X,K,Y are fields and X,K are incomparable field extensions of Y (in Ringop). But does anyone know any positive results we can state here?