Revisions to What are the epimorphisms in the category of schemes?

Link to @KevinVentullo's answer while this was on the front page; removed "which I learned from" from @KReiser's edit, since it was not in the original

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edited Mar 8, 2022 at 23:07

LSpice

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Is there a known characterization of epimorphisms in the category of schemes?

It is easy to see that a morphism $f : X \to Y$ such that the underlying map $|f|$$\lvert f\rvert$ is surjective and the homomorphism $f^\# : \mathcal{O}_Y \to f_* \mathcal{O}_X$ is injective, is an epimorphism. But there are other examples, too: if $Y$ is reduced and locally of finite type over a field, the obvious morphism from $X=\bigsqcup_{y\in Y_{cl}} \operatorname{Spec}(k(y))$$X=\bigsqcup_{y\in Y_\text{cl}} \operatorname{Spec}(k(y))$ to $Y$ is an epimorphism, which I learned from (see #8(b) in Mark Haiman's thisHomework Set 9 for Math 256AB problem sheet).

If this is is not possible, what about regular, extremal, or effective epimorphisms? Here, again, I know only some examples.

My background is that I want to know if there is a categorical characterization of the spectra of fields in the category of schemes. In the full subcategory of affine schemes, they are characterized by the property: $X$ is non-initial and every morphism from a non-initial object to $X$ is an epimorphism. But I doubt that this characterization takes over to the category of schemes. EDIT: Kevin Ventullo has shown belowbelow that the characterization takes over. Thus my original question has been answered (and I wonder if itsit's appropriate to accept it as an answer). But of course every other hint about the characterization of epimorphisms of schemes is appreciated.

Is there a known characterization of epimorphisms in the category of schemes?

It is easy to see that a morphism $f : X \to Y$ such that the underlying map $|f|$ is surjective and the homomorphism $f^\# : \mathcal{O}_Y \to f_* \mathcal{O}_X$ is injective, is an epimorphism. But there are other examples, too: if $Y$ is reduced and locally of finite type over a field, the obvious morphism from $X=\bigsqcup_{y\in Y_{cl}} \operatorname{Spec}(k(y))$ to $Y$ is an epimorphism, which I learned from this problem sheet.

If this is is not possible, what about regular, extremal, or effective epimorphisms? Here, again, I know only some examples.

My background is that I want to know if there is a categorical characterization of the spectra of fields in the category of schemes. In the full subcategory of affine schemes, they are characterized by the property: $X$ is non-initial and every morphism from a non-initial object to $X$ is an epimorphism. But I doubt that this characterization takes over to the category of schemes. EDIT: Kevin Ventullo has shown below that the characterization takes over. Thus my original question has been answered (and I wonder if its appropriate to accept it as an answer). But of course every other hint about the characterization of epimorphisms of schemes is appreciated.

Is there a known characterization of epimorphisms in the category of schemes?

It is easy to see that a morphism $f : X \to Y$ such that the underlying map $\lvert f\rvert$ is surjective and the homomorphism $f^\# : \mathcal{O}_Y \to f_* \mathcal{O}_X$ is injective, is an epimorphism. But there are other examples, too: if $Y$ is reduced and locally of finite type over a field, the obvious morphism from $X=\bigsqcup_{y\in Y_\text{cl}} \operatorname{Spec}(k(y))$ to $Y$ is an epimorphism (see #8(b) in Mark Haiman's Homework Set 9 for Math 256AB).

If this is is not possible, what about regular, extremal, or effective epimorphisms? Here, again, I know only some examples.

My background is that I want to know if there is a categorical characterization of the spectra of fields in the category of schemes. In the full subcategory of affine schemes, they are characterized by the property: $X$ is non-initial and every morphism from a non-initial object to $X$ is an epimorphism. But I doubt that this characterization takes over to the category of schemes. EDIT: Kevin Ventullo has shown below that the characterization takes over. Thus my original question has been answered (and I wonder if it's appropriate to accept it as an answer). But of course every other hint about the characterization of epimorphisms of schemes is appreciated.

replaced broken link with wayback machine version; added explicit example from link in to body of post

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edit approved Mar 8, 2022 at 22:07

KReiser

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Is there a known characterization of epimorphisms in the category of schemes?

It is easy to see that a morphism $f : X \to Y$ such that the underlying map $|f|$ is surjective and the homomorphism $f^\# : \mathcal{O}_Y \to f_* \mathcal{O}_X$ is injective, is an epimorphism. But there are other examples, too: if $Y$ is reduced and locally of finite type over a field, see for examplethe obvious morphism from this$X=\bigsqcup_{y\in Y_{cl}} \operatorname{Spec}(k(y))$ to $Y$ is an epimorphism, which I learned from this problem sheet.

If this is is not possible, what about regular, extremal, or effective epimorphisms? Here, again, I know only some examples.

My background is that I want to know if there is a categorical characterization of the spectra of fields in the category of schemes. In the full subcategory of affine schemes, they are characterized by the property: $X$ is non-initial and every morphism from a non-initial object to $X$ is an epimorphism. But I doubt that this characterization takes over to the category of schemes. EDIT: Kevin Ventullo has shown below that the characterization takes over. Thus my original question has been answered (and I wonder if its appropriate to accept it as an answer). But of course every other hint about the characterization of epimorphisms of schemes is appreciated.

Is there a known characterization of epimorphisms in the category of schemes?

It is easy to see that a morphism $f : X \to Y$ such that the underlying map $|f|$ is surjective and the homomorphism $f^\# : \mathcal{O}_Y \to f_* \mathcal{O}_X$ is injective, is an epimorphism. But there are other examples, too, see for example this problem sheet.

If this is is not possible, what about regular, extremal, or effective epimorphisms? Here, again, I know only some examples.

My background is that I want to know if there is a categorical characterization of the spectra of fields in the category of schemes. In the full subcategory of affine schemes, they are characterized by the property: $X$ is non-initial and every morphism from a non-initial object to $X$ is an epimorphism. But I doubt that this characterization takes over to the category of schemes. EDIT: Kevin Ventullo has shown below that the characterization takes over. Thus my original question has been answered (and I wonder if its appropriate to accept it as an answer). But of course every other hint about the characterization of epimorphisms of schemes is appreciated.

Is there a known characterization of epimorphisms in the category of schemes?

It is easy to see that a morphism $f : X \to Y$ such that the underlying map $|f|$ is surjective and the homomorphism $f^\# : \mathcal{O}_Y \to f_* \mathcal{O}_X$ is injective, is an epimorphism. But there are other examples, too: if $Y$ is reduced and locally of finite type over a field, the obvious morphism from $X=\bigsqcup_{y\in Y_{cl}} \operatorname{Spec}(k(y))$ to $Y$ is an epimorphism, which I learned from this problem sheet.

If this is is not possible, what about regular, extremal, or effective epimorphisms? Here, again, I know only some examples.

My background is that I want to know if there is a categorical characterization of the spectra of fields in the category of schemes. In the full subcategory of affine schemes, they are characterized by the property: $X$ is non-initial and every morphism from a non-initial object to $X$ is an epimorphism. But I doubt that this characterization takes over to the category of schemes. EDIT: Kevin Ventullo has shown below that the characterization takes over. Thus my original question has been answered (and I wonder if its appropriate to accept it as an answer). But of course every other hint about the characterization of epimorphisms of schemes is appreciated.