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Let $\phi:A\to B$ be an epimorphism in the category of commutative rings, we can find that the induced continuous map $\phi^*$ from Spec$B$ to Spec$A$ is injective as a map between sets,

I want to know if $\phi^*$ is also an immersion of topological spaces, that is if Spec$B$ is homeomorphic to $\phi^*(\mathrm{Spec}(B))$ under the map $\phi^*$ ?

Taking localization and quotient are the just the special cases of epimorphisms, how far are they away from epimorphisms?

Is there an explicit construction of epimorphisms in CRings ?


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Do you mean by your last question, is there an explicit characterization of when a homomorphism is right cancellable? See this previous question for the statement of the Silver-Mazet-Isbell Zigzag theorem for rings; the same characterization holds for commutative rings (see this MO question). – Arturo Magidin Oct 13 '11 at 4:33
@Arturo Magidin : thanks, I hope the answer of first question is yes, then Spec$B$ is a closed set in the constructible topology of Spec$A$, and ....I wish $B$ is just taking localization and quotient by many times of $A$, but it seems there is a long distance to my wish :).... – wxu Oct 13 '11 at 4:44
@Arturo Magidin: Wonderful links! thanks again..:)+100 – wxu Oct 13 '11 at 4:51
up vote 3 down vote accepted

The answer is no.

If $X_1,X_2 \to Y$ are two monomorphisms of schemes (in the sense of category theory) with disjoint images (in the sense of set theory), then it is easy to verify that $X_1 \sqcup X_2 \to Y$ is also a monomorphism. For example, $X_1$ might be a closed subscheme and $X_2$ might be an open subscheme, and then $X_1 \sqcup X_2 \to Y$ is almost never a homeomorphism onto its image. Algebraically: For every ring $A$ and every $a \in A$ the canonical map $A \to A/aA \times A_a$ is an epimorphism in the category of commutative rings.

In contrast to that, flat monomorphisms are better behaved. For more on epimorphisms of commutative rings / monomorphisms of schemes, see this Seminar from the 60's. They did give interesting characterizations and properties of epimorphisms, but on the other hand no "structural classification" is known.

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Dear @Martin, I am looking for a characterization of monomorphisms in the category of schemes (or varieties) I looked up at the link you proposed but did not find anything. Do you know whether they have a neat description? – Brenin Jul 4 '13 at 19:49

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