Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $f:Y\to Z$ be an etale morphism, $g_1, g_2:X\to Y$ be morphisms. Assume that for all closed points $x\in X$ we have $g_1(x)=g_2(x)$ and that $f\circ g_1=f\circ g_2$. Can I conclude (under some reasonable conditions on $X$, $Y$, and $Z$) that $g_1=g_2$?

share|improve this question
2  
Take $X=Y=\mathrm{Spec}(\mathbb{C})$, $Z=\mathrm{Spec}(\mathbb{R})$, $g_1=$ identity, $g_2=$ complex conjugation. Perhaps the question is: what do you exactly mean by $g_1(x)=g_2(x)$? –  Laurent Moret-Bailly Nov 8 '12 at 21:50
    
Damian: If $f$ is the identity then $f \circ g_1 =f \circ g_2$ implies $g_1=g_2$, so I'm not sure what you mean by that. –  Will Sawin Nov 8 '12 at 22:46
    
Laurent: agreed. –  Roman Fedorov Nov 9 '12 at 4:44
    
@Will Savin: you are right, I misunderstood the statement. –  Damian Rössler Nov 9 '12 at 18:07

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.