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Residue field does not change under restriction to closed subschemes or fibres

By dunghoangnguyen

It does not change when restricting to closed subschemes. It is because the residue ring does not change : \frac{O}{p} = \frac{O/I}{p/I}.
When taking fibre of a morphism : f : X \to Y, f(x) = y
Consider a scheme Spec T = k(x). Then by the universal property of fibre product, there are ( (natural) morphisms
T \to X_y \to latex $ and since the composition induces isomorphism on residue fields, so does the first map.

This entry was posted on May 8, 2008 at 12:01 am and is filed under AG- Foundations. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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