Janiyah Hays
2022-04-03
Examples for when the quotient ring is necessarily / not necessarily an extension of the residue field.
Let R be an integral domain with unique maximal ideal . Let be the field of fraction of R.
etsahalen5tt
Beginner2022-04-04Added 8 answers
Step 1
I can at least explain the situation you linked to.
Let R be a finitely generated algebra over an algebraically closed field k. Let be a maximal ideal of R, and let
Now the quotient is a field which is also a finitely generated k-algebra, so by Zariski's Lemma is a finite extension of k. But now since k is algebraically closed, this forces to just be k.
So, embeds in B, and thus also in
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