[Merged by Bors] - feat(ring_theory/ideal/over): algebra structure on R/p → S/P for P lying over p
#9858
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…` lying over `p`
Co-Authored-By: Eric Wieser <wieser.eric@gmail.com>
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… lying over `p` (#9858) This PR shows `P` lies over `p` if there is an injective map completing the square `R/p ← R —f→ S → S/P`, and conversely that there is a (not necessarily injective, since `f` doesn't have to be) map completing the square if `P` lies over `p`. Then we specialize this to the case where `P = map f p` to provide an `algebra p.quotient (map f p).quotient` instance. This algebra instance is useful if you want to study the extension `R → S` locally at `p`, e.g. to show `R/p : S/pS` has the same dimension as `Frac(R) : Frac(S)` if `p` is prime.
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… lying over `p` (#9858) This PR shows `P` lies over `p` if there is an injective map completing the square `R/p ← R —f→ S → S/P`, and conversely that there is a (not necessarily injective, since `f` doesn't have to be) map completing the square if `P` lies over `p`. Then we specialize this to the case where `P = map f p` to provide an `algebra p.quotient (map f p).quotient` instance. This algebra instance is useful if you want to study the extension `R → S` locally at `p`, e.g. to show `R/p : S/pS` has the same dimension as `Frac(R) : Frac(S)` if `p` is prime.
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P lying over pP lying over p
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This PR shows
Plies overpif there is an injective map completing the squareR/p ← R —f→ S → S/P, and conversely that there is a (not necessarily injective, sincefdoesn't have to be) map completing the square ifPlies overp. Then we specialize this to the case whereP = map f pto provide analgebra p.quotient (map f p).quotientinstance.This algebra instance is useful if you want to study the extension
R → Slocally atp, e.g. to showR/p : S/pShas the same dimension asFrac(R) : Frac(S)ifpis prime.simplemmas forideal.quotient.mk+algebra_map#9829