English
Let S be a local R-algebra presented by 0 → I → P → S → 0 with P formally smooth over R, Ω[P/R] finite free over P, and I finitely generated. Then S is formally smooth iff the map k ⊗_S I/I^2 → k ⊗_P Ω[P/R] is injective, where k is the residue field of S.
Русский
Пусть S — локальное R-алгебра с представлением 0 → I → P → S → 0, где P формально гладна над R, Ω[P/R] конечно свободна над P, и I конечно порожден. Тогда S формально гладна тогда и только тогда, когда карта k ⊗_S I/I^2 → k ⊗_P Ω[P/R] инъективна, где k — остаточное поле S.
LaTeX
$$$\\text{S is formally smooth over } R \\iff k \\otimes_S I/I^2 \\rightarrow k \\otimes_P \\Omega_{P/R} \\text{ is injective}$ with the stated hypotheses.$$
Lean4
theorem localization_map [FormallySmooth R S] : FormallySmooth Rₘ Sₘ :=
by
haveI : FormallySmooth S Sₘ := FormallySmooth.of_isLocalization (M.map (algebraMap R S))
haveI : FormallySmooth R Sₘ := FormallySmooth.comp R S Sₘ
exact FormallySmooth.localization_base M
