theorem smooth_iff_completion_power_series (R : Type u_1) [CommRing R] [IsLocalRing R] [IsNoetherianRing R] (k : Type u_2) [Field k] [Algebra k R] (d : ℕ) (hdim : ringKrullDim R = ↑d) (hres : IsLocalRing.ResidueField R ≃+* k) : IsRegularLocalRing R ↔ Nonempty (MvPowerSeries (Fin d) k ≃+* AdicCompletion (IsLocalRing.maximalIdeal R) R)

Proposition 32 (Lecture 19). A Noetherian local k-algebra R of dimension d with residue field k is regular (smooth) iff its m-adic completion is isomorphic to the formal power series ring k[[t_1,…,t_d]].

theorem smooth_iff_cotangent_dim (R : Type u_1) [CommRing R] [IsLocalRing R] [IsNoetherianRing R] : IsRegularLocalRing R ↔ ↑(Module.finrank (IsLocalRing.ResidueField R) (IsLocalRing.CotangentSpace R)) = ringKrullDim R

A Noetherian local ring is regular iff the dimension of its Zariski cotangent space matches its Krull dimension.

theorem smooth_iff_cotangent_dim_nat (R : Type u_1) [CommRing R] [IsLocalRing R] [IsNoetherianRing R] (d : ℕ) (hdim : ringKrullDim R = ↑d) : IsRegularLocalRing R ↔ Module.finrank (IsLocalRing.ResidueField R) (IsLocalRing.CotangentSpace R) = d

Numeric form of the regularity criterion: a Noetherian local ring of Krull dimension d is regular iff its cotangent space has dimension d.