Normalization in a field, and resolution of curves #
Let K be a field and f : Spec K ⟶ Y a quasi-compact quasi-separated morphism. Over an affine
open U ⊆ Y the relative normalization f.normalization of Y in Spec K is the spectrum of
the integral closure of Γ(Y, U) in Γ(Spec K, f ⁻¹ U), which is either K or zero. An
integral closure in a field is an integrally closed domain, so f.normalization is a normal
scheme.
The main case is the normalization of an integral scheme Y in its function field, along
Spec k(Y) ⟶ Y. When Y is locally Noetherian of dimension at most one and this normalization is
finite over Y, it is again locally Noetherian of dimension at most one, so its local rings are
fields or discrete valuation rings and it is a regular scheme. This is the regularity half of the
resolution of curves by normalization. The finiteness hypothesis carries the Japanese/Nagata
input; for one-dimensional Y it is equivalent to the existence of a resolution of
singularities (Stacks, Lemma 54.15.1 (0BI4)).
Main results #
TauCeti.AlgebraicGeometry.isNormalScheme_normalization_of_field: the normalization of a scheme in the spectrum of a field is normal.TauCeti.AlgebraicGeometry.isRegular_normalization_of_field: the normalization of a locally Noetherian scheme of dimension at most one in the spectrum of a field is regular when it is finite over the scheme.TauCeti.AlgebraicGeometry.isRegular_normalization_fromSpecStalk_genericPoint: resolution of curves, regularity half. The normalization of an integral locally Noetherian scheme of dimension at most one in its function field is regular when it is finite over the scheme.
References #
The normalization of a scheme Y in the spectrum of a field K, along a quasi-compact
quasi-separated morphism f : Spec K ⟶ Y, is a normal scheme: its sections over the preimage of
an affine open of Y are an integral closure in K, or zero.
The normalization of a locally Noetherian scheme Y of dimension at most one in the spectrum
of a field, along f : Spec K ⟶ Y, is a regular scheme provided it is finite over Y: it is
normal, locally Noetherian and of dimension at most one, so its local rings are fields or discrete
valuation rings.
Resolution of curves: regularity of the normalization. Let Y be an integral locally
Noetherian scheme of dimension at most one, and suppose that the normalization of Y in its
function field, along Spec k(Y) ⟶ Y, is finite over Y. Then this normalization is a regular
scheme.