Blowing up affine schemes commutes with flat base change #
Let I be an ideal of a commutative ring R and let S be an R-algebra, with extended ideal
IS = I.map (algebraMap R S). The blowup of Spec R along V(I) is Proj R[It], with structure
morphism Proj.toSpecCoeff to Spec R. The base change of the Rees algebra
(reesAlgebra.baseChangeGradedAlgHom) is a surjective graded homomorphism
S ⊗[R] R[It] ⟶ S[(IS)t], and Proj commutes with arbitrary extension of the coefficient ring
(AlgebraicGeometry.Proj.baseChangeIso). Together they give a comparison morphism over Spec S
Proj S[(IS)t] ⟶ Spec S ×_{Spec R} Proj R[It]
from the blowup of Spec S along the inverse image V(IS) of the centre to the base change of
the blowup of Spec R. It is always a closed immersion, and it is an isomorphism when S is flat
over R: then the blowup of Spec S along V(IS) is the base change of the blowup of Spec R
along V(I). In particular blowing up commutes with localizations R ⟶ R_f and with étale
R-algebras.
Over a local ring A, let I = (a, b) with b² ∈ a I, so that the chart Spec A[I/a] is the
whole blowup. If the closed fibre of A[I/a] is a finite étale algebra over the residue field
κ, then the base change of the blowup to a field-valued point of the closed point of Spec A
is finite étale of rank dim_κ (κ ⊗[A] A[I/a]).
Main definitions #
TauCeti.AlgebraicGeometry.affineBlowupBaseChangeHom I S: the comparison morphism from the blowup ofSpec SalongV(IS)to the base change of the blowup ofSpec RalongV(I).
Main results #
TauCeti.AlgebraicGeometry.toSpecCoeff_reesAlgebra: the coefficient structure morphism of the blowup is the structure morphism to the spectrum of the degree zero partR.TauCeti.AlgebraicGeometry.affineBlowupBaseChangeHom_snd: the comparison lies overSpec S.TauCeti.AlgebraicGeometry.isClosedImmersion_affineBlowupBaseChangeHom: the comparison is a closed immersion.TauCeti.AlgebraicGeometry.isIso_affineBlowupBaseChangeHom: for flatS, it is an isomorphism.TauCeti.AlgebraicGeometry.isPullback_affineBlowup_of_flat: for flatS, the square formed by the two blowups overSpec S ⟶ Spec Ris cartesian.TauCeti.AlgebraicGeometry.etale_isFinite_finrank_of_isPullback_toSpecCoeff: when a single chart is the whole blowup of a local ring and its closed fibre is finite étale, the base change of the blowup to a field-valued point of the closed point is finite étale of the same rank.
References #
- The Stacks Project, Section Blowing up, the lemma comparing the blowup of the inverse image of the centre with the base change of the blowup: it is a closed subscheme, equal to it after flat base change.
The coefficient structure morphism Proj R[It] ⟶ Spec R of the blowup is the structure
morphism to the spectrum of the degree zero part of R[It], identified with R by
reesAlgebra.gradeZeroEquiv.
The comparison of blowups under base change. The morphism from the blowup Proj S[(IS)t]
of Spec S along V(IS) to the base change Spec S ×_{Spec R} Proj R[It] of the blowup of
Spec R along V(I), induced by the base change S ⊗[R] R[It] ⟶ S[(IS)t] of Rees algebras.
It is a closed immersion (isClosedImmersion_affineBlowupBaseChangeHom), and an isomorphism for
flat S (isIso_affineBlowupBaseChangeHom).
Instances For
The projection of the comparison to the blowup of Spec R is the morphism of projective
spectra induced by the base change of Rees algebras, followed by the coefficient projection.
The projection of the comparison to the blowup of Spec R is the morphism of projective
spectra induced by the base change of Rees algebras, followed by the coefficient projection.
The comparison of blowups is a morphism over Spec S.
The comparison of blowups is a morphism over Spec S.
The blowup of the inverse image of the centre is a closed subscheme of the base change of
the blowup: the comparison Proj S[(IS)t] ⟶ Spec S ×_{Spec R} Proj R[It] is a closed
immersion, for every R-algebra S.
Blowing up commutes with flat base change: for flat S, the comparison
Proj S[(IS)t] ⟶ Spec S ×_{Spec R} Proj R[It] is an isomorphism.
The blowup of Spec S along V(IS) is the base change of the blowup of Spec R along
V(I), for flat S: the square formed by the projection of the comparison to Proj R[It]
and the structure morphisms of the two blowups over Spec S ⟶ Spec R is cartesian.
Let I = (a, b) with b² ∈ a I, so that the chart Spec A[I/a] is the whole blowup of
Spec A along V(I). If the closed fibre κ ⊗[A] A[I/a] is a finite étale κ-algebra, then the
base change of the blowup along a field-valued point of the closed point of Spec A is finite
étale, of the same rank.