The pushforward of the structure sheaf under flat base change #
A morphism of schemes f : X ⟶ S satisfies f_* 𝒪_X = 𝒪_S when every map
f.app V : Γ(S, V) ⟶ Γ(X, f⁻¹ V) is an isomorphism; it suffices to ask this for affine V. This
file shows that for a quasi-compact quasi-separated f this condition is stable under flat base
change: for every flat g : T ⟶ S, the projection p : T ×_S X ⟶ T again satisfies
p_* 𝒪_{T ×_S X} = 𝒪_T. On an
affine open U of T lying over an affine open V of S, the sections of T ×_S X over
p⁻¹ U are Γ(X, f⁻¹ V) ⊗_{Γ(S, V)} Γ(T, U) (Mathlib's
AlgebraicGeometry.isIso_pushoutSection_of_isQuasiSeparated_of_flat_right), which is Γ(T, U)
when Γ(S, V) ⟶ Γ(X, f⁻¹ V) is an isomorphism. Such opens form a basis of T, and a morphism of
sheaves that is an isomorphism on a basis is an isomorphism.
Over a field every base change is flat, so a quasi-compact quasi-separated scheme X over a field
K with Γ(X, 𝒪_X) = K satisfies p_* 𝒪_{X_T} = 𝒪_T for every scheme T over K. This holds
for a proper (more generally, universally closed and quasi-separated) integral scheme with a
K-rational point, whose global functions are constant
(TauCeti.AlgebraicGeometry.appTop_bijective_of_section). It is the hypothesis "f_* 𝒪_X = 𝒪
universally" under which line bundles rigidified along a section of X have no automorphisms
other than the identity, after every base change.
Main results #
AlgebraicGeometry.Scheme.Hom.isIso_app_of_isBasis: a morphism of schemes induces isomorphisms on the sections over all opens once it does so over a basis of opens.TauCeti.AlgebraicGeometry.isIso_app_pullback_fst_of_flat:f_* 𝒪_X = 𝒪_Sis stable under flat base change for quasi-compact quasi-separatedf.TauCeti.AlgebraicGeometry.isIso_app_pullback_fst_of_isIso_appTop: over a field, a quasi-compact quasi-separated scheme with constant global functions satisfiesp_* 𝒪_{X_T} = 𝒪_Tafter every base change.TauCeti.AlgebraicGeometry.isIso_app_pullback_fst_of_section: the same for a universally closed quasi-separated integral scheme with a rational point, such as a proper one.
References #
- The Stacks Project, Lemma 30.5.2, flat base change.
- S. Bosch, W. Lütkebohmert, M. Raynaud, Néron Models, Section 8.1.
A morphism of schemes p : Y ⟶ X induces isomorphisms Γ(X, U) ≅ Γ(Y, p⁻¹ U) for all opens
U once it does so for the opens of a basis of X.
A morphism of schemes into a scheme with at most one point induces isomorphisms on the sections over all opens once it induces one on global sections.
Flat base change for f_* 𝒪_X = 𝒪_S. If f : X ⟶ S is quasi-compact and
quasi-separated and induces isomorphisms Γ(S, V) ≅ Γ(X, f⁻¹ V) for all affine opens V, then
for every flat g : T ⟶ S the projection T ×_S X ⟶ T induces isomorphisms on the sections over
all opens of T.
f_* 𝒪_X = 𝒪 universally over a field. If X is quasi-compact and quasi-separated over a
field K and its global functions are the constants, then for every scheme T over K the
projection T ×_K X ⟶ T induces isomorphisms on the sections over all opens of T.
f_* 𝒪_X = 𝒪 universally for a proper integral scheme with a rational point. If X is
integral, universally closed and quasi-separated over a field K (for instance proper) and has a
K-rational point, then for every scheme T over K the projection T ×_K X ⟶ T induces
isomorphisms on the sections over all opens of T.