Pullback of the ideal sheaf of an effective Cartier divisor

Let D be an effective Cartier divisor on a scheme Y, with ideal sheaf I, and let g : X ⟶ Y be a morphism such that the inverse image g⁻¹ D, with ideal sheaf I.comap g, is again an effective Cartier divisor. This holds for flat g, and for the base change of a relative effective Cartier divisor along any morphism of bases. Then the pullback g^* I of the line bundle I is the ideal sheaf of g⁻¹ D: the comparison morphism g^* I ⟶ I.comap g is an isomorphism (Scheme.IdealSheafData.isIso_pullbackSheafHom). Consequently, pulling back the line-bundle class of I gives the line-bundle class of I.comap g (Scheme.IdealSheafData.IsEffectiveCartier.pullback_mk_toInvertibleSheaf); dually, 𝒪(D) pulls back to 𝒪(g⁻¹ D).

Near a point of X, the ideal I is generated on an affine open U of Y by a nonzerodivisor a, and I.comap g is then generated by g^♯ a on small affine opens V (Scheme.IdealSheafData.ideal_comap_of_le). Since I.comap g is effective Cartier, g^♯ a is a nonzerodivisor there, so it is a basis of I.comap g over V. The comparison morphism sends the image of the section a of I in g^* I to g^♯ a, and a morphism out of a line bundle sending a local section onto a local basis is an isomorphism (AlgebraicGeometry.Scheme.Modules.isIso_of_forall_exists_bijective_smul).

References

• The Stacks Project, Divisors, Section Effective Cartier divisors (Tag 01WQ): the pullback of an effective Cartier divisor, and its invertible ideal sheaf.

theorem AlgebraicGeometry.Scheme.IdealSheafData.isIso_pullbackSheafHom {X Y : Scheme} (I : Y.IdealSheafData) (g : X ⟶ Y) (hI : I.IsEffectiveCartier) (hI' : (I.comap g).IsEffectiveCartier) : CategoryTheory.IsIso (I.pullbackSheafHom g)

The ideal sheaf of an effective Cartier divisor commutes with pullback. If I is an effective Cartier divisor on Y whose inverse image along g : X ⟶ Y is again an effective Cartier divisor, then pullbackSheafHom : g^* I ⟶ I.comap g is an isomorphism.

theorem AlgebraicGeometry.Scheme.IdealSheafData.IsEffectiveCartier.pullback_mk_toInvertibleSheaf {X Y : Scheme} {I : Y.IdealSheafData} {g : X ⟶ Y} (hI : I.IsEffectiveCartier) (hI' : (I.comap g).IsEffectiveCartier) : TauCeti.AlgebraicGeometry.LineBundleClass.pullback g (TauCeti.AlgebraicGeometry.LineBundleClass.mk hI.toInvertibleSheaf) = TauCeti.AlgebraicGeometry.LineBundleClass.mk hI'.toInvertibleSheaf

The class of the ideal sheaf of an effective Cartier divisor commutes with pullback. If I and its inverse image along g are effective Cartier divisors, then pulling back the line-bundle class of I along g gives the line-bundle class of I.comap g.