The complement of an effective Cartier divisor is scheme-theoretically dense

Let D be an effective Cartier divisor on a scheme X, given by an ideal sheaf I. Its open complement X ∖ D is scheme-theoretically dense in X: the inclusion of X ∖ D is scheme-theoretically dominant, so the scheme-theoretic closure of X ∖ D is all of X, with its nilpotents. Locally, on an affine open U where I is generated by a nonzerodivisor a, the complement contains the basic open D(a), and restriction from U to D(a) is the localization Γ(X, U) ⟶ Γ(X, U)_a, which is injective because a is a nonzerodivisor. No reducedness, Noetherianity or finiteness hypothesis on X is needed.

This is what makes the complement of the exceptional divisor of a blowup dense in the blowup, and so identifies the strict transform of a closed subscheme with a scheme-theoretic closure.

Main results

• IsEffectiveCartier.isSchemeTheoreticallyDominant_support_compl_ι, in the namespace AlgebraicGeometry.Scheme.IdealSheafData: the inclusion of the complement of an effective Cartier divisor is scheme-theoretically dominant.

References

• The Stacks Project, Section 01WQ, Effective Cartier divisors: the lemma that the complement of an effective Cartier divisor is scheme-theoretically dense.

theorem AlgebraicGeometry.Scheme.IdealSheafData.IsEffectiveCartier.isSchemeTheoreticallyDominant_support_compl_ι {X : Scheme} {I : X.IdealSheafData} (hI : I.IsEffectiveCartier) : IsSchemeTheoreticallyDominant (Opens.ι I.support.compl)

The complement of an effective Cartier divisor is scheme-theoretically dense: the inclusion of the open complement of the support of an effective Cartier divisor is scheme-theoretically dominant.