Incidence and equality loci of closed subschemes #
Let f : X ⟶ S be a morphism of schemes and let I, J be ideal sheaves on X whose closed
subscheme V(J) is finite locally free over S (finite, flat and locally of finite presentation).
For a base change X' = X ×_S S' along t : S' ⟶ S, the inverse image of V(J) may or may not
be contained in the inverse image of V(I). This file shows that the condition is closed on the
base: there is an ideal sheaf incidenceLocus f I J on S such that V(J)_{S'} ⊆ V(I)_{S'},
that is I.comap g ≤ J.comap g, exactly when t factors through its closed subscheme
(AlgebraicGeometry.Scheme.IdealSheafData.incidenceLocus_le_ker_iff_of_isPullback). It is the
Weil restriction along V(J) ⟶ S of the inverse image of V(I) in V(J).
When both V(I) and V(J) are finite locally free over S, the equality locus
equalityLocus f I J, the supremum of the two incidence loci, is the closed subscheme of S over
which the two closed subschemes agree (equalityLocus_le_ker_iff_of_isPullback). Both loci are
compatible with arbitrary base change (comap_incidenceLocus, comap_equalityLocus).
The main application is to relative effective Cartier divisors D, D' on a proper relative
curve C ⟶ S, which are finite locally free over S
(AlgebraicGeometry.Scheme.IdealSheafData.IsRelativeEffectiveCartier.isFinite together with
flatness and finite presentation). Then incidenceLocus is the universal incidence locus
D' ≤ D and equalityLocus the universal equality locus D' = D: the closed subschemes of the
base over which one divisor is contained in, respectively equal to, the other. No hypothesis
is placed on S, which may be nonreduced: equality is tested scheme-theoretically, not on
geometric fibres.
Main definitions #
AlgebraicGeometry.Scheme.IdealSheafData.incidenceLocus f I J: the ideal sheaf onSof the locus over whichV(J)is contained inV(I).AlgebraicGeometry.Scheme.IdealSheafData.equalityLocus f I J: the ideal sheaf onSof the locus over whichV(I)andV(J)agree.
Main results #
incidenceLocus_le_ker_iff_of_isPullback,incidenceLocus_le_ker_iff: the universal property of the incidence locus;incidenceLocus_eq_bot_iffis its value overSitself.comap_le_comap_pullback_snd_subschemeι_incidenceLocus: the universal family of the incidence locus.comap_incidenceLocus: the incidence locus commutes with arbitrary base change.equalityLocus_le_ker_iff_of_isPullback,equalityLocus_le_ker_iff,equalityLocus_eq_bot_iff,comap_eq_comap_pullback_snd_subschemeι_equalityLocusandcomap_equalityLocus: the corresponding statements for the equality locus.
References #
- N. M. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, Annals of Mathematics Studies 108, Princeton University Press, 1985, Chapter 1, for relative effective Cartier divisors on relative curves and the closed loci where they are contained in, or equal to, one another.
The ideal sheaf on S of the incidence locus of the closed subschemes V(J) ⊆ X and
V(I) ⊆ X, where V(J) is finite locally free over S along f : X ⟶ S. A morphism S' ⟶ S
factors through its closed subscheme exactly when the inverse image of V(J) in the base change
X ×_S S' is contained in that of V(I)
(AlgebraicGeometry.Scheme.IdealSheafData.incidenceLocus_le_ker_iff_of_isPullback). It is the
Weil restriction along V(J) ⟶ S of the inverse image of V(I) in V(J).
Instances For
The incidence locus is the Weil restriction along V(J) ⟶ S of the inverse image of V(I) in
V(J).
The universal property of the incidence locus. For a pullback square with g : X' ⟶ X
over t : S' ⟶ S, the morphism t factors through the closed subscheme of
incidenceLocus f I J exactly when the inverse image of V(J) in X' is contained in that of
V(I).
The universal property of the incidence locus, for the base change T ×_S X along
t : T ⟶ S.
The incidence locus is all of S exactly when V(J) is contained in V(I).
The incidence locus of V(J) in itself is all of S.
The incidence locus of V(J) in X is all of S.
The universal family of the incidence locus. Over the closed subscheme Z of
incidenceLocus f I J, the inverse image of V(J) in Z ×_S X is contained in that of
V(I).
The incidence locus commutes with base change. For a pullback square with g : X' ⟶ X
over t : S' ⟶ S, the inverse image along t of the incidence locus of V(J) in V(I) is the
incidence locus of their inverse images in X'.
The closed subscheme of J.comap g is the base change of that of J
(AlgebraicGeometry.Scheme.IdealSheafData.isPullback_subschemeMap_comap), so the instance
arguments follow from those for J by MorphismProperty.of_isPullback.
The ideal sheaf on S of the equality locus of the closed subschemes V(I) and V(J) of
X, both finite locally free over S along f : X ⟶ S. A morphism S' ⟶ S factors through
its closed subscheme exactly when the inverse images of V(I) and V(J) in the base change
X ×_S S' agree
(AlgebraicGeometry.Scheme.IdealSheafData.equalityLocus_le_ker_iff_of_isPullback). It is the
supremum of the two incidence loci, so its closed subscheme is their intersection.
Instances For
The equality locus is the supremum of the two incidence loci.
The equality locus is symmetric.
The universal property of the equality locus. For a pullback square with g : X' ⟶ X
over t : S' ⟶ S, the morphism t factors through the closed subscheme of
equalityLocus f I J exactly when the inverse images of V(I) and V(J) in X' agree.
The universal property of the equality locus, for the base change T ×_S X along
t : T ⟶ S.
The equality locus is all of S exactly when V(I) and V(J) agree.
The equality locus of V(I) with itself is all of S.
The universal family of the equality locus. Over the closed subscheme Z of
equalityLocus f I J, the inverse images of V(I) and V(J) in Z ×_S X agree.
The equality locus commutes with base change. For a pullback square with g : X' ⟶ X
over t : S' ⟶ S, the inverse image along t of the equality locus of V(I) and V(J) is the
equality locus of their inverse images in X'.