Smooth divisors are a stalkwise notion: restriction and gluing along open immersions #
IsSmoothDivisor D (Hironaka.Scheme.Snc.SmoothDivisor) asks that V(D) be regular and that at
every point x ∈ V(D) the stalk D_x be generated by an element of 𝔪_x ∖ 𝔪_x². Both clauses are
conditions on the stalks of X at the points of the support — regularity of V(D) at z is
regularity of 𝒪_{X,x} / D_x for x = ι(z) (stalkQuotientEquiv) — so the notion is local: it
restricts along open immersions (IsSmoothDivisor.comap_of_isOpenImmersion of
Hironaka.Resolution.Algebraic.MaximalContact.Restrict) and it holds as soon as it holds after
restriction to the members of a covering family of open immersions
(isSmoothDivisor_of_forall_comap, this module). The stalk of D.comap f at y is the image of
the stalk of D at f y under the stalk map of f (stalkIdeal_comap), an isomorphism of local
rings when f is an open immersion, and both clauses transport along a ring isomorphism
(isSmoothDivisorAt_map).
Two further facts on ideal sheaves along open immersions, used with these by
Hironaka.Resolution.Algebraic.Kol07.MaximalContactGlobalization to glue the hypersurfaces of
maximal contact of the pieces of a disjoint union: the inverse image along an open immersion
commutes with finite intersections (comap_iInf_of_isOpenImmersion; on each affine open it is the
inverse image of ideals under a ring isomorphism), and an inclusion of ideal sheaves may be checked
after restriction to a covering family of open immersions (le_of_forall_comap_le).
Source: Step 3 of the proof of [Kol07, Theorem 103] ("the disjoint union H^* := ∐ H^(j) ⊂ ∐ X^(j)
is a smooth hypersurface of maximal contact"), which uses this locality tacitly.
The smooth-ideal condition on an ideal I of a local ring R — the quotient R / I is a
regular local ring and I is generated by an element of 𝔪 ∖ 𝔪². At a stalk 𝒪_{X,x} with
I = D_x these are the two clauses of IsSmoothDivisor at x.
Equations
- Hironaka.Snc.IsSmoothIdealAt I = (IsRegularLocalRing (R ⧸ I) ∧ ∃ a ∈ IsLocalRing.maximalIdeal R, a ∉ IsLocalRing.maximalIdeal R ^ 2 ∧ I = Ideal.span {a})
Instances For
The stalkwise smooth-divisor condition at x — the smooth-ideal condition for the stalk
D_x ⊆ 𝒪_{X,x}.
Equations
Instances For
D is a smooth divisor iff the stalkwise condition holds at every point of its support —
regularity of V(D) at z is regularity of 𝒪_{X, ι z} / D_{ι z}.
The smooth-ideal condition transports along an isomorphism of local rings — the maximal ideal, its square and principal ideals are carried to their counterparts.
The smooth-ideal condition is invariant under an isomorphism of local rings.
Along an open immersion f, the stalkwise condition for D.comap f at y is the condition for
D at f y — the stalk map of an open immersion is an isomorphism of local rings carrying
D_{f y} to (f^* D)_y (stalkIdeal_comap).
An ideal sheaf whose restrictions along a covering family of open immersions are smooth divisors is a smooth divisor.
The inverse image along an open immersion commutes with finite intersections of ideal sheaves — on each affine open it is the inverse image of ideals under the ring isomorphism of sections, which commutes with intersections.
An inclusion of ideal sheaves may be checked after restriction to a covering family of open immersions — stalkwise, through the stalk isomorphisms.
The inverse image, along f, of the direct image of an ideal sheaf along a closed immersion g
whose image is disjoint from the image of f is the unit ideal — its support is the preimage under
f of a closed subset of the image of g.