Claim 71.2 when a component of X lies inside Y #
Kollár's proof of Claim 71.2 [Kol07, 108] takes Y ⊂ X to be the bottom of a chain of
hypersurfaces; the chain has length 0 on a component of X lying inside Y, where Y ⊂ X is a
local isomorphism and "every local equation of Y is in I" says nothing (max-ord(ker j) = ⊤).
This module settles that case for the stage functors of the tower, so that the induction of
Hironaka.Resolution.Algebraic.Stage.ClosedEmbeddingStep covers every closed embedding:
stalkIdeal_ker_eq_bot_of_two_le_ord— if the ideal ofYhas order≥ 2at a point, its stalk there is0: the stalk𝒪_{X,x}is regular with regular quotient𝒪_{Y,y}, so the kernel is generated by part of a regular system of parameters ([TSPA, Tag 00NR];IsRegularLocalRing.exists_span_eq_maximalIdeal_and_eq_span_image_lt), none of which lies in𝔪²(notMem_sq_of_span_eq) — the part is empty.exists_isClosed_stalkIdeal_ker_eq_bot— such a point specializes to a closed point ofYwhere the stalk ofker jis still0(upper semicontinuity of the order,ord_le_ord_of_specializes; the Jacobson property of schemes of finite type over a field).smoothOfRelativeDimension_eq_of_stalkIdeal_ker_eq_bot— theredim Y = dim X: the Krull dimensions of the stalks at the closed point agree, the stalk ofYbeing the quotient by0.seq_eq_pushforward_of_not_maxOrd_ker_le_one— the identity of Claim 71.2 in this case, by a two-hypersurface argument, not in the sources: inX⁺ := X ×_k 𝔸¹bothX(the zero sections, the graph of a morphism over the separated𝔸¹ → Spec k, a closed immersion) andY(alongj ≫ s) are smooth hypersurfaces —dim X⁺ = dim X + 1 = dim Y + 1(smoothOfRelativeDimension_Spec_map_algebraMap_mvPolynomial,isSmoothDivisor_of_smoothOfRelativeDimension) — and(X⁺, s_* I, 1, ∅)is a marked triple with(X, I, 1, ∅)and(Y, J, 1, ∅)closed embeddings of empty boundary. The hypersurface case (Hironaka.Resolution.Algebraic.Stage.ClosedEmbeddingHypersurface) twice, at the stagen + 2, givess_* BMO(X) = BMO(X⁺) = (j ≫ s)_* BMO(Y) = s_* j_* BMO(Y);s_*is injective (pushforward_injective), and the coherence of the tower brings both sides back to the stagen + 1.
The case is the degenerate end of Kollár's chain; the two-hypersurface argument is bookkeeping on the hypersurface case of Claim 71.2.
The image of a closed embedding of smooth k-schemes is smooth over k: V(ker j) ≅ Y.
Where the ideal of Y has order ≥ 2, its stalk is 0: the kernel of the surjection of
regular local rings 𝒪_{X,x} → 𝒪_{Y,y} is generated by part of a regular system of parameters
([TSPA, Tag 00NR];
IsRegularLocalRing.exists_span_eq_maximalIdeal_and_eq_span_image_lt),
no member of which lies in 𝔪²
(notMem_sq_of_span_eq).
A point where the stalk of ker j is 0 specializes to a closed point of X in Y where it
is still 0 (upper semicontinuity of the order and the Jacobson property).
At a closed point of X in Y where the stalk of ker j is 0, the Krull dimensions of the
stalks of X and of Y agree, so dim Y = dim X.
The degenerate end of the chain of [Kol07, 108] (c = 0: a component of X inside Y,
max-ord(ker j) = ⊤), by the two-hypersurface argument. With Y of the dimension of X
(smoothOfRelativeDimension_eq_of_stalkIdeal_ker_eq_bot), embed X into X⁺ := X ×_k 𝔸¹ by the
zero section s; both s and j ≫ s are closed immersions with smooth hypersurfaces as images,
(X⁺, s_* I, 1, ∅) is a marked triple, and the hypersurface case at the stage n + 2 gives
s_* BMO(X) = BMO(X⁺) = (j ≫ s)_* BMO(Y) = s_* (j_* BMO(Y)); s_* is injective and the coherence
of the tower returns to the stage n + 1.