Orders of products, the weak transform of a product, the factors along an irreducible center #
Hironaka's Corollary 1 applies Main Theorem II to the product J = ∏ J_j of finitely many ideal
sheaves and reads off its condition (1) for the factors: for every pair (i, j), the order
ν(J_j(i)_y) "is a positive constant for the points y of D(i)" Hir64, Corollary 1,
pp. 143–144. The three
lemmas behind that sentence, which the source does not state:
ord_mul,ord_prod: at every point of a smooth scheme over a field of characteristic zero the order of a product is the sum of the orders. The stalk is a regular local ring containingℚ(isRegularLocalRing_stalk,stalkAlgebraRat), on which the order of elements is additive (ordElem_mul_of_algebraRat, through the Cohen structure theorem); the order of an ideal is the least order of an element (ord_eq_iInf), attained on a finite generating set (exists_ord_span_eq_ordElem), which givesord_mul_of_ordElem_mulfor ideals; the stalk of a product is the product of the stalks (stalkIdeal_mul). Compare the properties of the cosupport in [Kol07, Definition 59].weakTransform_prod: along a regular centerDon which every factor has constant orderc_j, the weak transform of the product is the product of the weak transforms. Each weak transform is the marked transform with markc_j(weakTransform_eq_markedTransform_of_smooth),π^* J_j = F^{c_j} · W_j[Kol07, Definition 60, (60.1)], soπ^* (∏ J_j) = F^{∑ c_j} · ∏ W_j, and the colon by the invertibleF^{∑ c_j}recovers∏ W_j(colon_pow_eq_of_mul_eq).ord_factors_const_on_center: on an irreducible center on which the order of the product is the constantd, every factor has constant order: eachν(J_j)is upper semicontinuous with its least value at the generic pointη, the sum of theevalues at anyy ∈ Dand atηis the samed, so the termwise inequalities are equalities.
The local lemma: the order of a product of ideals #
ord I ≤ ordElem f for f ∈ I.
The order of a nonzero ideal of a Noetherian local ring is attained on an element.
When the order of elements is additive, the order of a product of ideals is the sum of the orders.
The order of a product on a smooth scheme #
The Corollary's setting #
On a smooth scheme over a field of characteristic zero the order of a product is the sum of
the orders at every point: the stalk is a regular local ring containing ℚ, where the order of
elements is additive (ordElem_mul_of_algebraRat).
ord_mul_of_smooth for finite products.
The weak transform of a product along a smooth center on which every factor has constant
order is the product of the weak transforms (general form over the structure morphism f): each
inverse image is F^{c_j} times the weak transform (the marked transform with mark c_j,
[Kol07, Definition 60, (60.1)]), so the inverse image of the product is F^{Σ c_j} times the
product of the weak transforms, and the colon by the invertible F^{Σ c_j} recovers it. Not
stated in [Hir64, Corollary 1], whose proof uses it.
On an irreducible closed subscheme on which the order of the product is constant, the order of
every factor is constant (general form over f): each order is upper semicontinuous with its
least value at the generic point, the sums agree, and termwise ≤ with equal finite sums forces
equality. This is condition (1) of [Hir64, Corollary 1] for the factors.
The order of a product is the sum of the orders at every point of a smooth k-scheme; used
without comment in the proof of [Hir64, Corollary 1].
The weak transform of a product along a regular center on which every factor has constant
order is the product of the weak transforms (weakTransform_prod_of_smooth at the structure
morphism; a regular closed subscheme of a smooth k-scheme is smooth over k).
On an irreducible center on which the order of the product is constant, the order of every
factor is constant (ord_factors_const_on_center_of_smooth at the structure morphism); condition
(1) of [Hir64, Corollary 1] for the factors.