Saturations under a ring map that becomes flat after a descent #
The transport lemma behind the local description of the strict transform of a closed analytic
subspace, I_{X',a'} = {f ∈ 𝒪_{M',a'} : y_exc^k f ∈ I_{σ^{-1}(X),a'} for some k}
[BM97, Proposition 3.13]: for a ring map χ : R → S and a further map ι : S → T such that
ι ∘ χ is flat and ideals of S descend along ι (ι⁻¹(J T) = J), the saturation
⋃_k (I : e^k) of an ideal I ⊆ R by the powers of an element e is carried by χ onto the
saturation of I S by the powers of χ(e):
⋃_k (I S : χ(e)^k) = (⋃_k (I : e^k)) S.
The inclusion ⊇ needs nothing. For ⊆, an element g with χ(e)^k g ∈ I S is sent by ι
into (I T : (ιχ)(e)^k) = (I : e^k) T — colon ideals by a principal ideal commute with flat base
change, AlgebraicGeometry.map_colon_singleton_of_flat — and (I : e^k) T ∩ S = (I : e^k) S by
the descent hypothesis. Bierstone and Milman prove Proposition 3.13 through the diagram of initial
exponents; the argument here uses flatness instead, with S the local ring of the blown-up
manifold at a', T its completion, and R a polynomial chart ring of the blow-up: the
flatness of R → T comes from identifying the completions of the chart ring and of the local
ring, and the descent from the faithful flatness of completion (I 𝒪̂ ∩ 𝒪 = I). Hironaka's
definition of the strict transform of a sheaf of ideals is [Hir64, Ch. 0, §2, pp. 129–130].
A flat descent datum for χ : R → S: a ring T with a map ι : S → T along which every
ideal of S descends (ι⁻¹(J T) = J) and such that ι ∘ χ : R → T is flat. For χ the map
from a polynomial chart ring of a blow-up to the local ring 𝒪_{M',a'} of the blown-up manifold,
the datum is the completion T = 𝒪̂_{M',a'}.
Equations
Instances For
The saturation ⋃_k (I : e^k) is carried by χ onto the saturation of I S by χ(e), given
a map ι : S → T with ι ∘ χ flat along which ideals of S descend. The transport step of
[BM97, Proposition 3.13]; not stated there in this form.
The transport lemma with the datum as a hypothesis.