The reduced closed subspace, as an ideal sheaf #
Bierstone–Milman define the reduced space X_red corresponding to X using the coherent sheaf
of ideals I_{X_red} = √I_X [BM97, Remark 3.14]. For an ideal sheaf J of finite type on a sheaf
of rings, radicalSubspace J is the ideal sheaf whose stalks are the radicals √J_x, taken when
that stalk family has local generators (the coherence of the radical, Cartan's theorem on the
radical of a coherent ideal, provided by hasLocalGenerators_radical_of_cartan), the unit ideal
sheaf otherwise: the pattern of the weak transform and of strictTransformSubspace. It is the
reduction of a closed subspace at the level of ideal sheaves on a manifold, used to show that the
strict transform of a reduced subspace is reduced
(Hironaka.Manifold.BlowUp.Transform.StrictSubspaceReduced); for analytic spaces the
reduction is defined in Hironaka/AnalyticSpace/.
Boundary cases: for J with radical stalks (a reduced subspace), radicalSubspace J has the
same stalks as J, given the witness; for J = ⊤, the radical of ⊤ is ⊤, and both branches
give ⊤.
The reduced closed subspace X_red of the closed subspace with ideal sheaf J
[BM97, Remark 3.14]: the ideal sheaf √J with the radicals √J_x as stalks, when that family has
local generators (Cartan's theorem on the radical of a coherent ideal), the unit ideal sheaf
otherwise.
Equations
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