Locally finitely generated ideal sheaves #

The ideal sheaves of the Hironaka library: for a sheaf of commutative rings 𝒪 on a topological space X (the structure sheaf of an analytic manifold in the applications), a locally finitely generated ideal sheaf IdealSheaf 𝒪 is a sub-presheaf of ideals J(U) ⊆ 𝒪(U) satisfying the sheaf condition (a section locally in the ideal is in the ideal) and local finite generation: every point has an open neighbourhood U and finitely many sections f_1, …, f_k ∈ J(U) whose germs generate the stalk ideal J_b at every b ∈ U. This is Hironaka's "coherent sheaf of ideals" on a non-singular analytic space [Hir64, Main Theorem II''(N), p. 158], coherence being replaced by local finite generation, and Bierstone–Milman's "sheaf of ideals of finite type" [BM97, §3].

The stalk ideal stalkIdeal J b is the set of germs at b of sections of J (stalkIdealOf).
Any family of stalk ideals with local generators (HasLocalGenerators) is the family of stalks of an ideal sheaf (ofStalks); this is how the ideal sheaf ofGlobal 𝒪 f generated by finitely many global sections, the unit and zero ideal sheaves ⊤, ⊥ and the product I * J are built, the last with the stalkwise product.
IsNonzeroEverywhere J: every stalk is nonzero, Hironaka's "coherent sheaf of non-zero ideals" [Hir64, Main Theorem II''(N), p. 158].
support J = {a | J_a ≠ 𝒪_a}, the set of points of the closed subspace defined by J (Kollár's cosupp; Bierstone–Milman's V(J) [BM97, Introduction]).
When the stalks are local rings, ord J x is the order ν_x(J) = max {p | J_x ⊆ 𝔪_x^p} of the stalk ideal (Hironaka's ν(J_x) [Hir64, p. 133]; [Kol07, Definition 47]; [BM97, §3]), the Hironaka library's IsLocalRing.ord.

On analytic manifolds:

For an analytic map φ : N → M and an ideal sheaf J of the structure sheaf of M, the pullback IdealSheaf.pullback J φ hφ is the ideal sheaf generated by the f ∘ φ, f a section of J: Hironaka's total transform f⁻¹(J), "the one generated by J with reference to f" [Hir64, pp. 141–142], and Bierstone–Milman's σ⁻¹(I) := σ^*(I) · 𝒪 [BM97, Theorem 1.10]. Its stalk at b is the image of J_{φ b} under the stalk map germMap φ hφ b.
The vanishing ideal vanishingStalk Z x of a set Z at x is the ideal of the germs of analytic functions at x whose function germ (stalkToGerm) vanishes on Z near x (Germ.VanishesOn, vanishingGermIdeal): the stalk of Hironaka's reduced structure red on the closed set Z [Hir64, Main Theorem II'(N) (iii), p. 156]; Bierstone–Milman's reduced space, 𝓘_{X_red} = √𝓘_X [BM97, Remark 3.14]. Whether these stalks are the stalks of a locally finitely generated ideal sheaf (coherence of the ideal of an analytic set) is a separate question, left as a guard in the definitions that use them.

For a bundled analytic manifold M, AnalyticManifold.IdealSheaf M is the type of the ideal sheaves of its structure sheaf; closed analytic subspaces of M (centres, boundaries) are given by their ideal sheaves. With the unit ideal ⊤, the inverse image J.pullback f f.contMDiff along an analytic map f, its case restrict for the inclusion of an open subset, the product I * J, the order J.ord x and J.IsNonzeroEverywhere, these are the operations the analytic main theorems use. stalkRing M x is the local ring 𝒪_{M,x}, and IsNonsingular J says that the closed subspace defined by J is non-singular: every one of its points is simple, the quotient 𝒪_{M,x}/J_x being a regular local ring [Hir64, p. 121]. IsReduced J says that the closed subspace is reduced (every stalk J_x is a radical ideal), and regularLocus J is its set of simple points, Włodarczyk's Reg(Y) [Wlo09, Theorem 2.0.2 (2)].

Locally finitely generated ideal sheaves #

The unit and zero ideal sheaves, products #

Nonzero stalks, the cosupport, the order #

The pullback along an analytic map #

The vanishing ideal of a set #

The ideal sheaves of an analytic manifold #