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Zr near a, such that S = {z : z1 = · · · = zr = 0}. The ideal spanned by z1 , z2 , . . , zr at the point a (the origin of the coordinates) is of rank r, and is obviously equal to its own radical: it is therefore ia (S). 6 Irreducibility The following definitions can just as well be applied locally (to germs of analytic sets) as globally (to analytic sets). An analytic set S is said to be irreducible if it cannot be written as the union of two analytic sets which are distinct from it. Otherwise, it is said to be reducible.

We can then pose the following algebraic problem: since C ∗ (X) is the Z-dual of C∗ (X) by | , can we deduce that H ∗ (X) is the Z-dual of H∗ (X) by | ? The answer is yes, modulo torsion. If, instead of taking coefficients in Z, we took them in a field, then we avoid any torsion problems and H ∗ (X) is the vector space which is dual to H∗ (X). 1 becomes particularly interesting in the light of de Rham’s theorem. This states that we can replace C ∗ (X) by Ω(X), δ by d, and | by integration in the definition of cohomology H ∗ (X) with coefficients in R or C (the formula δϕ | γ = ϕ | ∂γ then becomes Stokes’ formula).

1, so that we can define its cohomology: H ∗ (X, A) = Z ∗ (X, A)/B ∗ (X, A) [= Φ(X, A)/dΩ(X, A)], B ∗ (X, A) = Im δ. 4) that if A is a closed differentiable submanifold of X, the restriction homomorphism of differential forms is surjective. 4 De Rham duality 39 a way that ψ exists). Then δψ defines a cocycle on X, whose restriction to A is zero: δψ|A = i∗ δψ = δi∗ ψ = δϕ = 0, and is therefore a relative cocycle on (X, A). One checks that its relative cohomology class, denoted δ ∗ hp ∈ H p+1 (X, A), only depends on the original class hp .