Definition 13 ([Scalar rings and formal coefficients]). An algebraic parameter chart will mean a local ℂ -algebra essentially of finite type and etale over an affine coordinate space at the designated origin. Its completion is …

Definition 13 ([Scalar rings and formal coefficients]). An algebraic parameter chart will mean a local ℂ -algebra essentially of finite type and etale over an affine coordinate space at the designated origin. Its completion is B=ℂ[[v_1,…,v_N]] , with maximal ideal m . Finite collections of algebraic germs obtained by an implicit equation with invertible Jacobian can always be placed in one such chart, after replacing it by a common etale neighborhood. For a smooth manifold Z , smooth forms with formal coefficients mean Ω^j(Z)⊗B =_k(Ω^j(Z)⊗_ ℂ B/ m^k). For quotients of B , use the analogous inverse limit. The extension of a scalar ideal I to this space always means its closed extension. Differential operations in the Z -variables preserve that extension; multiplication respects its powers. Integration over a compact manifold is defined at each finite parameter quotient and then by inverse limit.