A Cohomological Property of Lagrange Multipliers

usual theorem of Lagrange multipliers says that a = (a1, . . . , an) ∈ Y is a critical point of f |Y if and only if there exists b = (b1, . . . , br) ∈ R such that (a;b) ∈ U×R is a critical point of the auxiliary function F = f+ ∑r i=1 yifi : U×R r → R. The point b is unique when it exists. We establish a closer relation between f and F for algebraic varieties over an arbitrary field K. Let X = Spec(A) be a smooth affine K-scheme of finite type, purely of dimension n, let f, f1, . . . , fr ∈ A and put I = (f1, . . . , fr) ⊆ A. Put B = A/I and let Y = Spec(B), a closed subscheme of X . We assume that Y is a smooth K-scheme, purely of codimension r in X . We write f̄ for the image of f ∈ A under the natural map A → B. Let y1, . . . , yr be indeterminates and considerX×K A = Spec(A[y1, . . . , yr]). We shall write A[y] for A[y1, . . . , yr]. Put F = f + ∑r i=1 yifi ∈ A[y]. Let Ω k B/K (resp. Ω k A[y]/K) be the module of differential k-forms of B (resp. A[y]) over K. Let dB/K f̄ ∈ Ω 1 B/K and dA[y]/KF ∈ Ω 1 A[y]/K be the exterior derivatives of f̄ and F , respectively. We consider the complexes (ΩB/K , φf̄ ) and (Ω · A[y]/K , φF ), where φf̄ : Ω k B/K → Ω k+1 B/K is the map defined by φf̄ (ω) = dB/K f̄ ∧ ω and φF : Ω k A[y]/K → Ω k+1 A[y]/K is the map defined by φF (ω) = dA[y]/KF ∧ ω.