By Kollar J., Lazarsfeld R., Morrison D. (eds.)
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Additional info for Algebraic Geometry Santa Cruz 1995, Part 1
1 in the Appendix) that there is a bijective correspondence between klinear symmetric maps V k → W , and k-homogeneous maps V → W , given by diagonalization of k-linear maps. This correspondence takes care of the alternative formulation in the following theorem. 2 (Taylor expansion) For any f : V → W , there exists a unique sequence of maps f0 , f1 , f2 , . . with fk : V k → W a symmetric k-linear map, such that, for each k = 0, 1, 2, . . f (x) = f0 + f1 (x) + f2 (x, x) + . . + fk (x, . . , x) for all x ∈ Dk (V ).
Also, for the contravariant determination, it is of interest to consider test functions φ which are only locally defined around a1 and a2 . Finally, it may be of interest to consider the category of those spaces where the covariant and contravariant determination of ∼ agree; this category will contain all manifolds. 3 Let W be a KL vector space. Let D(W ) denote the set of w ∈ W with w ∼ 0. Prove that w ∈ D(W ) iff there exists a finite-dimensional vector space V and a linear map f : V → W with w = f (d) for some d ∈ D(V ).
K! 11 Let M ⊆ U be a (formally) open subset of a finitedimensional vector space U. Let W1 , W2 and W3 be KL vector spaces; let ∗ : W1 ×W2 → W3 be a bilinear map. Let f1 : M → W1 and f2 : M → W2 . Then for x ∈ M, u ∈ U, we have d( f1 ∗ f2 )(x; u) = d f1 (x; u) ∗ f2 (x) + f1 (x) ∗ d f2 (x; u). Proof. Calculate the expression ( f1 ∗ f2 )(x + d · u) = f1 (x + d · u) ∗ f2 (x + d · u) in two ways, using the definition of directional derivative, and bilinearity of ∗. 12 Let V and V be KL vector spaces, and let M be a (formally) open subset of some finite-dimensional vector space U.
Algebraic Geometry Santa Cruz 1995, Part 1 by Kollar J., Lazarsfeld R., Morrison D. (eds.)
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