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Algebraic Geometry

By Drozd Yu.

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Then, if X → Ad is a finite dominant morphism, K[ X ] is integral over K[ x1 , . . , xd ] , thus, K(X) is algebraic over K( x1 , . . , xd ) and tr. deg(K(X)/K) = d . 5 follows from Noether’s Normalization Lemma (or its projective analogue, cf. 6. K. dim K[ x1 , . . , xn ] = n . 2. 7. If A ⊇ B is a finite extension of noetherian rings, then K. dim A = K. 7. It is a consequence of the following result. 8 (Going-Up Principle). Let A ⊇ B be a finite extension of noetherian rings, p ⊂ B be a prime ideal.

Vd such that the coordinates aki of vk are the following:   if i = k ≤ d , 1 aki = 0 if i = k ≤ d ,  (−1)d−i p if k > d . kd ) the Grassmann coordinates of V . dk for each k . kd for any k1 k2 . . kd . Denote by m the number of indices from k1 k2 . . kd which are greater than d and use the induction on m . The cases m ≤ 1 have just been considered. Suppose that the claim is valid for all d-tuples with the smaller value of m . Take, in the d-tuple k1 k2 . . kd = 12 . . d , some index kj > d .

Proof. Consider in the affine space of all d × n matrices the open subset U of the matrices of rank d . It is irreducible as Adn is irreducible. 1) define a surjective morphism U → Gr(d, n) . Hence, Gr(d, n) is also irreducible as the image of an irreducible space under a continuous mapping. 5. (1) Let W be an m-dimensional subspace in Kn . Prove that, for each r , { V ∈ Gr(d, n) | dim(V + W ) ≤ r } is closed in Gr(d, n) . In particular, the following subsets are closed: (a) { V ∈ Gr(d, n) | V + W = Kn } , (b) { V ∈ Gr(d, n) | V ∩ W = { 0 } } .

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