Download PDF by I. R. Shafarevich (editor), V.I. Danilov, V.V. Shokurov: Algebraic geometry 01 Algebraic curves, algebraic manifolds

By I. R. Shafarevich (editor), V.I. Danilov, V.V. Shokurov

ISBN-10: 3540519955

ISBN-13: 9783540519959

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Additional info for Algebraic geometry 01 Algebraic curves, algebraic manifolds and schemes

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0 . . 0⎥⎟ ⎢0 0 . . 0⎥ ⎜⎢ ⎥⎟ ⎢ ⎥ ξ ⎜⎢ . . ⎟ = ⎢. ⎥ . . ⎥ ⎝⎣ .. ⎦⎠ ⎣ .. . . ⎦ ∗ 0 ... 0 0 0 ... 0 Because ξ preserves rank one idempotents and adjacency, each subset P L( t y) ⊂ Pn (D) is mapped either into some P L( t w) or some P R(x), and the same is true for each subset P R(z) ⊂ Pn (D). By the last two equations the ξ-image of the set of all rank one idempotents is not a subset of some P R(x). Thus, we can now apply the induction hypothesis on the map ξ. Denote ⎡ 1 ⎢0 ⎢ ⎢0 ⎢ ⎢ ..

This completes the proof. 15. Let E ⊂ D be two division rings, k and n positive integers, 2 ≤ k ≤ n, and A = [aij ] ∈ Mn (E) a matrix such that rank A = k and aij = 0 whenever j > k (the matrix A has nonzero entries only in the first k columns). Let X Y ∈ Mn×2n (D) be a matrix of rank n with X and Y both n × n matrices. Assume that for every B = [bij ] ∈ Mn (E) satisfying • bij = 0 whenever j > k, • there exists an integer r, 1 ≤ r ≤ k, such that bir = 0, i = 1, . . , n (that is, one of the first k columns of B is zero), and • A and B are adjacent, the row spaces of matrices X Y and I B are adjacent.

Clearly, L( t x) is an adjacent subset of Mm×n (D) ∪ {0}. Moreover, if T ⊂ Mm×n (D) is an adjacent set, then φ(T ) is an adjacent set as well. 1 (D) ∪ {0}. It follows that φ(L( t x)) is an adjacent subset of Mp×q 1 (D) ∪ {0} Hence, all we need to show is that for every adjacent subset S ⊂ Mp×q t t p t there exists y ∈ D such that S ⊂ L( y), or there exists w ∈ Dq such that 1 (D) ∪ {0} be an adjacent subset. Assume that S ⊂ R(w). So, let S ⊂ Mp×q t t S ⊂ L( y) for every nonzero y ∈ t Dp . Then we can find A = t ab and B = t cd in S with t a and t c being linearly independent.

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Algebraic geometry 01 Algebraic curves, algebraic manifolds and schemes by I. R. Shafarevich (editor), V.I. Danilov, V.V. Shokurov

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