Download e-book for kindle: Algebraic Geometry Iv Linear Algebraic Groups Invariant by A.N. Parshin

By A.N. Parshin

Two contributions on heavily comparable matters: the idea of linear algebraic teams and invariant thought, through recognized specialists within the fields. The publication may be very important as a reference and examine consultant to graduate scholars and researchers in arithmetic and theoretical physics.

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6, N1 ∼ = N2 . 26]). Let Q1 , Q2 be some smooth projective quadrics, and α ∈ Hom(M (Q1 )(d1 )[2d1 ], M (Q2)(d2 )[2d2 ]), β ∈ Hom(M (Q2 )(d2 )[2d2 ], M (Q1)(d1 )[2d1 ]) be morphisms such that the composition degQ1 ◦β ◦ α : CHr (M (Q1 )(d1 )[2d1]|k ) → Z/2 is nonzero for some r. Then there exist indecomposable direct summands N1 of M (Q1 )(d1 )[2d1 ] and N2 of M (Q2 )(d2 )[2d2 ] such that N1 N2 , and Z(r)[2r] is a direct summand in Ni |k . See Sect. 4 for a proof. Here are two important cases of such a situation.

5 Proofs We start with some preliminary results. 1. Let N be a direct summand in M (Q) and ψ ∈ Hom(N, N ). (1) If ψ|k = 0, then ψn = 0 for some n. 44 Alexander Vishik (2) If ψ|k is a projector, then ψn is a projector for some n. (3) If ψ|k is an isomorphism, then ψ is an isomorphism. ψ0 , where ρ = 0 0ρ in cases (1) and (2), and ρ = idM in case (3). 1(2). Proof. Let M (Q) = N ⊕ M . 2. Let L and N be direct summands in M (Q) such that p L | k ◦ pN | k = pN | k ◦ pL | k = pL | k . ˜ in N such that L ˜ is isomorphic to L Then there exists a direct summand L and pL |k = pL˜ |k .

5. Let N be an indecomposable direct summand in M (Q), and ψ ∈ End N be an arbitrary morphism. Then either degN ◦ψ = degN , or degN ◦ψ = 0. In particular, to show that M (Q) is decomposable it is sufficient to exhibit a morphism ψ ∈ End M (Q) such that degQ = degQ ◦ψ = 0. 5 is in Sect. 2. Examples. e. the projective quadric Q has a rational point z. Then the cycle Q × z ⊂ Q × Q defines a morphism ρ ∈ End M (Q) such that ρ(0) = 1 and ρ(r) = 0, for all r = 0. So, degQ ◦ρ coincides with degQ on the group CH0 (Q|k ) and is zero on the other Chow groups.

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Algebraic Geometry Iv Linear Algebraic Groups Invariant Theory by A.N. Parshin


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