Rotman J.J.'s Advanced algebra PDF

By Rotman J.J.

ISBN-10: 0821847414

ISBN-13: 9780821847411

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We consider the square of pointed sets p H 1 Rn , Aut(G) ⏐ ⏐ −−−∗−→ H 1 Rn , Out(G) ⏐ ⏐ H 1 Rn , Aut(G) −−−−→ H 1 Rn , Out(G) . 1]. By construction, [p∗ E] comes from H 1 Rn , Out(G) , hence [E] comes from a class [F] ∈ H 1 Rn , Aut(G) . Our assumption is that the Rn –group G = E G = F G contains a maximal torus, so Gad = F Gad contains a maximal torus and F Aut(G) contains a maximal torus. In other words, F is a toral Rn –torsor under Aut(G) . 1 1 Rn , Aut(G) = Hloop Rn , Aut(G) , it follows that that From the equality Htoral F is a loop torsor under Aut(G) .

12. (g )θ is reductive, and therefore gθ = (g )θ × zθ is also reductive. Every Cartan subalgebra h of gθ is of the form h = h × zθ for some Cartan subalgebra h of (g )θ . Clearly zg (h) = zg (h ) × z. By [P3] theorem 9 the centralizer zg (h ) is a Cartan subalgebra of g , so the claim follows. We now return to the proof of the Theorem. Since H1 is normal in H we have ˜ algebra gθ . We an induced action (via φ) of H = H/H1 on the reductive k–Lie θ have induced group homomorphisms φ : H → Autk (g ) and ψ : H → Γ (this last since H1 ⊂ ker(ψ)).

T±1/m ˜ is a finite Galois ˜ = k[t R ] where m is a positive integers and k/k n 1 extension of fields containing all primitive m-th roots of unity of k. 3 that R 1 1 e ˜ where as follows: For e = (e1 , . . , en ) ∈ Zn we have e (λtjm ) = λξmj tjm for all λ ∈ k, − n n ˜ : Z → (Z/mZ) is the canonical map, while the Galois group Γ = Gal(k/k) ˜ ˜ through its action on k. 2). By assumption, we can assume that G is the twist of G0 = G0 ×k R by a loop cocycle ˜ ˜ → Aut(G0 )(k). u:Γ ˜ = (Z/mZ)n Γ → Γ is defined to be the natural The homomorphism ψ : Γ projection.

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Advanced algebra by Rotman J.J.


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