Unformatted text preview: b ∈ H t,c . Show that [˜ a, ˜ b ] ∈ tH t,c and that the element 1 t [˜ a, ˜ b ] modulo t depends only on a, b and lies in Z c . Let { a, b } be that element. Show that {· , ·} is the Poisson bracket. (2) Show that { Z 6 i c , Z 6 j c } ⊂ Z i + j2 c . Show that the induced bracket on gr Z c = S ( V ) Γ is a nonzero multiple of the standard bracket. Can you identify the scalar factor? Problem 9.4. Show that the scheme C p is irreducible and normal (and, well, CohenMacaulay and Gorenstein, if you know what these words mean). Problem 9.5. Show that if C p is smooth, then H p e is a locally free H pmodule. Problem 9.6. Let A be a ﬁltered algebra. Show that if gr A has ﬁnite global dimension, then A does. Problem 9.7. Prove that if C p is smooth, then p is spherical (we deal here with p ∈ P ). 1...
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 Fall '12
 IvanLosev
 Math, Algebra, Vector Space, Hamiltonian mechanics

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