MATH 7364 Lecture 19 problems

# MATH 7364 Lecture 19 problems - Problem 19.3 A...

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PROBLEMS ON SYMPLECTIC REFLECTION ALGEBRAS 19. KZ functor I Problem 19.1. Let M 1 , M 2 be D X -modules that are coherent sheaves. Show that dim Hom D X ( M 1 , M 2 ) < . Exercise 19.1. 1 Let M be an H c -module with locally nilpotent action of h . Show that M is finitely generated iff the action of h on M is locally finite and all generalized eigen-subspaces are finite dimensional. Problem 19.2. Show that Ext i (∆( E ) , ( E )) = C if E = E , i = 0 , and 0 else. Moreover, show that if Ext 1 (∆( E ) , M ) = 0 for all E , then M is -filtered, i.e., admits a filtration with successive quotients ( E ) . Problem 19.3.
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Unformatted text preview: Problem 19.3. A ∆-ﬁltered object M is projective iﬀ Ext 1 ( M, ∆( E )) = 0 for all E . Problem 19.4. (1) Show that the double centralizer property is equivalent to Ext 1 ( M,P ) = for any projective P and M ∈ O tor . (2) Use the naive duality to show that π is fully faithful on injectives. (3) Show that π has left adjoint π ! and that π ◦ π ! is the identity on the image of π . 1 This exercise and the next problem also appeared last time. 1...
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