Y would mean the existence of a functor J', right adjoint to the functor f*, that gives an isomorphism Hom(f*F, G) :::o Hom(F, J'G) for arbitrary sheaves Fon X and Gon Y.

T;:'" , where all mi < 0. The right-hand side is graded by the usual degree. This can be expressed in the following invariant form: H 0 (1P'n,O(m)) = Symm(V*) for m ~ 0, Hn(IP'n, 0( -n- 1- k)) ~ Symk(V) for k ~ 0; The remaining Hq (IP'n, 0( m)) are equal to zero. We collect the data in a table. q S2 V V K 0 ... 0 0 0 0 0 0 0 V* S 2 V* n 0 0 0 0 0 0 0 0 0 0 0 0 ... 1 K 0 0 -n -2 -n-1 -n -1 0 1 m The symmetry of the table hints for a possibility of the existence of a duality. In Sect. 5, we will discuss it in a more general setting.

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