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Math. 9 (1961) 23-64 [2] M. " Topology 1 (1962) 245-248 [3] M. " J. Diff. Geom. 3 (1969) 1-18 [4] M. L. -B. " American J. Math. " Bull. London Math. Soc. " Trans. American Math. Soc. Algebra 192 (1997) 678-700. " J. Pure and Applied Algebra 122 (1997) 41-53. " Inventiones Math. " (Submitted for publication) 126pp (With an appendix by A. Math. Helv. " Ann. of Math. Math. Helv. " (In preparation) 21 pp Local Cohomologyin Equivariant Topology 37 [21] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J.

J. American Math. Soc. Joachim (private communication) [45] M. " IHES Pub. Math. May and M. " Lecture notes in maths. Quillen "The spectrum of an equivariant cohomology ring I,II" Ann. of Math. " Math. Ann. " Bull. American Math. " J. Alg. " Math. Comp. Math. Math. P. Stanley "Hilbert functions of graded algebras. " Adv. in Maths. " Progress in Math. " QJM24 (1973) 499-526 DEPARTMENT OF PURE MATHEMATICS,HICKS BUILDING, SHEFFIELD $3 7RH. UK. E-mail address: j. greenleesOsheffield, ac. uk LECTURES ON LOCAL COHOMOLOGY AND DUALITY JosephLipman PurdueUniversity, WestLafayette, Indiana ABSTRACT.

Now the mapping cone of B is F jR A ~G where ~G is the mapping cone of EG+ ~ S °. Because (/~G) ~ = S°, there is unique map/~G ~4 S~y which is the identity in G-fixed points, where V is the reduced regular 32 Greenlees representation. It suffices to show that P jR A S~V is contractible. if we define Q by the co fibre sequence ~,G Indeed, ~ S ~V ~ Q, we see that Q is built from cells G/H+for proper subgroups H. It follows from the inductive hypothesis F jR A Q -~ ,. Since FjR A S~V is obviously H-contractible for all proper subgroups H it suffices to show its G-homotopy is zero.

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