Part I. The G-equivariant analogues of the stable homotopy groups of spheres are the equivariant homotopy groups of stable homotopy representations. Let G be a compact Lie group and A(G) the Burnside ring of G. Let Z be a stable homotopy representation with non-negative dimension function. We prove, with one extra hypothesis on Z, that pG0 (Z), as an A(G)-module, is isomorphic to a quotient of A(G) tensored with an invertible A(G)-module. This is the equivariant analogue of the non-positive stems.; Part II. We investigate the Picard group Pic( DM ) of isomorphism classes of invertible objects in the derived category of O -modules for a commutative unital ringed Grothendieck topos ( E,O ) with enough points. Let C(pt( E )) denote the additive group of continuous functions from the space of isomorphism classes of points of E to the integers. When the ring O p has connected prime ideal spectrum for all points p of E we show that Pic( DM ) is naturally isomorphic to the Cartesian product of C(pt( E )) with the Picard group of O -modules Also, for a commutative unital ring R, the group Pic( D R) is isomorphic to the Cartesian product of Pic(R) and the additive group of continuous functions from spec R to the integers.
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