Abstract. We introduce an equivariant algebraic kk-theory for G-algebras and G-graded algebras. We study some adjointness theorems related with crossed product, trivial action, induction and restriction. In particular we obtain an algebraic version of the Green-Julg Theorem which gives us a computational tool.

Abstract. Let G be a group and let E be a functor from small Z-linear categories to spectra. Also let A be a ring with a G-action. Under mild conditions on E and A one can define an equivariant homology theory of Gsimplicial sets H G

Let G be an algebraic quantum group. We introduce an equivariant algebraic kk-theory for G-module algebras. We study an adjointness theorem related with smash product and trivial action. We also discuss a duality property.

We calculate the endomorphism dga of Franke's exotic algebraic model for the K-local stable homotopy category at odd primes. We unravel its original abstract structure to give explicit generators, differentials and products. 2The authors thank the organizers of the WIT II conference, the Banff International Research Station for hosting us, and the AWM for providing travel support. The first author is grateful to the Universidad de la República -CSIC for its support and for travel funds. The second author would like to thank the University of Kent Faculty of Sciences Research Fund as well as SMSAS for travel funds, and would furthermore like to thank Andrew Baker, David Barnes and Sarah Whitehouse for interesting discussions.

Let [Formula: see text] be a strong [Formula: see text]-coherent ring such that each finitely [Formula: see text]-presented [Formula: see text]-module has finite projective dimension. We consider [Formula: see text] the full subcategory of [Formula: see text]-Mod of finitely [Formula: see text]-presented modules. We prove that [Formula: see text] is an exact category, [Formula: see text] for every [Formula: see text] and we obtain an expression of [Formula: see text].