A generalization of the atiyah-segal completion theorem

Derived Completions in Stable Homotopy theory

“…In order to prove this, it will suffice to show that the pro-A-module {T i A (M/I k M; B)} k≥0 is isomorphic, as pro-abelian groups, to the constant pro-abelian group with value T i A (M; B). By 4.2, this means that it will suffice to show that the pro-abelian group Recall (see [1]) that the category of pro-A-modules is itself an abelian category, and so the notion of exact sequence has meaning. The construction M → I M gives a functor from the category of finitely generated A-modules to the category of pro-A-modules, which is exact in the sense that it carries exact sequences to exact sequences.…”

Equivariant Stable Homotopy Theory

“…A geodesic route from Bott periodicity to the conclusion, basically a cohomological precursor of the homological argument sketched in the previous section, is given in [2]. That paper also gives the generalization of the result to arbitrary families of subgroups in G.…”

“…Therefore the techniques of the previous section do not apply in general. The arguments in [8] and [2] prove the isomorphism of Theorem 8.1 directly in cohomology. The trick that recovers enough exactness to make this work is to study pro-group valued cohomology theories.…”

The Atiyah–Segal completion theorem in twisted K–theory

“…Our argument for proving Theorem 8 is closely based on the one Adams, Haeberly, Jackowski and May present for proving a generalization of the Atiyah-Segal completion theorem in the untwisted case [1]. Their argument in turn builds on ideas due to Carlsson [5].…”

“…of G by the circle group T. For such twistings, twisted G-equivariant K-groups correspond to certain direct summands of untwisted G -equivariant K-groups, and the Adams-Haeberly-Jackowski-May argument contained in [1] goes through with these summands to prove the theorem in this case. It follows that the theorem holds when X is a point, and the general theorem then follows by a Mayer-Vietoris argument.…”