Atiyah–Segal completion theorem
The Atiyah–Segal completion theorem is a theorem in mathematics about equivariant K-theory in homotopy theory. Let G be a compact Lie group and let X be a G-CW-complex. The theorem then states that the projection map
induces an isomorphism of prorings
Here, the induced map has as domain the completion of the G-equivariant K-theory of X with respect to I, where I denotes the augmentation ideal of the representation ring of G.
In the special case of X a point, the theorem specializes to give an isomorphism between the K-theory of the classifying space of G and the completion of the representation ring.
The theorem can be interpreted as giving a comparison between the geometrical process of completing a G-space by making the action free and the algebraic process of completing with respect to an ideal.
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