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Is there a good reference that explains mod p K-theory and p-adic or p-complete K- theory? All I know about K-theory is the topological K-theory of "vector bundles and k-theory" in Switzer's book (and similar expositions found in a book by Hatcher, and a chapter in May's Concise course). There seems to me be a jump from that type of material to understanding statements like the " p-completion of $K(\mathbb{Z}/p)$ is $H\mathbb{Z}_p$", a result due to Quillen, according to this question. I looked at the titles of Quillen's papers and am not sure in which paper it appears. Also, perhaps there are now other good expositions introducing and computing basic things about mod p or p-complete k-theory.

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You have learned about topological K-theory (of topological spaces). The Quillen result from

"On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field", Daniel Quillen, Ann. Math., Vol. 96, No. 3 (Nov., 1972), pp. 552-586,

is about algebraic K-theory (of rings).

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