Rosenberg's proof of Bass-Heller-Swan - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T10:46:39Z http://mathoverflow.net/feeds/question/98104 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98104/rosenbergs-proof-of-bass-heller-swan Rosenberg's proof of Bass-Heller-Swan Martin Brandenburg 2012-05-27T12:14:04Z 2012-05-29T11:29:28Z <p>I'm reading the proof the Bass-Heller-Swan Theorem in Rosenberg's book <em>Algebraic K-Theory and Applications</em> (Theorem 3.2.22), which asserts $$K_1(R[t,t^{-1}]) \cong K_0(R) \oplus K_1(R) \oplus NK^+_1(R) \oplus NK_1^-(R).$$ First it is shown that there are homomorphisms</p> <p>$K_0(R) \oplus K_1(R) \xrightarrow{\Phi} K_1(R[t,t^{-1}]) \xrightarrow{\Psi} K_0(R) \oplus K_1(R)$</p> <p>such that $\Psi \circ \Phi = \mathrm{id}$. If necessary, I will include their definitions here. There is a canonical map $$\alpha : NK^+_1(R) \oplus NK_1^-(R) \to \mathrm{coker}(\Phi)$$ mapping $([1+Nt],[1+Mt^{-1}]) \mapsto [(1 + Nt)(1+Mt^{-1})]$ for nilpotent $N,M$, and it is left to prove that this is an isomorphism. As for this step, Rosenberg isn't really precise.</p> <p>Higman's trick shows that every element in the cokernel has the form $[1+(P+N)(t-1)]$ for commuting matrices $N,P$, where $N$ is nilpotent and $P$ is idempotent. Now Rosenberg seems to allude that $([1+PN t],[1+(1-P)N t^{-1}])$ is a preimage. But this is wrong, I think that $([1+(1-PN)^{-1} PN t],[1-(1+(1-P)N)^{-1} (1-P)N t^{-1}])$ is a preimage. So $\alpha$ is surjective. Anyway:</p> <p><strong>Why is $\alpha$ injective?</strong></p> <p>It is tempting to define a map in the other direction by the rule above, but then we have to show that this is well-defined, which is basically the same as to prove that $\alpha$ is injective. I've tried to use the isomorphism $\mathrm{coker}(\Phi) \cong \mathrm{ker}(\Psi)$ induced by $\mathrm{id} - \Phi \Psi$, according to which we "only" have to prove that a certain homomorphism $NK^+_1(R) \oplus NK_1^-(R) \to K_1(R[t,t^{-1}])$ is injective, but this didn't simplify the problem either.</p> <p>I've already looked at the original sources: Bass-Heller-Swan (<em>The Whitehead group of a polynomial extension</em>) prove their Theorem only for regular rings. The first appearance of the Theorem with Nil terms seems to be in Bass' <em>Algebraic K-Theory</em>. But his treatment is rather abstract and more machinery is developed in order to optain the Theorem. Instead, I would like to know if it is possible to fix / complete Rosenberg's proof.</p>