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The K-theory of semilinear endomorphisms. (English) Zbl 0656.16011
Given an automorphism $$\phi$$ of a ring k, the twisted polynomial ring over k in a variable T has multiplication given by $$aT=T\phi (a)$$. Let $$R^{\pm}$$ be the Laurent polynomial ring and let $$R^-$$ be the skew polynomial ring in $$T^{-1}$$. A module M over the “twisted” projective line X is given as a triple $$(M^+,M^-,\vartheta_ M)$$, with $$\vartheta_ M: M^+R^{\pm}\cong M^-R^{\pm}$$ an isomorphism. The “vector bundles” over X are those P with $$P^+$$ and $$P^-$$ both projective; these form a category $${\mathcal P}_ X.$$
Define the category Nil($$\phi)$$ to be that of pairs (V,f) with V a projective k-module and f a $$\phi$$-semilinear nilpotent endomorphism. Standard splittings give $$K_ i\underline{Nil}(\phi)=K_ ik\oplus Nil_ i(\phi)$$, $$K_ iR^+=K_ ik\oplus NK_ i(\phi)$$, defining the groups $$Nil_ i\phi$$ and $$NK_ i(\phi)$$. The main results are:
(a) $$K_ iX\cong K_ ik\oplus K_ ik$$ $$(K_ iX=K_ i{\mathcal P}_ X)$$, (b) $$NK_ i(\phi^{-1})\cong Nil_{i-1}(\phi)$$, (c) $$K_ iR^{\pm}\cong F_{i-1}(\phi)\oplus Nil_{i-1}(\phi)\oplus Nil_{i- 1}(\phi^{-1})$$, (d) There is an exact sequence $$...\to F_ i(\phi)\to K_ i(k)\to^{1-\phi_*}K_ i(k)\to$$. When k is a division ring, there is a localization sequence $$\to K_ iAut \phi \to K_ iX\to K_ iB\to$$, where B is a pull-back of localizations of $$R^+$$ and $$R^-$$ and Aut $$\phi$$ is the category of $$\phi$$-semilinear automorphisms on k- spaces.
Reviewer: M.E.Keating

##### MSC:
 16E20 Grothendieck groups, $$K$$-theory, etc. 16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras) 16W20 Automorphisms and endomorphisms 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects)
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