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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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