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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

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)
Full Text: DOI
[1] Bass, H, Algebraic K-theory, (1968), Benjamin New York/Amsterdam · Zbl 0174.30302
[2] Farrell, F; Hsiang, W, A formula for K1rx[T], (), 192-218
[3] Grayson, D, K-theory and localization of noncommutative rings, J. pure appl. algebra, 18, 125-127, (1980) · Zbl 0441.16020
[4] Grayson, D, Higher algebraic K-theory, II [after D. quillen], ()
[5] Grayson, D, The K-theory of endomorphisms, J. algebra, 48, 439-446, (1977) · Zbl 0413.18010
[6] Hiller, H, Λ-rings and algebraic K-theory, J. pure appl. algebra, 20, 241-266, (1981) · Zbl 0471.18007
[7] Karoubi, M, Localisation de formes quadratiques, I, Ann. sci. école norm. sup., 7, 359-404, (1974) · Zbl 0325.18011
[8] Lang, S, Algebraic groups over finite fields, Amer. J. math., 78, (1956) · Zbl 0073.37901
[9] Lazard, Autour de la platitude, Bull. soc. math. France, 97, 81-128, (1969) · Zbl 0174.33301
[10] Quillen, D, Higher algebraic K-theory, I, () · Zbl 0292.18004
[11] Quillen, D, On the cohomology and K-theory of the general linear groups over a finite field, Ann. of math., 96, 552-586, (1972) · Zbl 0249.18022
[12] Ranicki, A, Exact sequences in the algebraic theory of surgery, (1981), Princeton Univ. Press Princeton, NJ · Zbl 0471.57012
[13] Siebenmann, A total Whitehead torsion obstruction to fibering over the circle, Comm. math. helv., 45, 1-48, (1970) · Zbl 0215.24603
[14] Stenström, B, Rings of quotients, (1975), Springer-Verlag Heidelberg/Berlin
[15] Suslin, A, On the K-theory of algebraically closed fields, Invent. math., 73, 241-245, (1983) · Zbl 0514.18008
[16] Waldhausen, F, Algebraic K-theory of generalized free products, Ann. of math., 108, 135-256, (1978) · Zbl 0407.18009
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