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Homotopy algebraic K-theory. (English) Zbl 0669.18007
Algebraic \(K\)-theory and algebraic number theory, Proc. Semin., Honolulu/Hawaii 1987, Contemp. Math. 83, 461-488 (1989).
[For the entire collection see Zbl 0655.00010.]
The author constructs a version \(KH_*\) of algebraic K-theory of associative rings and schemes which is homotopy invariant, i.e. satisfies \(KH_*(A)=KH_*(A[t])\). For regular rings or schemes \(KH_*\)-theory is the same as \(K_*\)-theory, and under \(K_ 0\)-regularity \(KH_*\)-theory coincides with the \(KV_*\)-theory of Karoubi-Villamayor. \(KH_*\)-theory has many nice properies, including the validity of the Fundamental Theorem, excision and the existence of localization sequences - even without the usual assumptions on non-zero divisors. As a major application (due to Bob Thomason) the Brown-Gersten spectral sequence for quasi-projective spaces is deduced from the Mayer-Vietoris properties in \(KH_*\)-theory.
Reviewer: M.Kolster

18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
16E20 Grothendieck groups, \(K\)-theory, etc.