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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

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