## Projections, cycles, and algebraic $$K$$-theory.(English)Zbl 0358.18012

A theorem of D. Quillen [Theorem 5.11, Sect. 7, Higher Algebraic K-theory I, Lecture Notes Math. 341, 85–147 (1973; Zbl 0292.18004)] known as Gersten’s conjecture, the main step in the proof of Bloch’s formula for the group of cycles modulo rational equivalence on a non-singular variety, is proved in a different way. The difference consists of the use of linear projections in place of arbitrary ones. When the ground field is infinite a suitably chosen linear projection serves to prove the theorem. A trick of J. Roberts is used to extend the result to cover the case of a finite ground field.
Reviewer: Daniel R. Grayson

### MSC:

 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects)

Zbl 0292.18004
Full Text:

### References:

 [1] Grothendieck,A., Dieudonné,J.A.: Eléments de géométrie algébrique. EGA IV. Publications Mathématiques, IHES No. 20, 24, 28, 32, Paris, 1964, 65, 66, 67 [2] Grothendieck, A.: Revetements étales et groupe fondamental. SGA 1. Lecture Notes in Mathematics 224. Berlin, Heidelberg, New York: Springer 1971 · Zbl 0234.14002 [3] Quillen, D.: Higher algebraicK-theory. II. In: AlgebraicK-theory I. Lecture Notes in Mathematics 341. Berlin, Heidelberg, New York: Springer 1973 · Zbl 0292.18004 [4] Roberts, J.: Chow’s moving lemma. In: Algebraic geometry, Oslo 1970, ed. by F. Oort. Groningen: Wolters-Noordhoff 1972 [5] Shafarevich, I.: Basic algebraic geometry. Berlin, Heidelberg, New York: Springer 1974 · Zbl 0284.14001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.