Abstract
The Grothendieck group K 0(A) of a ring A was introduced in the sixties by Grothendieck in order to give a nice formulation of the Riemann-Roch theorem. Then it was recognized that the Grothendieck group is closely related to the abelianization K 1(A) of the general linear group, which had been studied earlier (1949) by J.H.C. Whitehead in his work on simple homotopy. The next step was the discovery of the K 2-group by Milnor in his attempt to understand the Steinberg symbols in arithmetic. At that point these three groups were expected to be part of a family of algebraic K-functors K n defined for all n ≥ 0. After several attempts by different people, Quillen came with a simple construction, the so-called plus-construction, which gives rise to higher algebraic K-theory.
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Loday, JL. (1998). Algebraic K-Theory. In: Cyclic Homology. Grundlehren der mathematischen Wissenschaften, vol 301. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-11389-9_11
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