Abstract
K-theory as an independent discipline is a fairly new subject, only about 35 years old. (See [Bak] for a brief history, including an explanation of the choice of the letter K to stand for the German word Klasse.) However, special cases of K-groups occur in almost all areas of mathematics, and particular examples of what we now call K0 were among the earliest studied examples of abelian groups. More sophisticated examples of the idea of the definition of K0 underlie the Euler-Poincaré characteristic in topology and the Riemann-Roch theorem in algebraic geometry. (The latter, which motivated Grothendieck’s first work on K-theory, will be briefly described below in §3.1.) The Euler characteristic of a space Xis the alternating sum of the Betti numbers; in other words, the alternating sum of the dimensions of certain vector spaces or free R-modules H i (X;R) (the homology groups with coefficients in a ring R). Similarly, when expressed in modern language, the Riemann-Roch theorem gives a formula for the difference of the dimensions of two vector spaces (cohomology spaces) attached to an algebraic line bundle over a non-singular projective curve. Thus both involve a formal difference of two free modules (over a ring R which can be taken to be ℂ). The group K1(R) makes it possible to define a similar formal difference of two finitely generated projective modules over any ring R
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© 1994 Springer-Verlag New York, Inc.
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Rosenberg, J. (1994). K0 of Rings. In: Algebraic K-Theory and Its Applications. Graduate Texts in Mathematics, vol 147. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4314-4_1
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DOI: https://doi.org/10.1007/978-1-4612-4314-4_1
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