K1 of Rings
Most courses in linear algebra begin with a discussion of vector spaces and dimension, and then go on to a study of automorphisms of vector spaces, i.e., linear transformations and their invariants (determinants, canonical forms, and so on). The usual development of K-theory for rings follows the same pattern. One begins by studying projective modules and their stable classification via K0, and then goes on to the study of the stable classification of automorphisms of free and projective modules, in other words, to invariants of (invertible) matrices, which are given by the functor K1.
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