Abstract
An exact category is an additive category together with a notion of short exact sequence which satisfies some familiar properties. In [3], Quillen defines the K-groups of an exact category n/r. He constructs from M 9 certain category QM. Its classifying space SOM is a CW-complex whose homotopy group rri+J3(&M is, by definition, the group KJ4. If R is a ring (with l), then KiZ? is defined to beKJ’(R), where P(R) is the category of finitely-generated projective left R-modules. If M’+ M is an exact functor from an exact category _M’ to an exact category M, then knowledge about the homotopy fiber F of the map BQM’-, SQM may yield useful information about the K-groups of n/r and M’, in light of the long exact sequence of homotopy groups:
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