nLab Eilenberg swindle

The Eilenberg swindle gives a means to prove that the Grothendieck group of many abelian categories is zero; it is the reason that, for instance, one has to restrict oneself to finitely generated projective modules in defining K 0K_0 of a ring.

The idea is essentially the Hilbert hotel paradox. Given an abelian category AA with countable direct sums, one can show that K(A)=0K(A)=0. If XAX \in A, then X i=1 X i=1 XX \oplus \bigoplus_{i=1}^{\infty} X \simeq \bigoplus_{i=1}^{\infty} X , which implies [X]=0[X]=0 in K(A)K(A).

Last revised on January 25, 2017 at 02:33:49. See the history of this page for a list of all contributions to it.