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 $K_0$ of a ring.

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

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