nLab Eilenberg swindle




In homological algebra, what came to be known [Bass (1963)] as the Eilenberg swindle (in honor of Samuel Eilenberg) is an argument (originally due to Mazur (1961)) proving that the Grothendieck group (the 0th algebraic K-theory group K 0K_0) of many abelian categories is trivial.

The content of the “swindle” is essentially the following Hilbert hotel-type argument:

In an abelian category AA with countable direct sums, we have for any object XAX \in A an isomorphism

X i=1 X i=1 X, X \oplus \bigoplus_{i=1}^{\infty} X \,\simeq\, \bigoplus_{i=1}^{\infty} X \,,

which implies [X]=0[X]=0 in K 0(A)K_0(A).

This is the reason that, for instance, one has to restrict oneself to categories of finitely generated (projective) modules (which lack infinite direct sums) in defining (nontrivial) algebraic K-theory groups K 0K_0 of a ring.


The earliest known appearance of Eilenberg’s swindle in writing is in Lemma 3 of

The earliest attribution to Samuel Eilenberg as well as the use of the term “swindle” is in §1 of

Last revised on April 8, 2023 at 19:15:50. See the history of this page for a list of all contributions to it.