nLab Eilenberg swindle

Idea

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 $K_0$) of many abelian categories is trivial.

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

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

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

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

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 $K_0$ of a ring.

References

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

• Barry C. Mazur, On embeddings of spheres, Acta Mathematica 105 1–2 (1961) 1–17 [doi:10.1007/BF02559532]

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

• Hyman Bass, Big projective modules are free, Illinois Journal of Mathematics 7 (1963) 24–31 [doi:10.1215/ijm/1255637479]