Eilenberg swindle

Summary

The Eilenberg swindle is an argument with infinite direct sums that cancels an object by regrouping an infinite sum. Its basic form shows that every projective module $P$ becomes free after adding a large enough free module: $P \oplus F \cong F$ for some free $F$. It explains why the Grothendieck group is built only from finitely generated modules: with infinite sums allowed, every class would be zero. The same trick, used by Mazur in topology, shows that no nontrivial knot has an inverse under connected sum.