K-theory with finite coefficients

Summary

K-theory with coefficients $\Z/\ell$ is obtained by dividing the K-theory spectrum by the integer $\ell$, in the homotopical sense of a cofibre. Its groups $K_n(R; \Z/\ell)$ sit in short exact sequences with $K_n(R)/\ell$ and the $\ell$-torsion of $K_{n-1}(R)$. Mod $\ell$ K-groups are often much easier to compute than integral ones. The rigidity theorems of Suslin and Gabber, and the comparison with étale cohomology, are statements about them.