Witt vectors with coefficients and characteristic polynomials over non-commutative rings

Abstract

AbstractFor a not-necessarily commutative ring $R$ we define an abelian group $W(R;M)$ of Witt vectors with coefficients in an $R$-bimodule $M$. These groups generalize the usual big Witt vectors of commutative rings and we prove that they have analogous formal properties and structure. One main result is that $W(R) := W(R;R)$ is Morita invariant in $R$. For an $R$-linear endomorphism $f$ of a finitely generated projective $R$-module we define a characteristic element $\chi _f \in W(R)$. This element is a non-commutative analogue of the classical characteristic polynomial and we show that it has similar properties. The assignment $f \mapsto \chi _f$ induces an isomorphism between a suitable completion of cyclic $K$-theory $K_0^{\mathrm {cyc}}(R)$ and $W(R)$.