Witt vectors with coefficients and characteristic polynomials over non-commutative rings. Issue 2 (26th February 2022) - Dotto, Emanuele; Krause, Achim; Nikolaus, Thomas; Patchkoria, Irakli
Abstract: For 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 $χ _f \in W(R)$ . This element is a non-commutative analogue of the classical characteristic polynomial and we show that it has similar properties. The assignment $f \mapsto χ _f$ induces an isomorphism between a suitable completion of cyclic $K$ -theory $K_0^{cyc}(R)$ and $W(R)$ .