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

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 thatFor an R-linear endomorphism f of a finitely generated projective R-module we define a characteristic element χ f ∈ 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 ↦ χ f induces an isomorphism between a suitable completion of cyclic K-theory K cyc 0 (R) and W (R).Finally, Kaledin defines in [Kal18a] (see also [Kal18b]) abelian groups W n (V ) of 'polynomial Witt vectors' for a vector space V over a perfect field k of characteristic p. We will also show in the forthcoming paper [DKNP] that his group W n (V ) is isomorphic to our group W p,n (k; V ) of truncated, p-typical Witt vectors with coefficients.

$K$-theory of endomorphisms, the $\mathit{TR}$-trace, and zeta functions

“…takes each endomorphism [ f ] ∈ K 0 End(A) to its characteristic polynomial (Theorem 9.9). Here W(A) ∼ = (1 + tA[[t]]) × is the ring of big Witt vectors of A, and the isomorphism π 0 TR(A) ∼ = W(A) is a result of Hesselholt and Madsen [HM97], see also [Hes97,DKNP20]. As a result, the TR-trace of this paper is a generalization of the characteristic polynomial map K 0 End(A) → (1 + tA[[t]]) × studied by Almkvist and others [Alm74].…”