Witt vectors with coefficients and characteristic polynomials over non-commutative rings. Issue 2 (26th February 2022)

• Record Type: Journal Article
• Authors: Dotto, Emanuele; Krause, Achim; Nikolaus, Thomas; Patchkoria, Irakli
• Is Part Of: Compositio mathematica. Volume 158:Issue 2(2022)
• Page Start: 366
• Page End: 408
• Publication Date: 2022-02-26
• Subjects: Witt vectors -- characteristic polynomial -- trace; 13F35 -- 19D55 -- 16E20 -- 15A15; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X22007254
• Languages: English
• ISSNs: 0010-437X

Abstract

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 $\chi _f \in W(R)$ . 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 $W(R)$ .