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

Emanuele Dotto, Achim Krause, Thomas Nikolaus, Irakli Patchkoria* (Corresponding Author)

*Corresponding author for this work

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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 χf ∈ 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 ↦ χf induces an isomorphism between a suitable completion of cyclic K-theory K cyc0 (R) and W(R).
Original languageEnglish
Pages (from-to)366-408
Number of pages43
JournalCompositio Mathematica
Issue number2
Early online date26 Apr 2022
Publication statusPublished - 26 Apr 2022

Bibliographical note

Funding Information:
The first and the fourth authors were supported by the German Research Foundation Schwerpunktprogramm 1786 and by the Hausdorff Center for Mathematics at the University of Bonn. The second and third authors were funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC 2044 390685587, Mathematics Münster: Dynamics–Geometry–Structure.


  • Witt vectors
  • characteristic polynomial
  • trace


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