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

Emanuele Dotto; Achim  Krause; Thomas  Nikolaus; Irakli Patchkoria

doi:10.1112/S0010437X22007254

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

Mathematical Science

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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 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 language	English
Pages (from-to)	366-408
Number of pages	43
Journal	Compositio Mathematica
Volume	158
Issue number	2
Early online date	26 Apr 2022
DOIs	https://doi.org/10.1112/S0010437X22007254
Publication status	Published - 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.

Keywords

Witt vectors
characteristic polynomial
trace
TRACE

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Dotto_etal_CM_Witt_Vectors_With_VoR
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. Compositio Mathematica is © Foundation Compositio Mathematica.
Final published version, 1.21 MBLicence: CC BY

Cite this

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title = "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 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).",

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author = "Emanuele Dotto and Achim Krause and Thomas Nikolaus and Irakli Patchkoria",

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 M{\"u}nster: Dynamics–Geometry–Structure. ",

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AU - Dotto, Emanuele

AU - Krause, Achim

AU - Nikolaus, Thomas

AU - Patchkoria, Irakli

N1 - 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.

PY - 2022/4/26

Y1 - 2022/4/26

N2 - 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).

AB - 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).

KW - Witt vectors

KW - characteristic polynomial

KW - trace

KW - TRACE

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JF - Compositio Mathematica

IS - 2

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

Abstract

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Keywords

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