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

Compositio mathematica

article

Irakli Patchkoria¹  Emanuele Dotto²  Achim Krause³  Thomas Nikolaus³ 

[1] Department of Mathematics, University of Aberdeen;Mathematics Institute, University of Warwick;Mathematisches Institut, Universität Münster

关键词: Witt vectors;    characteristic polynomial;    trace;    13F35;    19D55;    16E20;    15A15;   

DOI  :  10.1112/S0010437X22007254

学科分类：数学（综合）

来源: Cambridge University Press

【 摘 要 】

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 $W(R)$ .

【 授权许可】

CC BY