Operations in the K-theory of endomorphisms
type: Item
instance of: scholarly article
title: Operations in the K-theory of endomorphisms
zbMATH Open document ID: 0534.13010
DOI: 10.1016/0021-8693(83)90080-7
author: Michiel Hazewinkel
published in: Journal of Algebra
publication date: 1983-01-01T00:00:00Z
full work available at URL: https://ir.cwi.nl/pub/10210
review text: For a commutative ring with unity A, let \underbar{End} A be the category of all pairs (P,f), where P is finitely generated projective A-module and f an endomorphism of A. The K-group \(K_ 0(A)\) is a direct summand and ideal of \(K_ 0(\underline{End} A)\), and Almkvist showed that the quotient ring \(W_ 0(A)=K_ 0(\underline{End} A)/K_ 0(A)\) is a functorial subring of the ring of the big Witt vectors W(A). In this paper, I determine the ring of all continuous functorial operations on \(W_ 0(-)\), and the semiring of all operations (and all continuous operations) liftable to \underbar{End} (A. This solves some of the open problems listed by \textit{G. Almkvist} in J. Algebra 55, 308-340 (1978; Zbl 0414.18009). Let \({\mathbb{Z}}[X]={\mathbb{Z}}[X_ 1,X_ 2,X_ 3,...]\). Then the results involve the J-topology on \({\mathbb{Z}}[X]\) defined by the ideals \(J_ n\) generated by all the \((n+1)x(n+1)\) minors of the infinite matrix \((a_{ij})\) with \(a_{ij}=X_{i+j-2}\) (define \(X_ 0=1)\). A key result is that the \(J_ n\) are prime ideals.
Mathematics Subject Classification ID: 13D15
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 13K05
Mathematics Subject Classification ID: 14M12
Mathematics Subject Classification ID: 93B25
zbMATH DE Number: 3847538
zbMATH Keywords: K-theory of endomorphisms
zbMATH Keywords: determinantal varieties
zbMATH Keywords: ring of the big Witt vectors
MaRDI profile type: Publication
OpenAlex ID: W1970445796
cites work: K-theory of endomorphisms
cites work: The Grothendieck ring of the category of endomorphisms
cites work: The K-theory of endomorphisms
cites work: Q3901604
cites work: Q5727262
cites work: Algebraic Structure of Linear Dynamical Systems. III. Realization Theory Over a Commutative Ring
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description: scientific article; zbMATH DE number 3847538