Strong shift equivalence and algebraic K-theory

Mike Boyle, Scott Edward Schmieding

Research output: Contribution to journalArticle

Abstract

For a semiring R, the relations of shift equivalence over R (SE-R) and strong shift equivalence over R (SSE-R) are natural equivalence relations on square matrices over R, important for symbolic dynamics. When R is a ring, we prove that the refinement of SE-R by SSE-R, in the SE-R class of a matrix A, is classified by the quotient NK1(R)=E(A, R) of the algebraic K-theory group NK1(R) Here (E,A) R is a certain stabilizer group, which we prove must vanish if A is nilpotent or invertible. For this, we first show for any square matrix A over R that the refinement of its SE-R class into SSE-R classes corresponds precisely to the refinement of the GL (R[t]) equivalence class of I -tA into El (R[t]) equivalence classes. We then show this refinement is in bijective correspondence with NK1(R)/E(A, R). For a general ring R and A invertible, the proof that E(A, R) is trivial rests on a theorem of Neeman and Ranicki on the K-theory of noncommutative localizations. For R commutative, we show AE(A, R) = NSK1(R) the proof rests on Nenashev?s presentation of K1 of an exact category.

Original languageEnglish (US)
Pages (from-to)63-104
Number of pages42
JournalJournal fur die Reine und Angewandte Mathematik
Volume2019
Issue number752
DOIs
StatePublished - Jul 1 2019

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Algebraic K-theory
Equivalence classes
Equivalence
Refinement
Group theory
Square matrix
Equivalence class
Invertible
Ring
Symbolic Dynamics
Semiring
Bijective
K-theory
Equivalence relation
Vanish
Quotient
Trivial
Correspondence
Theorem
Class

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Boyle, Mike ; Schmieding, Scott Edward. / Strong shift equivalence and algebraic K-theory. In: Journal fur die Reine und Angewandte Mathematik. 2019 ; Vol. 2019, No. 752. pp. 63-104.
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Strong shift equivalence and algebraic K-theory. / Boyle, Mike; Schmieding, Scott Edward.

In: Journal fur die Reine und Angewandte Mathematik, Vol. 2019, No. 752, 01.07.2019, p. 63-104.

Research output: Contribution to journalArticle

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