### 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 NK_{1}(R)=E(A, R) of the algebraic K-theory group NK_{1}(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 NK_{1}(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 _{A}E(A, R) = NSK_{1}(R) the proof rests on Nenashev?s presentation of K_{1} of an exact category.

Original language | English (US) |
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Pages (from-to) | 63-104 |

Number of pages | 42 |

Journal | Journal fur die Reine und Angewandte Mathematik |

Volume | 2019 |

Issue number | 752 |

DOIs | |

Publication status | Published - Jul 1 2019 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Journal fur die Reine und Angewandte Mathematik*,

*2019*(752), 63-104. https://doi.org/10.1515/crelle-2016-0056