### 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) |
---|---|

Pages (from-to) | 63-104 |

Number of pages | 42 |

Journal | Journal fur die Reine und Angewandte Mathematik |

Volume | 2019 |

Issue number | 752 |

DOIs | |

State | Published - Jul 1 2019 |

### Fingerprint

### 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

}

*Journal fur die Reine und Angewandte Mathematik*, vol. 2019, no. 752, pp. 63-104. https://doi.org/10.1515/crelle-2016-0056

**Strong shift equivalence and algebraic K-theory.** / Boyle, Mike; Schmieding, Scott Edward.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Strong shift equivalence and algebraic K-theory

AU - Boyle, Mike

AU - Schmieding, Scott Edward

PY - 2019/7/1

Y1 - 2019/7/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85068310832&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85068310832&partnerID=8YFLogxK

U2 - 10.1515/crelle-2016-0056

DO - 10.1515/crelle-2016-0056

M3 - Article

VL - 2019

SP - 63

EP - 104

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

SN - 0075-4102

IS - 752

ER -