TY - JOUR

T1 - Strong shift equivalence and algebraic K-theory

AU - Boyle, Mike

AU - Schmieding, Scott

N1 - Funding Information:
Mike Boyle was supported by the Danish National Research Foundation, through the Centre for Symmetry and Deformation (DNRF92), and by the NSERC Discovery grants of David Handelman and of Thierry Giordano, at the University of Ottawa.
Publisher Copyright:
© De Gruyter 2019.

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.

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U2 - 10.1515/crelle-2016-0056

DO - 10.1515/crelle-2016-0056

M3 - Article

AN - SCOPUS:85068310832

SN - 0075-4102

VL - 2019

SP - 63

EP - 104

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

IS - 752

ER -