We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Strong shift equivalence and algebraic K-theory.
- Authors
Boyle, Mike; Schmieding, Scott
- Abstract
For a semiring ℛ \mathcal{R} , the relations of shift equivalence over ℛ \mathcal{R} ( SE- ℛ \textup{SE-}\mathcal{R}) and strong shift equivalence over ℛ \mathcal{R} ( SSE- ℛ \textup{SSE-}\mathcal{R}) are natural equivalence relations on square matrices over ℛ \mathcal{R} , important for symbolic dynamics. When ℛ \mathcal{R} is a ring, we prove that the refinement of SE- ℛ \textup{SE-}\mathcal{R} by SSE- ℛ \textup{SSE-}\mathcal{R} , in the SE- ℛ \textup{SE-}\mathcal{R} class of a matrix A, is classified by the quotient N K 1 (ℛ) / E (A , ℛ) NK_{1}(\mathcal{R})/E(A,\mathcal{R}) of the algebraic K-theory group N K 1 (ℛ) NK_{1}(\mathcal{R}). Here, E (A , ℛ) E(A,\mathcal{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 \mathcal{R} that the refinement of its SE- ℛ \textup{SE-}\mathcal{R} class into SSE- ℛ \textup{SSE-}\mathcal{R} classes corresponds precisely to the refinement of the GL (ℛ [ t ]) \mathrm{GL}(\mathcal{R}[t]) equivalence class of I - t A I-tA into El (ℛ [ t ]) \mathrm{El}(\mathcal{R}[t]) equivalence classes. We then show this refinement is in bijective correspondence with N K 1 (ℛ) / E (A , ℛ) NK_{1}(\mathcal{R})/E(A,\mathcal{R}). For a general ring ℛ \mathcal{R} and A invertible, the proof that E (A , ℛ) E(A,\mathcal{R}) is trivial rests on a theorem of Neeman and Ranicki on the K-theory of noncommutative localizations. For ℛ \mathcal{R} commutative, we show ∪ A E (A , ℛ) = N S K 1 (ℛ) \cup_{A}E(A,\mathcal{R})=NSK_{1}(\mathcal{R}) ; the proof rests on Nenashev's presentation of K 1 K_{1} of an exact category.
- Subjects
SYMBOLIC dynamics; MATHEMATICAL equivalence; EQUIVALENCE relations (Set theory); EQUIVALENCE classes (Set theory); K-theory
- Publication
Journal für die Reine und Angewandte Mathematik, 2019, Vol 2019, Issue 752, p63
- ISSN
0075-4102
- Publication type
Article
- DOI
10.1515/crelle-2016-0056