Strong shift equivalence and algebraic K-theory

For a semiring \mathcal{R} , the relations of shift equivalence over \mathcal{R} ( \textup{SE-}\mathcal{R} ) and strong shift equivalence over \mathcal{R} ( \textup{SSE-}\mathcal{R} ) are natural equivalence relations on square matrices over \mathcal{R} , important for symbolic dynamics. When \mathcal{R} is a ring, we prove that the refinement of \textup{SE-}\mathcal{R} by \textup{SSE-}\mathcal{R} , in the \textup{SE-}\mathcal{R} class of a matrix A, is classified by the quotient NK_{1}(\mathcal{R})/E(A,\mathcal{R}) of the algebraic K-theory group NK_{1}(\mathcal{R}) . Here, 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 \mathcal{R} that the refinement of its \textup{SE-}\mathcal{R} class into \textup{SSE-}\mathcal{R} classes corresponds precisely to the refinement of the \mathrm{GL}(\mathcal{R}[t]) equivalence class of I-tA into \mathrm{El}(\mathcal{R}[t]) equivalence classes. We then show this refinement is in bijective correspondence with NK_{1}(\mathcal{R})/E(A,\mathcal{R}) . For a general ring \mathcal{R} and A invertible, the proof that E(A,\mathcal{R}) is trivial rests on a theorem of Neeman and Ranicki on the K-theory of noncommutative localizations. For \mathcal{R} commutative, we show \cup_{A}E(A,\mathcal{R})=NSK_{1}(\mathcal{R}) ; the proof rests on Nenashev’s presentation of K_{1} of an exact category.