Strong shift equivalence and algebraic K-theory

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.