Strong shift equivalence and algebraic \(K\)-theory. (English) Zbl 1439.19003

Boyle, Mike; Schmieding, Scott

J. Reine Angew. Math. 752, 63-104 (2019).

Summary: For a semiring \(\mathcal{R}\), the relations of shift equivalence over \(\mathcal{R}\) (SE-\(\mathcal{R}\)) and strong shift equivalence over \(\mathcal{R}\) (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 SE-\(\mathcal{R}\) by SSE-\(\mathcal{R}\), in the 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 SE-\(\mathcal{R}\) class into SSE-\(\mathcal{R}\) classes corresponds precisely to the refinement of the \(\operatorname{GL}(\mathcal{R}[t])\) equivalence class of I-tA into \(\operatorname{El}(\mathcal{R}[t])\) 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.

MSC:

19D10	Algebraic \(K\)-theory of spaces
14C35	Applications of methods of algebraic \(K\)-theory in algebraic geometry
16E20	Grothendieck groups, \(K\)-theory, etc.
37B10	Symbolic dynamics

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