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

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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