Strong shift equivalence and algebraic K-theory
Let R be a ring. Let SSE-R be the equivalence relation on square matrices (allowed to have different size) over R generated by A ~ B if there exist matrices U,V over R such that A = UV and B = VU . An invariant of SSE-R is shift equivalence over R (SE-R); for example, A and B are SE-R iff the R[t]-modules cok(I-tA) and cok(I-tB) are isomorphic. We show that the refinement of SE-R by SSE-R is captured precisely by NK_1(R)/E(A,R), where the "elementary stabilizer" group E(A,R) depends on the SE-R class of A. E(A,R) is trivial if A is invertible, and in some related cases; for general R, our proof of this relies on the K-theory of noncommutative localization developed by Neeman and Ranicki. The result has application to symbolic dynamics. For R commutative, the union over A of the E(A,R) equals NSK_1(R); the proof uses a stabilization result of Fitting and (especially) the Nenashev characterization of K_1 of an exact category, applied to the exact category of f.g. modules over R[t] with a projective resolution of length at most 1.