Abstract
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.
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