A cyclic six-term exact sequence for block matrices over a PID

Let be a 2×2 upper triangular block matrix over a principal ideal domain D with square diagonal blocks A 1 and A 2. We define a cyclic six-term exact sequence e(A) in terms of the kernels and the cokernels of A 1 and A 2 with a connecting map defined by the off-diagonal block X. This cyclic sequence e(A), under a variety of block-preserving matrix equivalences, is an invariant strictly finer than the Smith normal forms of A 1 A 2 and A combined. As one example of how this new K-theoretic invariant is used in classical linear algebra, we prove that 2×2 upper triangular block matrices A and B over a field F are block-preserving similar if and only if , that is, there is a chain F[t]-module isomorphism between the two cyclic sequences. We conclude with an application e(A) to the flow equivalence classification of two-component shifts of finite type in symbolic dynamics. This paper is self-contained and presented in a matrix-theoretic form.