  In this note we study families of Gauss-Manin systems arising from Laurent
polynomials with parametric coefficients under projection to the parameter
space. For suitable matrices of exponent vectors, we exhibit a natural
four-term exact sequence for which we then give an interpretation via
generalized A-hypergeometric systems. We determine the extension groups from
the parameter sheaf to the middle term of this sequence and show that the
four-term sequence does not split. Auxiliary results include the computation of
Ext and Tor groups of A-hypergeometric systems against the parameter sheaf.

Algebraic Geometry