  In this short Note we show that the direct image sheaf R 1 $\pi$ * (O X)
associated to an analytic family of compact complex manifolds $\pi$ : X
$\rightarrow$ S parametrized by a reduced complex space S is a locally free
(coherent) sheaf of O S --modules. This result allows to improve a
semi-continuity type result for the algebraic dimension of compact complex
manifolds in an analytic family given in [B.15].
  AMS Classification 2010. 32G05-32A20-32J10.
