  In this paper, we present a complete proof of the so-called First On-Shell
Theorem that determines dynamical content of the unfolded equations for free
symmetric massless fields of arbitrary integer spin in any dimension and
arbitrary integer or half-integer spin in four dimensions. This is achieved by
calculation of the respective $\sigma_-$ cohomology both in the tensor language
in Minkowski space of any dimension and in terms of spinors in $AdS_4$. In the
$d$-dimensional case $H^p(\sigma_-)$ is computed for any $p$ and interpretation
of $H^p(\sigma_-)$ is given both for the original Fronsdal system and for the
associated systems of higher form fields.
