  We establish general theorems on the cohomology $H^*(s|d)$ of the BRST
differential modulo the spacetime exterior derivative, acting in the algebra of
local $p$-forms depending on the fields and the antifields (=sources for the
BRST variations). It is shown that $H^{-k}(s|d)$ is isomorphic to $H_k(\delta
|d)$ in negative ghost degree $-k\ (k>0)$, where $\delta$ is the Koszul-Tate
differential associated with the stationary surface. The cohomological group
$H_1(\delta |d)$ in form degree $n$ is proved to be isomorphic to the space of
constants of the motion, thereby providing a cohomological reformulation of
Noether theorem. More generally, the group $H_k(\delta|d)$ in form degree $n$
is isomorphic to the space of $n-k$ forms that are closed when the equations of
motion hold. The groups $H_k(\delta|d)$ $(k>2)$ are shown to vanish for
standard irreducible gauge theories. The group $H_2(\delta|d)$ is then
calculated explicitly for electromagnetism, Yang-Mills models and Einstein
gravity. The invariance of the groups $H^{k}(s|d)$ under the introduction of
non minimal variables and of auxiliary
