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 $H_k(δ|d)$ in negative ghost degree $-k\ (k>0)$, where $δ$ is the Koszul-Tate differential associated with the stationary surface. The cohomological group $H_1(δ|d)$ 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 $H_k(δ|d)$ in form degree $n$ is isomorphic to the space of $n-k$ forms that are closed when the equations of motion hold. The groups $H_k(δ|d)$ $(k>2)$ are shown to vanish for standard irreducible gauge theories. The group $H_2(δ|d)$ 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

48 pages LaTeX file, ULB-PMIF-94/06 NIKEF-H 94-13 (minor changes in section 10)