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We establish general theorems on the cohomology $H^*(s|d)$ of the BRSTdifferential modulo the spacetime exterior derivative, acting in the algebra oflocal $p$-forms depending on the fields and the antifields (=sources for theBRST 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-Tatedifferential associated with the stationary surface. The cohomological group$H_1(\delta |d)$ in form degree $n$ is proved to be isomorphic to the space ofconstants of the motion, thereby providing a cohomological reformulation ofNoether 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 ofmotion hold. The groups $H_k(\delta|d)$ $(k>2)$ are shown to vanish forstandard irreducible gauge theories. The group $H_2(\delta|d)$ is thencalculated explicitly for electromagnetism, Yang-Mills models and Einsteingravity. The invariance of the groups $H^{k}(s|d)$ under the introduction ofnon minimal variables and of auxiliaryComment: 48 pages LaTeX file, ULB-PMIF-94/06 NIKEF-H 94-13 (minor changes in  section 10

Local BRST cohomology in the antifield formalism: I. General theorems