LOCAL BRST COHOMOLOGY IN THE ANTIFIELD FORMALISM .1. GENERAL THEOREMS

We establish general theorems on the cohomology H*(s\d) of the BRST differential module the spacetime exterior derivative, acting in the algebra of local p-forms depending on the fields and the antifields (= sources fur 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), when delta is the Koszul-Tate differential associated with the stationary surface. The cohomology 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's 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 fields is also demonstrated. In a companion paper, the general formalism is applied to the calculation of H-k(s\d) in Yang-Mills theory, which is carried out in detail for an arbitrary compact gauge group.