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In the gauge-invariant construction of abelian chiral gauge theories on thelattice based on the Ginsparg-Wilson relation, the gauge anomaly is topologicaland its cohomologically trivial part plays the role of the local counter term.We give a prescription to solve the local cohomology problem within a finitelattice by reformulating the Poincar\'e lemma so that it holds true on thefinite lattice up to exponentially small corrections. We then argue that thepath-integral measure of Weyl fermions can be constructed directly from thequantities defined on the finite lattice.Comment: revised version, 35pages, using JHEP3.cl

Solving the local cohomology problem in U(1) chiral gauge theories within a finite lattice