Pontryagin duality and sheaves of profinite modules

The well-known theory of Pontryagin duality provides a strong connection between the homology and cohomology theories of a profinite group in appropriate categories. A construction for taking the `profinite direct sum’ of an infinite family of profinite modules indexed over a profinite space has been found to be useful in the study of homology of profinite groups, but hitherto the appropriate dual construction for studying cohomology with coefficients in discrete modules has not been studied. This paper remedies this gap in the theory.