  Recently, we introduced a configuration space with interaction structure and
a uniform local cohomology on it with co-authors in arXiv:2009.04699. The
notion is used to understand a common structure of infinite product spaces
appeared in the proof of Varadhan's non-gradient method. For this, the
cohomology of the configuration space with a group action is the main target to
study, but the cohomology is easily obtained from that of the configuration
space without a group action by applying a well-known property on the group
cohomology. In fact, the analysis of the cohomology of the configuration space
without a group action is the essential part of arXiv:2009.04699. In this
article, we give an elementary and direct proof to obtain the cohomology of a
space with a group action from that without a group action under a certain
condition including the setting of the configuration space with interaction
structure. In particular, no knowledge of group cohomology is required.
