We study the cohomology of symbolic dynamical systems called homshifts: they are the nearest-neighbour Zᵈ shifts of finite type whose adjacency rules are the same in every direction. Building on the work of Klaus Schmidt (Pacific J. Math. 170 (1995), no. 1, 237-269) we give a necessary and sufficient condition for their cohomological triviality. This condition is expressed in terms of the topology of a natural simplicial complex arising from the shift space which can be analyzed in many natural cases. However, we preove that in general, cohomological triviality is algorithmically undecidable for homshifts.
Chandgotia et al. (2025) studied this question.