Take a (multiplicative finite) group $H$ acting on the left (by automorphisms) on an (additive finite) abelian group $A$, and recall that the abelian (additive) group of crossed homomorphisms from $H$ to $A$ is defined by
$$C^1(H,A) = \{f \colon H \to A \ : \ \forall\ x,y \in H. \ f(xy) = f(x) + xf(y)\}. $$
A crossed homomorphism $f \in C^1(H,A)$ is called principal if there exists some $a \in A$ such that for every $h \in H$ we have $f(h) = ha - a$. The set of principal crossed homomorphisms is a subgroup of $C^1(H,A)$ which is denoted by $Z^1(H,A)$.
We say that $f$ is locally principal if for every $h \in H$ there exists an $a \in A$ such that $f(h) = ha - a$. Similarly, locally principal crossed homomorphisms form a subgroup of $C^1(H,A)$ which we denote by $L^1(H,A)$.
Has the notion of a locally principal crossed homomorphism been studied?
More generally,
Has someone studied cochains which are locally coboundaries?