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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?

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    $\begingroup$ You are asking about the kernel of $H^1(H,A)\to \prod_C H^1(C,A)$ where the product runs over all cyclic subgroups $C$ in $H$. In Galois cohomology these kernels appear often when studying local-to-global principles as the product can also be viewed as running also over all localisations at unramified places by Chebotarev. $\endgroup$ Mar 6, 2016 at 13:50
  • $\begingroup$ @ChrisWuthrich This is great! Could you please give a reference to some result of that flavor? I will gladly accept this as an answer. $\endgroup$
    – Pablo
    Mar 6, 2016 at 14:00
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    $\begingroup$ One reference : Cohomology of number fields, a version of yoru kernel is denoted by $Ш^1$ defined in 8.6.2. The book contains many results including Grunwald-Wang on it. $\endgroup$ Mar 9, 2016 at 21:34
  • $\begingroup$ @ChrisWuthrich it seems that this definiton focuses mainly on ramified primes so noncyclic subgroups may appear in the product, contrary to the situation here. $\endgroup$
    – Pablo
    Mar 12, 2016 at 21:23

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