  We generalize the cohomology theory for linear cycle sets introduced by Lebed
and Vendramin. Our cohomology classifies extensions of linear cycle sets by
trivial ideals, whereas the cohomology of Lebed and Vendramin only deals with
central ideals $I$ (which are automatically trivial). Therefore our theory
gives an analog to the theory of extensions of braces by trivial ideals
constructed by Bachiller, but from a cohomological point of view. We also study
the general notions of extensions of linear cycle sets and the equivalence of
extensions.
