Triviality criteria for cohomology classes arising from trivial extensions

Characterize the triviality of the cohomology classes constructed from the trivial extension A = B ⊕ D B of a finite-dimensional algebra B, specifically the class in H^1(Der(B, D B), A^×) and its inflation to H^1(HH_1(B)^*, A^× ∩ Z(A)). Provide necessary and sufficient conditions on B for these classes to vanish.

Background

In the trivial extension A = B ⊕ D B, the paper constructs cohomological invariants by restricting the Nakayama Jacobian cocycle to derivations and related groups, yielding classes in H^1(Der(B, D B), A^×) and, via inflation, H^1(HH_1(B)^*, A^× ∩ Z(A)). Examples show these classes are often non-trivial, though in some special cases (e.g., certain quivers without oriented cycles), triviality can occur.

After presenting a necessary condition derived directly from definitions, the authors explicitly state that they do not know how to characterize the triviality of these classes in general. A complete criterion would clarify when these invariants vanish and illuminate their structural role.