  Local Morse cohomology associates cohomology groups to isolating
neighborhoods of gradient flows of Morse functions on (generally non-compact)
Riemannian manifolds $M$. We show that local Morse cohomology is a module over
the cohomology of the isolating neighborhood, which allows us to define a
cup-length relative to the cohomology of the isolating neighborhood that gives
a lower bound on the number of critical points of functions on $M$ that are not
necessarily Morse. Finally, we illustrate by an example that this lower bound
can indeed be stronger than the lower bound given by the absolute cup-length.
