  The complex ac conductivity of thin highly disordered InOx films was studied
as a function of magnetic field through the nominal two-dimensional
superconductor-insulator transition. We have resolved a significant
finite-frequency superfluid stiffness well into the insulating regime, giving
direct evidence for quantum superconducting fluctuations around an insulating
ground state and a state of matter with localized Cooper pairs. A phase diagram
is established that includes the superconducting state, a transition to a
"Bose" insulator, and an eventual crossover to a "Fermi" insulating state at
high fields. We speculate on the consequences of these observations, their
impact on our understanding of the insulating state, and its relevance as a
prototype for other insulating states of matter that derive from
superconductors.
