  Features of the nuclear isothermal incompressibility $\kappa$ and adiabatic
incompressibility $\kappa_Q$ are investigated. The calculations are done at
zero and finite temperatures and non zero entropy and for several equations of
state. It is shown that $\kappa_Q$ decreases with increasing entropy while the
isothermal $\kappa$ increases with increasing $T$. A duality is found between
the adiabatic $\kappa_Q$ and the T=0 isothermal $\kappa$. Our isothermal
results are compared with a recent lattice Monte Carlo calculation done at
finite $T$. The necessity of including correlations is shown if $\kappa$ is to
have a peak with increasing $T$ as seen in the Monte Carlo calculations. A peak
in $\kappa$ is linked to attractive scattering correlations in two nucleons
channel in the virial expansion in our approach which are Pauli blocked at low
$T$.
