  We study Bass numbers of local cohomology modules supported on squarefree
monomial ideals paying special attention to Lyubeznik numbers. We build a
dictionary between local cohomology modules and minimal free resolutions that
allow us to interpret Lyubeznik numbers as the obstruction to the acyclicity of
the linear strands of the Alexander dual ideals. The methods we develop also
help us to give a bound for the injective dimension of the local cohomology
modules in terms of the dimension of the small support.
