Let
R
=
k
[
x
1
,
.
.
.
,
x
n
]
R=k[x_1,...,x_n]
be the polynomial ring in
n
n
independent variables, where
k
k
is a field. In this work we will study Bass numbers of local cohomology modules
H
I
r
(
R
)
H^r_I(R)
supported on a squarefree monomial ideal
I
⊆
R
I\subseteq R
. Among them we are mainly interested in Lyubeznik numbers. We build a dictionary between the modules
H
I
r
(
R
)
H^r_I(R)
and the minimal free resolution of the Alexander dual ideal
I
∨
I^{\vee }
that allows us to interpret Lyubeznik numbers as the obstruction to the acyclicity of the linear strands of
I
∨
I^{\vee }
. 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.