  In this work we introduce a new set of invariants associated to the linear
strands of a minimal free resolution of a $\mathbb{Z}$-graded ideal $I\subseteq
R=\Bbbk[x_1, \ldots, x_n]$. We also prove that these invariants satisfy some
properties analogous to those of Lyubeznik numbers of local rings. In
particular, they satisfy a consecutiveness property that we prove first for
Lyubeznik numbers. For the case of squarefree monomial ideals we get more
insight on the relation between Lyubeznik numbers and the linear strands of
their associated Alexander dual ideals. Finally, we prove that Lyubeznik
numbers of Stanley-Reisner rings are not only an algebraic invariant but also a
topological invariant, meaning that they depend on the homeomorphic class of
the geometric realization of the associated simplicial complex and the
characteristic of the base field.
