  Let $R=\mathbb{K}[x_1,\dots,x_n]$, a graded algebra $S=R/I$ satisfies
$N_{k,p}$ if $I$ is generated in degree $k$, and the graded minimal resolution
is linear the first $p$ steps, and the $k$-index of $S$ is the largest $p$ such
that $S$ satisfies $N_{k,p}$. Eisenbud and Goto have shown that for any graded
ring $R/I$, then $R/I_{\geq k}$, where $I_{\geq k}=I\cap M^k$ and
$M=(x_1,\dots,x_n)$, has a $k$-linear resolution (satisfies $N_{k,p}$ for all
$p$) if $k\gg0$. For a squarefree monomial ideal $I$, we are here interested in
the ideal $I_k$ which is the squarefree part of $I_{\geq k}$. The ideal $I$ is,
via Stanley-Reisner correspondence, associated to a simplicial complex
$\Delta_I$. In this case, all Betti numbers of $R/I_k$ for
$k>\min\{\text{deg}(u)\mid u\in I\}$, which of course is a much finer invariant
than the index, can be determined from the Betti diagram of $R/I$ and the
$f$-vector of $\Delta_I$. We compare our results with the corresponding
statements for $I_{\ge k}$. (Here $I$ is an arbitrary graded ideal.) In this
case we show that the Betti numbers of $R/I_{\ge k}$ can be determined from the
Betti numbers of $R/I$ and the Hilbert series of $R/I_{\ge k}$.
