# Castelnuovo-Mumford regularity and Betti numbers: an existence question

Let $k$ be a field and $M$ a finitely generated, graded module over the graded ring $S=k[x_1,\dots,x_n]$. Let $\cdots \rightarrow F_j \rightarrow F_{j-1} \rightarrow \cdots F_1 \rightarrow F_0 \rightarrow M \rightarrow 0$ be the minimal, free, graded resolution of $M$. Define $b_j$ to be the maximum among the degrees that appear in $F_j$, i.e. if $F_j = \oplus_{i=1}^{n_j} S(a_{i,j})$, then $b_j = \max_i a_{i,j}$. Suppose that $M$ is $m$-regular in the sense of Castelnuovo-Mumford, i.e. that $b_j - j \le m, \, \forall j$.

Question: Why is it true that there exists a maximal integer $j$ such that $b_j -j = m$?

Motivation: Proof of Proposition 20.16 in Eisenbud (CA with a view...).

In the proof of the Proposition 20.16 it is said: "let $m'$ be the regularity of $M$" and this is a maximum, so there is an $i$ such that $b_i−i=m'$ and now can choose such a maximal $i$ (since the resolution is finite) and call it $j$.