Trying to understand the definition of Hilbert functions

Solution 1:

Let $M$ be a finitely-generated graded $k[x_1,\dots,x_n]$-module and let the minimal degree of $M$ be $s$. If $a_1,a_2 \dots$ is a $k$-basis of $M_s$, then $a_1,a_2,\dots$ are part of a minimal generating set of $M$. Indeed, no $a_i$ can be written as a $k[x_1,\dots,x_n]$-linear combination of $\{a_j\}_{j \ne i}$ and elements of higher degree, for degree reasons.

We thus consider the submodule $\bigoplus^{\infty}_{s} M_i$ of $M$ in your context that we may argue from the minimal degree. If we don't do this, there may be elements in degree $s$ that can be produced as a linear combination involving also generators in smaller degree; they are not minimal generators of $M$ but are of $\bigoplus^{\infty}_{s} M_i$.