Corollary 1.5 in Eisenbud’s Commutative Algebra

Question

I have two questions on Corollary 1.5 from Eisenbud's Commutative Algebra with a view toward Algebraic Geometry. I included a screenshot of the corollary in question below the questions themselves. In addition, if any one has suggestions for other expositions of this result I would love to hear them!

1. How do we know that the generators of $\mathfrak{m}S$ can be chosen as homogeneous elements of $\mathfrak{m}$? It seems that this must follow from what we know: $\mathfrak{m}S$ is finitely generated by Hilbert's Basis Theorem, and that $\mathfrak{m}$ is generated by homogeneous elements, but I don't see it.
2. How do we know that we can write $f = \sum g_i f_i$ where the $g_i$ are homogeneous? Eisenbud has already shown that this decomposition holds when $f$ is homogeneous, but my understanding of the proof is that we are aiming to show that any element of $R$ is contained in $R'$, not just the homogeneous ones. Was the idea to assum $f$ is homogeneous, then see that $f \in R$, and use that any element of $R$ can be written as a sum of homogeneous elements of $R$?

Answer 1

1. $\mathfrak{m}S$ is an $S$-submodule of a finitely-generated $S$-module and so is finitely-generated because $S$ is Noetherian. In this case though $\mathfrak{m}S$ is actually an ideal in $S$, and one can consider the ideals $$\langle f_1\rangle\subset \langle f_1,f_2\rangle \subset\cdots\subset\mathfrak{m}S$$ defined by adding in new homogeneous elements $f_i$ of $\mathfrak{m}$. This chain must eventually stabilize, and so $\mathfrak{m}S$ is generated by finitely-many homogeneous elements of $\mathfrak{m}$.

2.You are exactly correct, $f$ is assumed to be homogeneous, otherwise $\mathrm{deg}(f)$ doesn't make sense. The fact that every element of $R$ can be written as a sum of homogeneous elements of $R$ follows from the corresponding fact for elements of $S$ and the properties of the map $\varphi$.