How is this proof correct in regard to a $k$-subalgebra (Eisenbud)?

Question

This is testing my understanding of algebras, subalgebras, & polynomial rings.

Page 31.

Corollary 1.5. Let $k$ be a field, and let $S = k[x_1, \dots, x_r]$ be a polynomial graded by degree. Let $R$ be a $k$-subalgebra of $S$. If $R$ is a summand of $S$, in the sense that there is a map of $R$-modules $\varphi: S \to R$ that preserves degrees and takes each element of $R$ to itself, then $R$ is a finitely generated $k$-algebra.

Proof. Let $\hat{m} \subset R$ be the ideal generated by the homogeneous elements of $R$ of strictly positive degree. Since $S$ is Noetherian, the ideal $\hat{m}S$ has a finite set of generators, which may be chosen to be homogeneous elements $f_1, \dots, f_s$ of $\hat{m}$. We shall show that these elements generate $R$ as a $k$-algebra.
To do this, let $R'$ be the $k$-subalgebra of $S$ generated by $f_1, \dots, f_s$, and suppose $f \in R$. We shall show that $f \in R'$ by induction on the degree of $f$. To start the induction, note that if degree $f = 0$, then $f \in k \subset R'$, as claimed.
Now suppose $\deg f \gt 0$, so that $f \in \hat{m}$. Since the $f_i$ generate $\hat{m}S$ as an ideal of $S$, we may write $f = \sum g_if_i$, where each $g_i$ is a homogeneous form of degree $\deg g_i = \deg f - \deg f_i \lt \deg f$. [...].

1. Is the bolded part an erratum in Eisenbud? Shouldn't it be $\hat{m}S$ there?
2. How is each $g_i$ homogeneous?

Answer 1

Thanks @Ken Duna.

According to first properties of a graded ring: A homogeneous ideal is the direct sum of its homogeneous parts.

Proof. Let $I \subset S$ be a homogeneous ideal (which clearly $\hat{m}S$ is).

Define the $n$th homogeneous part of $I$ to be $S_n \cap I$, where $S = S_0 \oplus S_1 \oplus \cdots$. I want to show that $I = \bigoplus$ of its homogeneous parts.

Clearly the intersection of the modules $S_n \cap I, n\geq 0$ since $S_n = $ polynomials of degree $n$. And clearly a finite sum of $s_0 + s_1 + \dots \in I$, where $s_i \in S_i \cap I$. Thus I want to show that $I \subset \bigoplus S_n \cap I$. This is also obvious.

In regard to my question, since $\hat{m}S = I$ is graded by degree we can write each $f \in I$ as $f = \sum_i^s g_i f_i$ where each $g_i$ is a homogeneous form. This is where I am stuck. Supposedly $g_i$ is homogeneous.

Conjecture: If $f \in \hat{m}$, then $f = \sum g_i f_i$ where each $g_i$ is homogeneous.