Comparison of projective and affine Hilbert functions ( Ideals, Varieties and Algorithms book, Theorem 9.3.12-(i) )

Let $k$ be an infinite field.

• Definition 1. ( Affine Hilbert function ). Let $R := k[x_1, \dots ,x_n]$ be a polynomial ring which can be viewed as a vector space over $k$. Let $R_{\le s} := k[x_1, \dots, x_n]_{\le s} $ denote the set of polynomials of total degree $\le s$ in $R$. Note that $\dim_{k} R_{\le s}= \begin{pmatrix}n+s \\ s \end{pmatrix} $ . And given an ideal $I \subseteq R$, we let $I_{\le s}:=I \cap R_{\le s}$. $I_{\le s}$ is a vector subspace of $R_{\le s}$. Then define the affine Hilbert function of $I$ as $$ ^{a}HF_{R/I}(s) = \dim R_{\le s}/ I_{\le s} = \dim R_{\le s} - \dim I_{\le s}. $$

• Definition 2. ( Projective Hilbert function ).Let $S:=k[x_0,\dots, x_n]$ be a polynomial ring. Let $S_s := k[x_0 ,\dots, x_n]_s$ denote the set of homogeneous polynomials of total degree $s$ in $S$, together with the zero polynomial. Note that $\dim_k S_s = \begin{pmatrix}n+s \\ s \end{pmatrix} $. If $I \subseteq S$ is a homogeneous ideal, we let $I_s := I \cap S_s$ denote the set of homogeneous polynomials in $I$ of total degree $s$ ( and the zero polynomial ). Then let's define the projective Hilbert function of $I$ by

$$ HF_{S/I}(s) = \dim S_s/I_s.$$

In the book, Cox's Ideals, Varieties, and algorithms, Theorem 9.3. 12- (i) ( p. 494 ) , the authors states that for $I\subseteq S:=k[x_0 ,\dots , x_n]$ a homogeneous ideal and for $s \ge 1$, we have $$ HF_{S/I}(s) \stackrel{?}{=} \ ^{a}HF_{S/I}(s) -\ ^{a}HF_{S/I}(s-1). \tag{1}$$

( And there is a similar relation between Hilbert polynomials ). And in the proof the authors wrote that " The first part of (i) follows easily by reducing to the case of a monomial ideal and using the results of $\S 2$."

Q. Why the equality $(1)$ is true? What is the argument the author intends to convey in the proof?

Note that we have following theorems :

Proposition 9.3.4. Let $I\subseteq R=k[x_1, \dots, x_n]$ be an ideal and let $>$ be a graded order on $R$. Then the monomial ideal $\langle \mathrm{LT}(I) \rangle$ has the same affine Hilbert function as $I$.

Proposition 9.3.9. Let $I\subseteq S=k[x_0,x_1, \dots, x_n]$ be a homogeneous ideal and let $>$ be a monomial order on $S$. Then the monomial ideal $\langle \mathrm{LT}(I) \rangle$ has the same projective Hilbert function as $I$.

By these propositions, in the question we may assume (?) that $I$ is a monomial ideal but in the monomial case how can we make progress for our reasoning, refering to $ \S 2$ ?

( Further progress ) Remark. There exist statements ( C.f. Proposition 9.3.3 - (i) ) that for $I \subseteq R:=k[x_1,\dots, x_n]$ a proper monomial ideal, for $s \ge0$, $^{a}HF_{R/I}(s)$ is the number of monomials not in $I$ of total degree $\le s$. And similarly for a monomial ideal $I \subseteq S:=k[x_0, \dots, x_n]$, $HF_{S/I}(s)$ is the number of monomials not in $I$ of total degree $s$.

Now let's return to our question. If $I \subseteq S:=k[x_0,\dots, x_n]$ is not a proper ideal, then the $(1)$ is automatically true. So let's assume that $I$ is a proper ideal of $S$. Note that

$$\{\operatorname{monomials not in} I \operatorname{of total degree} \le s \} = \{\operatorname{monomials not in} I \operatorname{of total degree} \le s-1 \} \biguplus \{\operatorname{monomials not in} I \operatorname{of total degree} = s \} $$

From this, by the remark above, we have $$ ^{a}HF_{S/I}(s) = ^{a}HF_{S/I}(s-1) + HF_{S/I}(s). $$

So, $$ HF_{S/I}(s) = ^{a}HF_{S/I}(s) - ^{a}HF_{S/I}(s-1)$$ and we are done. I don't think it is necessary to refer to the section 2. How do you think about this?