I did not understand this lemma, could someone explain it to me? Maybe with a concrete example.
We use the subscripts "pro" to distinguish the zero sets in affine and the projective setting.
Lemma. Let $f_1,\dots, f_r$ be a set of homogeneous polynomials from $S=k[x_0,\dots, x_n]$. Let $\mathfrak{a}$ be the homogeneous ideal defined by $\mathfrak{a}:=\langle f_1,\dots, f_r\rangle$. Then $$\boxed{Z_\text{pro}(\mathfrak{a})=Z_{\text{pro}}(f_1,\dots, f_r)}$$
Proof. By definition $Z_{\text{pro}}(\mathfrak{a})=Z_{\text{pro}}(\mathfrak{a}^h)$, where $\mathfrak{a}^h$ is the set of all homogeneous elements of $\mathfrak{a}$. Now, since $\{f_1,\dots, f_r\}\subseteq \mathfrak{a}^h$, we have $Z_{\text{pro}}(\mathfrak{a})\subseteq Z_{\text{pro}}(f_1,\dots f_r)\subseteq\mathbb{P}^n$.
Conversely, let $P\in Z_{\text{pro}}(f_1,\dots, f_r)$. Let $g\in\mathfrak{a}^h$ and write $$g=g_1f_1+g_2f_2+\cdots+g_rf_r$$ for some $g_1,\dots, g_r\in S$. Further decomposing the polynomials $g_1,\dots, g_r$ into homogeneous components we obtain $$g=g_{i_1}f_{i_1}+g_{i_2}f_{i_2}+\cdots +g_{i_N}f_{i_N}\tag1$$ where $g_{i_1},\dots, g_{i_N}\in S^h$
Question. Notice that in this expression there may be repetition of the polynomials $f_i$ for which we require the above indexing. Why this is true?
Using the direct sum decomposition $S=\oplus_{d\ge 0} S_d$, we may assume that $$\deg(g_{i_l}f_{i_l})=\deg g_{i_l}+\deg f_{i_l}=\deg g\quad\text{for each}\quad i_l.$$
Question. Why this is true?
Now from our assumption, we have $g(P)=0$. This implies $$g(P)=0\quad\text{for all}\quad g\in\mathfrak{a}^h$$ which implies $P\in Z_{\text{pro}}(\mathfrak{a}^h)$
Best Answer
About the first question: If say $g_1$ has homogeneous components $g_{1j}'s$ then for example $$g_1f_1=\sum_j g_{1j}f_1$$ so you can see $f_1$ is repeating. So the expression (1) takes care of this possibility.
For the 2nd question, $g$ is in the $\deg g$ component of $S$. So every term $g_if_j$ on the right hand side is in $\deg g$ component of $S$.