In the picture we discuss the Stanley-Reisner ring over a simplicial complex $\Delta$.
I do not understand the steps "(i) implies" and "(ii) implies", maybe I do not catch how to translate the complexes language into the element language. For example, which element in the $F$? Please help.
$F=\operatorname{supp}(x)$ and $\operatorname{supp}(x)\cup\operatorname{supp}(v)=\operatorname{supp}(xv)$; $xv\ne0$ (in the Stanley-Reisner ring) is equivalent to $xv\notin I_{\Delta}$. If $\operatorname{supp}(x)\cup\operatorname{supp}(v)=\operatorname{supp}(xv)\notin\Delta$ what can we say about $xv$?
$\deg v/x^l=a$ shows that $a_i=b_i-lc_i$ where $b_i$, resp. $c_i$ is the degree of variable $x_i$ in $v$, resp. in $x$. If $a_i<0$, then the variable $x_i$ appears necessarily in $x$ otherwise $c_i=0$, and thus $a_i\ge0$. Conclusion: $F\supset G_a$.
$a_i>0$ implies $b_i>lc_i$ and since $l\ge0$ and $c_i\ge 0$ we get $b_i>0$. Conclusion: $H_a\subset\operatorname{supp}(v)$.