Specific dimension calculation

Question

Let $k$ be an algebraically closed field with $\text{char }k>0$. Let $g\in k[x,y,z]$ be a homogeneous degree $4$ polynomial, and let $q\geq 2$. Consider the graded ring $R:=k[x,y,z]/(x^q,y^q,z^q)$ and let $\varphi_i:R_i\rightarrow R_{i+4}$ be the multiplication by $g$ map. I'm trying to deduce $$\dim(R/gR)=\sum\limits_{i=-4}^{3q-7} \dim(\text{coker}\varphi_i)$$ Edit: I've reduced to showing $\dim \text{coker}\varphi_i=0$ for $i\geq 3q-6$, but this isn't obvious to me.

Answer 1

If $i\geq 3q-6$, $i+4\geq 3q-2$. So, it is enough to show that $R_i=0$ for $i>3q-3$. It is well known that the socle of $R$ is generated by $x^{q-1}y^{q-1}z^{q-1}\in R_{3q-3}$ and thus you get what you need. If you have not seen this fact, I suggest you look up a commutative algebra book dealing with socles. More generally, if $R=k[x_1,\ldots, x_n]/(x_1^{d_1},\ldots, x_n^{d_n})$, $d_i>0$, then the socle is generated by $x_1^{d_1-1}\cdots x_n^{d_n-1}$ and can be proved using an induction.