Exercise 17.2 of Matsumura about CM rings

Question

I have two questions about this exercise of Matsumura:

Question 1: Why $y^3$ is $R/(x^3)$ regular?

Question 2: I hardly (in 20 lines) can prove that $k[x^4,x^3y,xy^3,y^4]$ is not CM. Is there a short way or intuition for this part ?

Answer 1

For the second question, note that $x^4, y^4$ is a SOP (system of parameters). Since $R=k[x^4, x^3y,xy^3, y^4]$ is a domain, $x^4$ is a non-zero divisor on $R$. But observe that $y^4$ is a zero divisor on $R/(x^4)$ because $y^4(x^3y)^2=x^4(xy^3)^2$. Of course, one needs to note that $(x^3y)^2\notin (x^4)$. Hence $R$ is not CM.

Added Later: For the first part, you can write $A$ as $$k[X,Y,Z,W]/(XW-YZ,Z^3-XW^2,Y^3-X^2W,XZ^3-Y^3W).$$ Now note that $X, W$ form a SOP, and that $W$ is regular on $$A/(X)=k[Y,Z,W]/(YZ,Z^3,Y^3,Y^3W).$$