Poincaré series of quotient module

Question

I am trying to calculate the Poincaré series $P(M,t)$ with respect to the standard degree grading of the graded $\mathbb{C} [x,y,z,w]$-module $ M=\mathbb{C}[x,y,z,w]/I$, where $I = (x,w) \cap (z,w) $.

Is there a nice abstract way to do it using short exact sequences?

Answer 1

The monomial ideal $I$ equals $(xz,w)$, so $M=\mathbb C$[$x,y,z$]$/(xy)$. From the short exact sequence $$0\to\mathbb C[x,y,z](-2)\stackrel{\cdot xy}\to\mathbb C[x,y,z]\to\mathbb C[x,y,z]/(xy)\to 0$$ we get $H_{\mathbb C[x,y,z]/(xy)}=\dfrac{1-t^2}{(1-t)^3}=\dfrac{1+t}{(1-t)^2}$.