Let $S=k[x,y,z,w]$ be the polynomial ring over a field $k$ and $P=(x,y,z,w)S$. We set $R=S_{P}/[(x,y)\cap (z,w)]S_{P}$. Prove that $dim(R)=2$ but $depth(R)=1$.

I try to solve this exercise by myself but I truly get stuck. I think it starts from this point: $$0\to R/(I\cap J)\to R/I \times R/J \to R/(I+J)$$

Replace $R=S_{P}$, $I=(x,y)S_{P}$, $J=(z,w)S_{P}$. However, that is all I have.

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  • What's the height of the ideal $(x,y)\cap (z,w)$? – user26857 May 24 '17 at 16:26
  • This looks like a duplicate of https://math.stackexchange.com/questions/1413766/prove-that-kx-1-ldots-x-4-langle-x-1x-2-x-2x-3-x-3x-4-x-4x-1-rangle-is – user26857 May 24 '17 at 16:27
  • Actually, it is quite similar. I do not understand why $dim(R)=2$ and what is the meaning of mod $x_{1}-x_{3}$ in these case. Could you help me say it specifically? – Soulostar May 24 '17 at 16:47

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