Can I use Auslander-Buchsbaum formula for polynomial rings?

Question

Let $R=k[x,y]$ be a polynomial ring over a field $k$ and set $T=R/\langle xy,y^2‎‎\rangle $‎. ‎Let $f:R\to T$ be the natural ring epimorphism‎. ‎Is $\operatorname{pd}_RT=2$? we know that $\operatorname{depth} R=2$ and $\operatorname{depth}_R T=0$.

$R$ is not local. Can I use the Auslander-Buchsbaum formula?

Answer 1

Yes, Auslander-Buchsbaum formula is true also in the graded case and here T is a finitely generated graded R-module. For the proof in this case you may look at Corollary A.4.3 of the book "Monomial Ideals" by Herzog and Hibi.