Socle degrees and last shift of free resolution

Question

I have seen in several references that the degrees of the socle of an Artinian graded algebra $k[x_1,\ldots,x_d]/I$ can be computed by looking at the shifts of the end of its graded free resolution. Does anybody know the proof of this? or a reference of the proof? Thanks!

Answer 1

Let $S=k[x_1,\dots,x_d]$ and $I\subset S$ a graded ideal such that $\dim S/I=0$. Then, from Auslander-Buchsbaum formula we get $\operatorname{pd}_SS/I=d$. The last graded free $S$-module in a graded minimal free resolution of $S/I$ has the form $\bigoplus_jS(-j)^{\beta_{dj}}$. It's easy to see that $\beta_{dj}=\dim_k\operatorname{Tor}_d^S(k,S/I)_j$. On the other hand, $\operatorname{Tor}_d^S(k,S/I)\simeq H_d(\underline x,S/I)$, where $H_d$ denotes the $d$th Koszul homology. Moreover, it's well known that $H_d(\underline x,S/I)=(0:_{S/I}\underline x)=\operatorname{Soc}S/I$ and we are done.