Edge ideals and DG algebra resolutions

Abstract

Let $R=S/I$ where $S=k[T_1,\dots ,T_n]$ and $I$ is a homogeneous ideal in $S$. The acyclic closure $R⟨Y⟩$ of $k$ over $R$ is a DG algebra resolution obtained by means of Tate's process of adjoining variables to kill cycles. In a similar way one can obtain the minimal model $S[X]$, a DG algebra resolution of $R$ over $S$. By a theorem of Avramov there is a tight connection between these two resolutions. In this paper we study these two resolutions when $I$ is the edge ideal of a path or a cycle. We determine the behavior of the deviations ${\epsilon }_i(R)$, which are the number of variables in $R⟨Y⟩$ in homological degree $i$. We apply our results to the study of the $k$-algebra structure of the Koszul homology of $R$.