Edge ideals and DG algebra resolutions

  • Adam Boocher
  • Alessio D'Alì
  • Eloisa Grifo
  • Jonathan Montano
  • Alessio Sammartano
Keywords: DG algebra resolution, Koszul homology, acyclic closure, minimal model, deviations, Poincaré series, Hilbert series, Koszul algebra, edge ideal, paths and cycles


Let R = S/I where S = k[T_1, . . . , 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 ε_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.