Dualizing Complex of a Toric Face Ring

Abstract

Abstract. A toric face ring, which generalizes both Stanley-Reisner rings and affine semigroup rings, is studied by Bruns, Römer and their coauthors recently. In this paper, under the "normality" assumption, we describe a dualizing complex of a toric face ring R in a very concise way. Since R is not a graded ring in general, the proof is not straightforward. We also develop the squarefree module theory over R, and show that the Cohen-Macaulay, Buchsbaum, and Gorenstein* properties of R are topological properties of its associated cell complex.

Dualizing complexes of seminormal affine semigroup rings and toric face rings

“…R −→ · · · −→ I 0 R −→ 0 is a cochain complex of finitely generated R-modules. The following is the main result of [13]. [13] is long and technical.…”

“…The following is the main result of [13]. [13] is long and technical. So we summarize it here for the reader's convenience.…”

“…For example, k[M σ ] and R itself are squarefree R-modules. In [13], squarefree modules over a cone-wise normal toric face ring play a key role. Many properties are lost in the non-normal case.…”

“…Acknowledgements I am grateful to Professor Yuji Yoshino for giving me valuable comments in the very early stage of this study (even before the publication of [13]). I also thank Professor Ryota Okazaki for stimulating discussion.…”

On the Koszul property of toric face rings

“…On the other hand, if all the cones of Σ are simplicial, and M C = C ∩Z d is generated by exactly dim C elements for every C ∈ Σ, then k[M] is a StanleyReisner ring. Starting with the work of Stanley [24], several authors have considered toric face rings [5], [8], [18], [22]. For an algebraic treatment of affine monoid rings and Stanley-Reisner rings, see Bruns-Herzog [7] or Bruns-Gubeladze [5]; for a more combinatorial treatment of Stanley-Reisner rings see Stanley [25].…”