On the Koszul property of toric face rings

Details On the Koszul property of toric face rings On the Koszul property of toric face rings

2011-07-22

arxiv.org/abs/1107.4503v1

Mathematics

Commutative Algebra

14 pages, comments welcome

Scientific paper

Toric face rings is a generalization of the concepts of affine semigroup rings and Stanley-Reisner rings. We characterize toric face rings having the Koszul, strongly Koszul or initially Koszul property. Firstly, we compute the graded Betti numbers of the underlying field as a module over the toric face ring. We ask whether given two conditions, that the defining ideal of the toric face ring has the monomial part generated in degree 2, and that for each cone of the supporting fan, the corresponding monoid ring is Koszul, we can conclude the ring itself is Koszul. Then we give a full characterization of strongly Koszul toric face rings. We also prove that initially Koszul homogeneous toric face rings are in fact affine semigroup rings.

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