## Nagoya Mathematical Journal

### On canonical modules of toric face rings

#### Abstract

Generalizing the concepts of Stanley-Reisner and affine monoid algebras, one can associate to a rational pointed fan $\Sigma$ in $\mathbb{R}^{d}$ the $\mathbb{Z}^{d}$-graded toric face ring $K[\Sigma]$. Assuming that $K[\Sigma]$ is Cohen-Macaulay, the main result of this paper is to characterize the situation when its canonical module is isomorphic to a $\mathbb{Z}^{d}$-graded ideal of $K[\Sigma]$. From this result several algebraic and combinatorial consequences are deduced. As an application, we give a relation between the cleanness of $K[\Sigma]$ and the shellability of $\Sigma$.

#### Article information

Source
Nagoya Math. J., Volume 194 (2009), 69-90.

Dates
First available in Project Euclid: 17 June 2009

https://projecteuclid.org/euclid.nmj/1245209125

Mathematical Reviews number (MathSciNet)
MR2536527

Zentralblatt MATH identifier
1194.13012

#### Citation

Ichim, Bogdan; Römer, Tim. On canonical modules of toric face rings. Nagoya Math. J. 194 (2009), 69--90. https://projecteuclid.org/euclid.nmj/1245209125

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