Nagoya Mathematical Journal

On canonical modules of toric face rings

Bogdan Ichim and Tim Römer

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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

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

First available in Project Euclid: 17 June 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13C14: Cohen-Macaulay modules [See also 13H10] 13D45: Local cohomology [See also 14B15]
Secondary: 05E99: None of the above, but in this section


Ichim, Bogdan; Römer, Tim. On canonical modules of toric face rings. Nagoya Math. J. 194 (2009), 69--90.

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