Rectangular simplicial semigroups. (English) Zbl 0952.20050
Eisenbud, David (ed.), Commutative algebra, algebraic geometry, and computational methods. Proceedings of the conference on algebraic geometry, commutative algebra, and computation, Hanoi, Vietnam, 1996. Singapore: Springer. 201-213 (1999).
Let \(\lambda_i\) be positive integers, \(1\leq i\leq n\), and let \(\Delta=\Delta(\lambda_1,\dots,\lambda_n)\subset\mathbb{R}^n\) be the simplex with vertices \((0,0,\dots,0)\), \((\lambda_1,0,\dots,0)\), \((0,\lambda_2,\dots,0),\dots,(0,0,\dots,\lambda_n)\). Let \(S_\Delta\) be the subsemigroup of \(\mathbb{Z}^{n+1}\) generated by the images of the vertices under the embedding \(\iota\colon\mathbb{R}^n\to\mathbb{R}^{n+1}\), \(\iota(x)=(x,1)\), and let \(K[S_\Delta]\) be the semigroup ring (\(K\) a field). The article contains several necessary and several sufficient conditions for the normality of \(K[S_\Delta]\). A class of \(\Delta\) with \(K[S_\Delta]\) Koszul is determined. Finally, the class group of \(K[S_\Delta]\) is determined in the normal case.
For the entire collection see [Zbl 0921.00043].
For the entire collection see [Zbl 0921.00043].
Reviewer: Ralf Fröberg (Stockholm)
MSC:
| 20M25 | Semigroup rings, multiplicative semigroups of rings |
| 13C20 | Class groups |
| 13F50 | Rings with straightening laws, Hodge algebras |
| 20M14 | Commutative semigroups |
| 52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |
| 16S36 | Ordinary and skew polynomial rings and semigroup rings |
| 13F55 | Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes |
| 52C07 | Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry) |
| 11H06 | Lattices and convex bodies (number-theoretic aspects) |
