# THE SHORT RESOLUTION OF A SEMIGROUP ALGEBRA

@article{Ojeda2015THESR, title={THE SHORT RESOLUTION OF A SEMIGROUP ALGEBRA}, author={Ignacio Ojeda and Alberto Vigneron-Tenorio}, journal={Bulletin of the Australian Mathematical Society}, year={2015}, volume={96}, pages={400 - 411} }

This work generalises the short resolution given by Pisón Casares [‘The short resolution of a lattice ideal’, Proc. Amer. Math. Soc. 131(4) (2003), 1081–1091] to any affine semigroup. We give a characterisation of Apéry sets which provides a simple way to compute Apéry sets of affine semigroups and Frobenius numbers of numerical semigroups. We also exhibit a new characterisation of the Cohen–Macaulay property for simplicial affine semigroups.

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## References

SHOWING 1-10 OF 20 REFERENCES

### On the Apéry sets of monomial curves

- Mathematics
- 2012

In this paper, we use the Apéry table of the numerical semigroup associated to an affine monomial curve in order to characterize arithmetic properties and invariants of its tangent cone. In…

### Finitely generated commutative monoids

- Mathematics
- 1999

There is a lack of effective methods for studying properties of finitely generated commutative monoids. This was one of the main reasons for developing a self-contained book on finitely generated…

### The short resolution of a lattice ideal

- Mathematics
- 2002

The short resolution of a lattice ideal is a free resolution over a polynomial ring whose number of variables is the number of extremal rays in the associated cone. A combinatorial description of…

### Gröbner Basis

- Computer Science, MathematicsEncyclopedia of Cryptography and Security
- 2011

This process generalizes three familiar techniques: Gaussian elimination for solving linear systems of equations, the Euclidean algorithm for computing the greatest common divisor of two univariate polynomials, and the Simplex Algorithm for linear programming.

### On Cohen-Macaulay rings

- Mathematics
- 1994

In this paper, we use a characterization of R-modules N such that fdRN = pdRN to characterize Cohen-Macaulay rings in terms of various dimensions. This is done by setting N to be the dth local…

### Combinatorial Commutative Algebra

- Mathematics
- 2004

Monomial Ideals.- Squarefree monomial ideals.- Borel-fixed monomial ideals.- Three-dimensional staircases.- Cellular resolutions.- Alexander duality.- Generic monomial ideals.- Toric Algebra.-…

### SINGULAR: a computer algebra system for polynomial computations

- Mathematics, Computer ScienceACCA
- 2009

SINGULAR is a specialized computer algebra system for polynomial computations with emphasize on the needs of commutative algebra, algebraic geometry, and singularity theory, which features one of the fastest and most general implementations of various algorithms for computing standard resp.

### numericalsgps, a GAP package for numerical semigroups

- MathematicsACCA
- 2016

The package numericalsgps performs computations with and for numerical and affine semigroups. This manuscript is a survey of what the package does, and at the same time intends to gather the trending…