Toric ideals generalized by quadratic binomials.

*(English)*Zbl 0943.13014Let \(K\) be a field and \(K[{\mathbf t}]=K[t_1,\dots,t_d]\) the polynomial ring in \(d\) variables over \(K\). Let \({\mathcal A}\) be a finite set of monomials belonging to \(K[{\mathbf t}]\), all of the same degree, and \(K[{\mathcal A}]\) the subalgebra of \(K[{\mathbf t}]\) generated by the monomials in \({\mathcal A}\). It is known that the presentation ideal \(I_{\mathcal A}\) of \(K[{\mathcal A}]\) is generated by binomials. The ideal \(I_{\mathcal A}\) is called the toric ideal associated with the affine semigroup ring \(K[{\mathcal A}]\).

In this paper the case is considered when \(\mathcal A\) consists of squarefree quadratic monomials, that is, squarefree monomials of degree \(2\). It is very natural to associate a finite graph \(G\) to \({\mathcal A}\). A combinatorial criterion on \(G\) is given for the toric ideal \(I_{\mathcal A}\) to be generated by quadratic binomials. As a corollary one has that every Koszul algebra generated by squarefree quadratic monomials is normal. Moreover the authors give an example of a normal non-Koszul squarefree semigroup ring whose toric ideal is generated by quadratic binomials, as well as an example of a non-normal Koszul squarefree semigroup ring whose toric ideal possesses no quadratic Gröbner basis.

In addition all the affine semigroup rings \(K[{\mathcal A}]\) which have 2-linear resolution are classified, under the assumption that \({\mathcal A}\) is a finite set of squarefree quadratic monomials; in case \({\mathcal A}\) consists of (not necessarily squarefree) quadratic monomials, it is proved that if \(K[{\mathcal A}]\) is normal and the convex polytope associated with \({\mathcal A}\) is simple, then \(K[{\mathcal A}]\) is generated by quadratic binomials.

In this paper the case is considered when \(\mathcal A\) consists of squarefree quadratic monomials, that is, squarefree monomials of degree \(2\). It is very natural to associate a finite graph \(G\) to \({\mathcal A}\). A combinatorial criterion on \(G\) is given for the toric ideal \(I_{\mathcal A}\) to be generated by quadratic binomials. As a corollary one has that every Koszul algebra generated by squarefree quadratic monomials is normal. Moreover the authors give an example of a normal non-Koszul squarefree semigroup ring whose toric ideal is generated by quadratic binomials, as well as an example of a non-normal Koszul squarefree semigroup ring whose toric ideal possesses no quadratic Gröbner basis.

In addition all the affine semigroup rings \(K[{\mathcal A}]\) which have 2-linear resolution are classified, under the assumption that \({\mathcal A}\) is a finite set of squarefree quadratic monomials; in case \({\mathcal A}\) consists of (not necessarily squarefree) quadratic monomials, it is proved that if \(K[{\mathcal A}]\) is normal and the convex polytope associated with \({\mathcal A}\) is simple, then \(K[{\mathcal A}]\) is generated by quadratic binomials.

Reviewer: Emanuela De Negri (Genova)

##### MSC:

13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |

13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |

20M25 | Semigroup rings, multiplicative semigroups of rings |

##### Keywords:

toric ideals associated with the affine semigroup ring; Koszul algebras; Gröbner bases; binomials
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\textit{H. Ohsugi} and \textit{T. Hibi}, J. Algebra 218, No. 2, 509--527 (1999; Zbl 0943.13014)

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