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Toric ideals generalized by quadratic binomials. (English) Zbl 0943.13014
Let \(K\) 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.

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
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