Minimal systems of generators for ideals of semigroups

On the structure of the value semigroup of a valuation

“…We shall call graph associated to C to a graph whose vertex set is the set of connected components of α which contain the support of a monomial belonging to a binomial in C. Two connected components, those associated to the monomials X a and X b , are adjacent by an edge whenever X a − X b ∈ C. C will be a generating tree for α if the graph associated to C is, in fact, a tree. This theorem is analogous to the stated in [2] for finitely generated semigroups and the proof runs similarly. It is based on the fact that two monomials M and M of degree α ∈ S satisfy M − M ∈ (M v I 0 ) α if, and only if, Supp (M ) and Supp (M ) are in the same connected component of α .…”

“…This set need not be finite. In the first part of this subsection, we shall use [2] to give a method to compute a minimal homogeneous generating set of I 0 , B, formed by binomials of the type described above. This method uses the structure of graph of the simplicial complex α .…”

“…It is based on the fact that two monomials M and M of degree α ∈ S satisfy M − M ∈ (M v I 0 ) α if, and only if, Supp (M ) and Supp (M ) are in the same connected component of α . Furthermore, it is possible to decide whether dim K H 0 ( α ) = 0 by a close method to that given in [2,Th. 3.11].…”

“…When S is finitely generated, this property is equivalent to S ∩ (−S) = {0}. There exists an extensive literature [5,2,1,4], which studies the graded algebra K[S] of cancellative commutative finitely generated semigroups S such that S ∩ (−S) = {0} (this study includes ideals defining toric varieties). Therefore, we devote Section 3 to extend to the S-graded algebra K[S], now S being the value semigroup of a valuation, the ideas of the toric case.…”

Minimal Resolutions of Lattice Ideals and Integer Linear Programming

“…Another application of the condition S ∩ (−S) = (0), is Nakayama's lemma for S-graded k[X]-modules (see [8]). Thus, there exists an S-graded minimal free resolution of k [S], which is unique up to isomorphism.…”