On the Structure of the Value Semigroup of a Valuation

Let v be a valuation of the quotient field of a noetherian local domain R. Assume that v is centered at R. This paper studies the structure of the value semigroup of v, S. Ideals defining toric varieties can be defined from the graded algebra K[T ] of cancellative commutative finitely generated semigroups such that T ∩ (−T ) = {0}. The value semigroup of a valuation S need not be finitely generated but we prove that S ∩ (−S) = {0} and so, the study in this paper can also be seen as a generalization to infinite dimension of that of toric varieties. In this paper, we prove that K[S] can be regarded as a module over an infinitely dimensional polynomial ring Av. We show a minimal graded resolution of K[S] as Av-module and we give an explicit method to obtain the syzygies of K[S] as Av-module. Finally, it is shown that free resolutions of K[S] as Av-module can be obtained from certain cell complexes related to the lattice associated to the kernel of the map Av → K[S].