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].
