Abstract

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 A_v. We show a minimal graded resolution of K[S] as A_v-module and we give an explicit method to obtain the syzygies of K[S] as A_v-module. Finally, it is shown that free resolutions of K[S] as A_v-module can be obtained from certain cell complexes related to the lattice associated to the kernel of the map A_v → K[S].

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