Toric Structure of the Graded Algebra Relative to a Valuation

Throughout this paper, v will denote a valuation of the quotient field F of a Noetherian local domain (R, M). Also, we shall assume that v is centered at R. Denote by K the residue field of R and by S (:= v(R\{0}) the value semigroup of v. For each m ∈ S, set P _m (P _m +) := { F ∈ R|v(f ≥ (>) m}). P _m and P _m + are ideals of R and we call the graded algebra of R relative to v to the S-graded K-algebra $$ gr_v R = \mathop \oplus \limits_{m \in S} P_m /P_{^m }^ + . $$