Takuma Aihara

of finitely generated modules over a numerical semigroup ring k[[H]] over a field k, in termsof the conductor of the semigroup and the multiplicity of the ring.Corollary 9. Let k be a field and H be a numerical semigroup. Let R be the numericalsemigroup ring k[[H]], that is, the subring k[[t h |h ∈ H]] of S = k[[t]]. Then D b (mod R) =〈S ⊕ k〉 2 ⌈ holds. Hence c(H)e(R) ⌉+2⌈ ⌉ c(H)dim D b (mod R) ≤ 2 + 1.e(R)4. FinitenessIn this section, we consider finiteness of the dimension of the bounded derived categoryof finitely generated modules over a complete local ring. Let R be a commutative algebraover a field k. Rouquier [14] proved the finiteness of the dimension of D b (mod R) whenR is an affine k-algebra, where the fact that the enveloping algebra R ⊗ k R is Noetherianplayed a crucial role. The problem in the case where R is a complete local ring is that onecannot hope that R ⊗ k R is Noetherian. Our methods instead use the completion of theenveloping algebra, that is, the complete tensor product R ̂⊗ k R, which is a Noetherianring whenever R is a complete local ring with coefficient field k.Let R and S be commutative Noetherian complete local rings with maximal ideals mand n, respectively. Suppose that they contain fields and have the same residue field k,i.e., R/m ∼ = k ∼ = S/n. Then Cohen’s structure theorem yields isomorphismsR ∼ = k[[x 1 , . . . , x m ]]/(f 1 , . . . , f a ),S ∼ = k[[y 1 , . . . , y n ]]/(g 1 , . . . , g b ).We denote by R ̂⊗ k S the complete tensor product of R and S over k, namely,R ̂⊗ k S = lim ←− i,j(R/m i ⊗ k S/n j ).For r ∈ R and s ∈ S, we denote by r ̂⊗ s the image of r ⊗ s by the canonical ringhomomorphism R ⊗ k S → R ̂⊗ k S. Note that there is a natural isomorphismR ̂⊗ k S ∼ = k[[x 1 , . . . , x m , y 1 , . . . , y n ]]/(f 1 , . . . , f a , g 1 , . . . , g b ).Details of complete tensor products can be found in [15, Chapter V].Recall that a ring extension A ⊆ B is called separable if B is projective as a B ⊗ A B-module. This is equivalent to saying that the map B ⊗ A B → B given by x ⊗ y ↦→ xy isa split epimorphism of B ⊗ A B-modules.Now, let us prove our main theorem.Theorem 10. Let R be a Noetherian complete local commutative ring containing a fieldwith perfect residue field. Then there exist a finite number of prime ideals p 1 , . . . , p n ∈Spec R and an integer m ≥ 1 such thatHence one has dim D b (mod R) < ∞.D b (mod R) = 〈R/p 1 ⊕ · · · ⊕ R/p n 〉 m.–10–