MEMOIRS

92 6. INSERTION OF WEIGHTSIf all d i =1,wewriteL(p) instead. Note that these groups can have a non-trivialtorsion part.Let R = k[X 1 ,X 2 ] be the polynomial algebra graded by total degree. Letλ 1 ,...,λ t be pairwise different elements in k ∪{∞}. Without loss of generalitywe assume λ 1 = ∞ and λ 2 =0. Letπ i ∈ R be the homogeneous prime elementX 2 + λ i X 1 for i =3,...,t and π 1 = X 1 , π 2 = X 2 . (In this case, all d i =1.)Successive insertion of weights p i > 1 into the primes π i leads to the L(p)-gradedalgebrak[X 1 ,X 2 ,...,X t ]/(X p ii − X p 22 − λ iX p 11 | i =3,...,t),which are just the projective coordinate algebras of the weighted projective linesdescribed in [34].Theorem 6.2.4. Let π ∈ R be a central prime element and x ∈ X be theassociated point. Let p ≥ 2 be an integer.(1) R = R[π 1/p ] is an H-graded factorial domain of Krull dimension two. Moreprecisely, the homogeneous prime ideals in R of height one are P = Rπ and P = Rq,where q ∈ R is prime and not associated to π. ( )(2) There is an equivalence modH (R) p ≃H .mod H 0 (R) xProof. (1) We have an embedding R ⊂ R, andR can be considered alsoas H-graded algebra. R = R[π] is a finite centralizing extension, since R = R ⊕Rπ ⊕···⊕Rπ p−1 . Hence, the intersection of a homogeneous prime ideal in R withR gives a homogeneous prime ideal in R, proper inclusion is preserved, and eachhomogeneous prime ideal in R is of this form (see [77, 10.]). Consequently, R is ofgraded Krull dimension two.By the definition of the grading, every homogeneous element a ∈ R has theform a = aπ l ,witha ∈ R homogeneous and 0 ≤ l ≤ p − 1. Hence, R is a gradeddomain like R.It is easy to see that πR ∩ R = πR, hence there is an isomorphism R/πR ≃R/πR, and it follows, that π is a central prime element in R.By the form of the homogeneous elements it follows easily that for a homogeneousprime ideal Rq ⊂ R different from Rπ, theidealRq is prime in R. Moreover,since the map P ↦→ P ∩ R preserves proper inclusions, we see that every homogeneousprime ideal of height one in R different from Rπ is of the form Rq, whereqis prime in R and not associated to π.(2) Denote by H the category of p-cycles concentrated in x. Let ˜· and Γ +be the functors as defined in 2.1.5. Extending this, we construct an exact functor˜· :mod H (R) −→ H with kernel mod H 0 (R). Denote by r :mod H (R) −→ mod H (R)the exact functor, given by restricting an H-graded module to the subgroup H.Obviously, r(R) =R. Moreover,M is of finite length over R if and only if r(M(ih))is of finite length over R for all i =0,...,p− 1. Hence r induces a functor r on thequotient categories. For M ∈ mod H (R) andi =0,...,p define(6.2.1) E i = ˜ r(M(ih)) ∈H.Then E p = E 0 (x) and˜M def= [ E 0·π−→ E 1·π−→ E 2 →···→E p−1·π−→ E 0 (x) ]