MEMOIRS

62 3. TUBULAR SHIFTS AND PRIME ELEMENTSProof. Let M ∈H + be of rank r>0. Let S = S x be the simple objectconcentrated in x. There is the universal extension0 −→ M α M−→ σ x (M) −→ Ext 1 (S, M) ⊗ End(S) S −→ 0.The morphism M ·π x−→ M(d) is a monomorphism and its cokernel is concentrated inthe point x and of length r · e(x): using a line bundle filtration of M and induction,it suffices to show this for line bundles L ′ . If L ′ is a shift of L then the cokernelis Sxe(x) . In orbit case III we have to consider also the case L ′ = L. There is anirreducible map L −→ uL, and the cokernel S 0 is a simple object. One can assumethat S 0 ≄ S x . It then follows that the cokernel of the map L ·π x−→ L(d) isalsoisomorphic to Sx e(x) .We have to show that the cokernel C of M ·π x−→ M(d) is semisimple: The mapC ·π x−→ C(d) is zero, which follows from the commutative exact diagram0 M ·π xM(d)·π x·π x (d)0 M(d) ·π x(d)M(2d)p(d)pC·π x0C(d) 0.Applying the exact functor φ ◦ q x from 2.2.6 we get a short exact sequence ofright R P -modules:·π0 −→ MxP −→ M(d)P −→ C P −→ 0.Since the map C ·π x−→ C(d) is zero, we see that C P is a graded R P /P P -module,hence semisimple by 2.2.8. It follows that C is a direct sum of copies of S.Hence we get a commutative, exact diagram0 M α MM(x)β MM x 0i M0 M ·π xM(d)j M0,Cwith isomorphisms i M and j M (compare 0.4.2 (2)). With the uniqueness property0.4.2 (1) of σ x (f) it follows easily that for each f ∈ Hom(M,N)(withN ∈H + )we have σ x (f)◦i M = i N ◦f(d). Therefore, the functors σ d and σ x are naturally isomorphicon H + . (Actually, the argument shows, that this holds everywhere outsideU x .) With Lemma 1.2.2, presenting each object in H 0 as cokernel of a monomorphismbetween objects from H + , the result follows by diagram chasing. □Corollary 3.1.3. Let x, y ∈ X such that the corresponding prime elementsπ x and π y are central of the same degree in R. Then the tubular shifts σ x and σ yare isomorphic.□3.2. Non-central prime elements and ghosts3.2.1 (Normal multiplication). We generalize Theorem 3.1.2 to arbitrary primeelements. Therefore, assume that r ∈ R is a (non-zero) normal element of degree n.Then r induces the automorphism γ = γ r of the graded algebra R, by the formula