MEMOIRS

0.4. TUBULAR SHIFTS 19(which is equivalent to finite k-dimension). This is a Serre subcategory of mod H (R),that is, it is closed under subobjects, quotients and extensions. The quotient categorymod H (R)/ mod H 0 (R) is taken in the Serre-Grothendieck-Gabriel sense. Werefer to [85].Then the graded algebra R is called a projective coordinate algebra for X ifthere is an equivalence of categoriesH≃ modH (R)mod H 0 (R) .Each exceptional curve admits a projective coordinate algebra, even a Z-graded one(see 6.2.1). Thus, in the terminology of [2], H is a (noncommutative) noetherianprojective scheme.Note that a projective coordinate algebra is not uniquely determined by X.One of the main aims of this article is to show that there is a projective coordinatealgebra with “good” ringtheoretical properties.0.4. Tubular shiftsOne of the most important concepts we will use in these notes is that of shiftautomorphisms as developed in [70], which is a particular class of tubular mutations[71, 79, 80]. For the details we refer to [70]. Since we will also deal withthe degree shift of graded objects, we will call a shift automorphism in the senseof [70] atubular shift or just shift associated to a point.0.4.1. Let X be an exceptional curve with associated hereditary category Hand tubular family H 0 = ∐ x∈X U x, with connected uniserial length categories U xwhich are pairwise orthogonal. We fix a point x ∈ X of weight p(x). Let S x be asimple object in U x ,denotebyS x additive closure of the Auslander-Reiten orbit ofS x , which consists of the semisimple objects from U x .Let M be an object. By the semisimplicity of the category S x , for the objectp(x)⊕M x = Ext 1 (τ j S x ,M) ⊗ End(Sx ) τ j S xj=1there is a natural isomorphism of functors∼(0.4.1) η M :Hom(−,M x )| Sx −→ Ext 1 (−,M)| Sx ,which by the Yoneda lemma can be viewed as short exact sequenceη M :0−→ M α M−→ M(x) β M−→ M x −→ 0such that the Yoneda composition Hom(U, M x ) −→ Ext 1 (U, M), f ↦→ η M · f isan isomorphism for each U ∈ S x . η M is called the S x -universal extension ofM. (If p(x) = 1 we also call it S x -universal.) By means of the identificationHom(−,M x )| Sx =Ext 1 (−,M)| Sx the assignment M ↦→ M x extends to a functoru ↦→ u x for each u : M −→ N such that u · η M = η N · u x .ThenM x (u x ) is calledthe fibre of M (of u, resp.)inx.Similarly, letp(x)⊕xM = Hom(τ j S x ,M) ⊗ End(Sx ) τ j S x .j=1