By Chazelle B., Goodman J.E., Pollack R. (eds.)

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The family Q B contains all modules from the component Q(A), and each of the indecomposable injective Bmodules belongs to Q B . Every indecomposable projective B-module P lies in P(B)∪TT B , and the radical rad P of P lies in add (P(B) ∪ T B ). 3). pd X ≤ 1, for each module X in P(B) ∪ T B , and id Y ≤ 1, for each module Y in Q B . dim B ≤ 2. 48 Chapter XV. Tubular extensions and coextensions Proof. Assume that A is a concealed algebra of Euclidean type. 4) that the Auslander–Reiten quiver Γ(mod A) of A has a disjoint union decomposition Γ(mod A) = P(A) ∪ T A ∪ Q(A) where P(A) is the unique postprojective component containing all the indecomposable projective A-modules, Q(A) is the unique preinjective component containing all the indecomposable injective A-modules, and T A = {TλA }λ∈P1 (K) is a P1 (K)-family of pairwise orthogonal hereditary standard stable tubes TλA .

0 0 0 1 1 00 0 0 0 .. Obviously, the rank of the component C3 is 7. Our ﬁnal objective in this section is to determine how the bound quiver of the algebra C, the tubular extension we start from, is modiﬁed. We know that the quivers of one-point extensions have the eﬀect of adding a source, and the quivers of line extensions have the eﬀect of adding an oriented line. We now see the eﬀect of the quiver QA of a tubular extension A of an algebra C. For this purpose, we need a deﬁnition. 38 Chapter XV.

S} consisting of all j such that the module Ej lies on TλA . (b) Q B = Q(B) = Q(A) is the unique preinjective component of Γ(mod B). An indecomposable B-module M belongs to Q(B) if the restriction resA M of M to A admits an indecomposable direct summand from Q(A). (c) P B is a family of components of Γ(mod B) consisting of all Bmodules X such that the restriction resA X of X to A is a module in add P(A). The family P B contains all modules from the component P(A), and each of the indecomposable projective B-modules belongs to P B .