Basic Representation Theory of Algebras by Ibrahim Assem & Flávio U. Coelho

Author:Ibrahim Assem & Flávio U. Coelho
Language: eng
Format: epub
ISBN: 9783030351182
Publisher: Springer International Publishing

where p is the inclusion. Identifying paths to their classes modulo the binding ideal, the basis of A as a k-vector space is {e 1, e 2, α, β, αβ, βα, αβα, βαβ, αβαβ}, that of P 1 is {e 1, α, αβ, αβα, αβαβ} and that of P 2 is {e 2, β, βα, βαβ}. We see easily that the morphism p is given by left multiplication by α. Applying the Nakayama functor ν yields an exact sequence

Denoting by the dual basis to the above basis of A, we see that the basis of I 2 is , see Lemma I.​2.​17. Because of the previous lemma, νp maps f ∈D(Ae 2) to the linear form ae 1↦f(aα). Thus,

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