SLIDE 9 Let k be an algebraically closed field.
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Let A be the k-algebra associated to the quiver
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1
a
/2
b
/3 with the relation ba = 0.
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The module T = 2
3 ⊕ 1 2 ⊕ 1 is a classical 2-tilting module.
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HomA(T, 2) ' Ext1
A(T, 2) 6= 0
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Consider the simple left A-module 2.
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= ⇒
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2 /
∈ KE`,` = 0,1,2.
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There is no way to filtrate the simple left A-module 2 with composition factors in the Miyashita classes.
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[T. 2002]
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A left A-module M admits a filtration
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M = Mn ≥ Mn−1 ≥ Mn−2 ≥ ··· ≥ M0 ≥ M−1 = 0
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such that for every i = 0,...,n the quotient Mi/Mi−1 belongs to KEi
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if and only if
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for every i 6= j 0,
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TorB
i (T,Extj A(T,M)) = 0.
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