# Download e-book for kindle: Algebraic surfaces and golomorphic vector bundles by Friedman.

By Friedman.

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**Example text**

This shows that εa (KQ)εb = 0. Similarly, εb (KQ)εa = 0. 6), KQ is not connected. Suppose now that Q is connected but KQ is not. 6), there exists a ˙ 0 such that, if x ∈ Q0 and y ∈ Q0 , then disjoint union partition Q0 = Q0 ∪Q εx (KQ)εy = 0 = εy (KQ)εx . Because Q is connected, there exist a ∈ Q0 and b ∈ Q0 that are neighbours. Without loss of generality, we may suppose that there exists an arrow α : a → b. But then we have α = εa αεb ∈ εa (KQ)εb = 0, a contradiction that completes the proof of the lemma.

Then A is a connected algebra if and only if there does not ˙ of the set {1, 2, . . , n} such that i ∈ I and exist a nontrivial partition I ∪J j ∈ J imply ei Aej = 0 = ej Aei . ej . Proof. Assume that there exists such a partition and let c = j∈J Because the partition is nontrivial, c = 0, 1. Because the ej are orthogonal idempotents, c is an idempotent. Moreover, cei = ei c = 0 for each i ∈ I, and cej = ej c = ej for each j ∈ J. Let now a ∈ A be arbitrary. 1. Quivers and path algebras hypothesis, ei aej = 0 = ej aei whenever i ∈ I and j ∈ J.

Ej . Proof. Assume that there exists such a partition and let c = j∈J Because the partition is nontrivial, c = 0, 1. Because the ej are orthogonal idempotents, c is an idempotent. Moreover, cei = ei c = 0 for each i ∈ I, and cej = ej c = ej for each j ∈ J. Let now a ∈ A be arbitrary. 1. Quivers and path algebras hypothesis, ei aej = 0 = ej aei whenever i ∈ I and j ∈ J. Consequently ca = ( ej )a = j∈J j∈J ej aek = ej a) · 1 = ( ( = j,k∈J ( ej a)( j∈J ej + j∈J ei )a( i∈I ei + i∈I ek ) k∈J ek ) = ac.

### Algebraic surfaces and golomorphic vector bundles by Friedman.

by Ronald

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