 Standard projective resolution of a quiver representation

ID: standard-projective-resolution-of-a-quiver-representation

A = k Q

P_{i} = A e_{i}

X_{i} = e_{i} X

0 \to ⨁_{ρ : i \to j} P_{j} \otimes_{k} X_{i} d ⨁_{i} P_{i} \otimes_{k} X_{i} \to X \to 0,

where

d (p \otimes x) = pρ \otimes x - p \otimes f_{ρ} (x)

in the source and target summands, and the augmentation is

p \otimes x \mapsto p x

. Each

P_{i}

is a summand of

A

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