Ext_{Q}^{1} (V, W)

0 \to W \to E \to V \to 0

, with the zero class represented by a split sequence and addition given by the Baer sum. The extension complex of quiver representations gives

0 \to Hom_{Q} (V, W) \to ⨁_{i} Hom_{k} (V_{i}, W_{i}) γ_{V, W} ⨁_{ρ : i \to j} Hom_{k} (V_{i}, W_{j}) \to Ext_{Q}^{1} (V, W) \to 0.

The printed map has

γ (u)_{ρ} = u_{j} f_{ρ} - g_{ρ} u_{i}

, so its kernel is

Hom_{Q} (V, W)

and its cokernel is

Ext_{Q}^{1} (V, W)

. Reversing the overall differential sign changes neither identification.

v, w

⟨ v, w ⟩_{Q} = i \sum v_{i} w_{i} - ρ : i \to j \sum v_{i} w_{j} = dim Hom_{Q} (V, W) - dim Ext_{Q}^{1} (V, W) .

The first expression makes its dependence only on the dimension vectors explicit.

For the one-loop representation

V = k

with loop scalar

λ

γ (u) = u λ - λ u = 0

k

. Both cochain spaces have dimension one, so $dim Ext_{Q}^{1} (V, V) = 1$ , for every

λ

. Concretely, a self-extension has loop matrix

(λ 0 c λ)

, with

c

the extension parameter.

 Ancestors (10)

 Discussion (0)

There are no discussions about this article yet.

There are currently no matching articles.