For the pair factor
R = Z_3[eta]/(eta^2 - 3 eta),the three indecomposable torsion-free / CM modules are
B0 = R/(eta), R, B3 = R/(eta-3).| Hom | value |
|---|---|
Hom(B0,B0) |
Z_3 |
Hom(B0,R) |
(eta-3)R ≅ Z_3 |
Hom(B0,B3) |
0 |
Hom(R,B0) |
B0 ≅ Z_3 |
Hom(R,R) |
R |
Hom(R,B3) |
B3 ≅ Z_3 |
Hom(B3,B0) |
0 |
Hom(B3,R) |
eta R ≅ Z_3 |
Hom(B3,B3) |
Z_3 |
The cross-branch morphisms vanish:
Hom(B0,B3) = Hom(B3,B0) = 0.So the two branches do not talk to each other directly; they only communicate through the node module R.
The node sits between the two branches through the two quotient sequences
0 -> B3 -> R -> B0 -> 0,
0 -> B0 -> R -> B3 -> 0.More concretely:
eta R is the kernel of R -> R/(eta) = B0, and eta R ≅ B3.(eta-3)R is the kernel of R -> R/(eta-3) = B3, and (eta-3)R ≅ B0.So the node module R is the unique indecomposable object that contains both branches and presents each one as a quotient by the other.
Since
O_sing ≅ Z_3(V7) ⊕ R,the full indecomposable CM modules are
Z_3(V7), B0(V1), R(node), B3(V5).The V7 branch is completely orthogonal to the pair factor at the module level: all cross-Homs vanish by the central idempotent splitting.
That is the finite category we were circling: one isolated smooth branch and one three-object nodal CM category.