Stable Ext Algebra of the Hecke Node

In the singularity category of the pair factor, the two nonfree indecomposables are

B0 = R/(eta),
B3 = R/(eta-3).

From the 2-periodic matrix factorizations, the graded Ext groups are one-dimensional in the unique parity where they can be nonzero:

• Ext^{2m}(B0,B0) ≅ F_3 for m >= 0
• Ext^{2m}(B3,B3) ≅ F_3 for m >= 0
• Ext^{2m+1}(B0,B3) ≅ F_3 for m >= 0
• Ext^{2m+1}(B3,B0) ≅ F_3 for m >= 0

Choose generators

u ∈ Ext^2(B0,B0) and Ext^2(B3,B3),
x ∈ Ext^1(B0,B3),
y ∈ Ext^1(B3,B0).

Since every relevant graded piece is one-dimensional, the multiplication is forced up to rescaling. After normalizing x and y, one gets

y x = u e0,
x y = u e3,

where e0,e3 are the vertex idempotents.

Therefore

Ext^*(B0 ⊕ B3, B0 ⊕ B3)
≅
[ e0, e3, x, y, u ] /
(e_i^2=e_i, e0e3=e3e0=0, yx=u e0, xy=u e3),

with

deg(x)=deg(y)=1,   deg(u)=2.

Equivalently, the stable Ext algebra is the 2-periodic two-vertex algebra with one odd arrow each way and one central degree-2 periodicity generator.

This is the sharpest stable-categorical form of the Hecke node so far: the free module disappears, and what remains is a two-object 2-periodic category whose entire algebra is generated by the two branch modules and the periodicity class.