Singularity Category of the Hecke Node

For the pair factor

R = Z_3[eta]/(eta^2 - 3 eta),

the CM indecomposables are

B0,   R,   B3.

The only projective/free one is R, so in the singularity category D_sg(R) it vanishes.

Therefore the singularity category has exactly two indecomposable objects:

B0,   B3.

Periodicity

From the matrix factorizations / syzygies,

Omega(B0) = B3,
Omega(B3) = B0.

Hence in the triangulated singularity category

Sigma(B0) = B3,
Sigma(B3) = B0,
Sigma^2 = Id.

So D_sg(R) is literally a two-object 2-periodic category.

Auslander-Reiten quiver

The non-split CM exact sequences

0 -> B3 -> R -> B0 -> 0,
0 -> B0 -> R -> B3 -> 0

descend to the singularity category as the two irreducible odd arrows between B0 and B3.

So the AR quiver is

B0  <-->  B3

with the translation swapping the two vertices.

Grothendieck group

In K_0(D_sg(R)), the exact sequence 0 -> B3 -> R -> B0 -> 0 gives

[B3] + [B0] = [R] = 0,

since R is perfect/projective and vanishes in the singularity category. Thus

[B3] = -[B0].

But Sigma(B0)=B3 and Sigma acts by negation on K_0, so

[B3] = -[B0]

again, and therefore the group is generated by one class of order 2:

K_0(D_sg(R)) ≅ Z/2Z.

Full singular Hecke order

For

O_sing ≅ Z_3(V7) × R,

the extra V7 branch is smooth/projective, so it disappears in the singularity category as well.

Therefore the full singular Hecke node has the same singularity category as the pair factor:

D_sg(O_sing) ≃ D_sg(R).

This is the most compressed form of the object so far: after modding out the smooth directions, the Hecke node is exactly a two-object 2-periodic singularity with K_0 = Z/2.