The q-Euler values of graded quiver vertex simples

For the path-length grading on a path algebra, the first-arrow resolution computes

χ_q(Sᵢ, Y) = gdim(eᵢ Y) - q⁻¹ ∑_{a : i ⟶ j} gdim(eⱼ Y).

In particular, χ_q(Sᵢ, Sⱼ) = δᵢⱼ - q⁻¹ # {a : i ⟶ j}. The inverse power of q comes from the shift by one of the source projectives in the arrow kernel: the form is q-antilinear in its first variable. The general formula needs a finite quiver with finitely many arrows and a finite-dimensional target. The simple-pair specialization uses finite-dimensionality of the path algebra to obtain finite arrow sets and finite-dimensional vertex simples.

These are object-level values, computed with explicit Euler-admissibility witnesses. They supply the entries for comparing the graded categorical pairing with the graded quiver form.

References

• I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras I, Chapter III, Sections 2--3, for the first-arrow resolution and Euler form.
• Z. Dancso and A. Licata, "Koszul algebras and flow lattices", Section 2.2, for the q-antilinear/q-linear convention.

@[simp]

theorem TauCeti.PathAlgebra.gradedExtEuler_gradedVertexProjective (k : Type w) (Q : Type u) [Field k] [Quiver Q] [Finite Q] [CategoryTheory.HasExt (GradedModuleCat (integerGrade k Q))] (i : Q) (Y : GradedModuleCat (integerGrade k Q)) [Module.Finite k Y.carrier] (h : IsGradedEulerAdmissible k (GradedModuleCat.shift (integerGrade k Q)) (gradedVertexProjective k Q i) Y) : gradedExtEuler k (GradedModuleCat.shift (integerGrade k Q)) h = GradedModuleCat.smulGradedDimension (vertexIdempotent k i) Y

A vertex projective evaluates the q-Euler characteristic at its idempotent graded dimension, for any finite-dimensional graded target.

theorem TauCeti.PathAlgebra.gradedExtEuler_gradedArrowSum (k : Type w) (Q : Type u) [Field k] [Quiver Q] [Fintype Q] [(a b : Q) → Fintype (a ⟶ b)] [CategoryTheory.HasExt (GradedModuleCat (integerGrade k Q))] (i : Q) (Y : GradedModuleCat (integerGrade k Q)) [Module.Finite k Y.carrier] (h : IsGradedEulerAdmissible k (GradedModuleCat.shift (integerGrade k Q)) (gradedArrowSum k Q i) Y) : gradedExtEuler k (GradedModuleCat.shift (integerGrade k Q)) h = LaurentPolynomial.T (-1) * ∑ a : (j : Q) × (i ⟶ j), GradedModuleCat.smulGradedDimension (vertexIdempotent k a.fst) Y

The q-Euler value of the arrow kernel is q⁻¹ times the sum of the idempotent graded dimensions at the targets of the arrows. Each arrow is counted separately.

theorem TauCeti.PathAlgebra.gradedExtEuler_gradedVertexSimple (k : Type w) (Q : Type u) [Field k] [Quiver Q] [Fintype Q] [(a b : Q) → Fintype (a ⟶ b)] [CategoryTheory.HasExt (GradedModuleCat (integerGrade k Q))] (i : Q) (Y : GradedModuleCat (integerGrade k Q)) [Module.Finite k Y.carrier] (h : IsGradedEulerAdmissible k (GradedModuleCat.shift (integerGrade k Q)) (gradedVertexSimple k Q i) Y) : gradedExtEuler k (GradedModuleCat.shift (integerGrade k Q)) h = GradedModuleCat.smulGradedDimension (vertexIdempotent k i) Y - LaurentPolynomial.T (-1) * ∑ a : (j : Q) × (i ⟶ j), GradedModuleCat.smulGradedDimension (vertexIdempotent k a.fst) Y

The graded first-arrow Euler formula. Pairing a vertex simple against a finite-dimensional graded module subtracts the shifted outgoing-arrow coordinates from its own vertex coordinate.

@[simp]

theorem TauCeti.PathAlgebra.gradedExtEuler_gradedVertexSimple_gradedVertexSimple (k : Type w) (Q : Type u) [Field k] [Quiver Q] [Finite Q] [CategoryTheory.HasExt (GradedModuleCat (integerGrade k Q))] [FiniteDimensional k (pathAlgebra k Q)] [DecidableEq Q] (i j : Q) (h : IsGradedEulerAdmissible k (GradedModuleCat.shift (integerGrade k Q)) (gradedVertexSimple k Q i) (gradedVertexSimple k Q j)) : gradedExtEuler k (GradedModuleCat.shift (integerGrade k Q)) h = (if i = j then 1 else 0) - LaurentPolynomial.T (-1) * ↑(Nat.card (i ⟶ j))

The q-Euler matrix entries in the graded vertex-simple basis are δᵢⱼ - q⁻¹ # {arrows i ⟶ j}. Rows are source vertices and columns are target vertices.