The graded quiver q-Euler form at q = 1

Let Q be a finite quiver with finite-dimensional path algebra kQ, graded by path length. For finite-dimensional graded kQ-modules X and Y, evaluating the q-Euler characteristic χ_q(X, Y) at q = 1 gives the ordinary Ext-Euler characteristic χ(X, Y) of the underlying ungraded modules. On Grothendieck groups, the q-Euler form specialized at q = 1 is the ungraded Ext-Euler pairing pulled back along the isomorphism G₀^gr(mod kQ) ⊗ ℤ[q = 1] ≃ G₀(mod kQ) induced by forgetting the grading. The isomorphism and the vertex-dimension comparison are supplied by TauCeti.RepresentationTheory.GrothendieckGroup.Quiver.ForgetGrading.

On objects, both sides are computed by Ringel formulas, so no comparison of graded and ungraded Ext groups is used. The graded formula TauCeti.PathAlgebra.gradedExtEuler_eq_sum_smulGradedDimension has the idempotent graded dimensions gdim(eᵢ • X) as coordinates. At q = 1 these become the vertex dimensions dim_k(eᵢ X) of the underlying module, and the factor q⁻¹ on each arrow becomes 1. The result is the ungraded formula TauCeti.extEuler_eq_eulerForm_dimVector.

The graded/ungraded Ext comparison hypothesis TauCeti.IsGradedExtComparison gives the same numerical identity in general. The identity on Grothendieck groups is the path-algebra case of TauCeti.gradedExtEulerSpecialized_one_eq_extEulerPairing_gradedFiniteModules, since a finite-dimensional path algebra is Noetherian.

Main results

• TauCeti.PathAlgebra.laurentEval_one_gradedExtEuler_eq_extEuler: χ_q(X, Y) at q = 1 is the Ext-Euler characteristic of the underlying modules.
• TauCeti.PathAlgebra.laurentSpecialize_gradedExtEulerForm_eq_extEulerPairing: the q-Euler form on G₀^gr(mod kQ) specialized at q = 1 is the ungraded Ext-Euler pairing of the classes with their grading forgotten.

References

• I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras I, Chapter III, Section 3, for the Ringel form.
• Z. Dancso and A. Licata, Koszul algebras and flow lattices, Section 2.2, for the graded Grothendieck group, the q-Euler form and its specialization at q = 1.

theorem TauCeti.PathAlgebra.laurentEval_one_gradedExtEuler_eq_extEuler (k : Type (max u v)) (Q : Type u) [Field k] [Quiver Q] [Finite Q] [FiniteDimensional k (pathAlgebra k Q)] [CategoryTheory.HasExt (GradedModuleCat (integerGrade k Q))] (X Y : GradedModuleCat (integerGrade k Q)) [Module.Finite k X.carrier] [Module.Finite k Y.carrier] (h : IsGradedEulerAdmissible k (GradedModuleCat.shift (integerGrade k Q)) X Y) (h' : IsEulerAdmissible k (GradedModuleCat.toModuleCat.obj X) (GradedModuleCat.toModuleCat.obj Y)) : (laurentEval 1) (gradedExtEuler k (GradedModuleCat.shift (integerGrade k Q)) h) = extEuler k h'

The q-Euler characteristic of graded path-algebra modules at q = 1 is the ungraded Ext-Euler characteristic. For finite-dimensional graded kQ-modules X and Y, evaluating χ_q(X, Y) at q = 1 gives χ(X, Y) of the underlying modules. The two admissibility witnesses are arbitrary.

theorem TauCeti.PathAlgebra.laurentSpecialize_gradedExtEulerForm_eq_extEulerPairing (k : Type (max u v)) (Q : Type u) [Field k] [Quiver Q] [Finite Q] [FiniteDimensional k (pathAlgebra k Q)] [CategoryTheory.HasExt (GradedModuleCat (integerGrade k Q))] (x y : LaurentSpecialization 1 (LaurentK0 (gradedFiniteModulesExactStructure (integerGrade k Q)))) : (((gradedExtEulerForm k Q).laurentSpecialize ⋯) x) y = ((extEulerPairing ⋯ ⋯ ⋯) ((finiteModulesExactK0Equiv (pathAlgebra k Q)) ((gradedFiniteModulesForgetK0 (integerGrade k Q)) x))) ((finiteModulesExactK0Equiv (pathAlgebra k Q)) ((gradedFiniteModulesForgetK0 (integerGrade k Q)) y))

The graded quiver q-Euler form at q = 1 is the ungraded Ext-Euler pairing. On G₀^gr(mod kQ) specialized at q = 1, the q-Euler form is the Ext-Euler pairing on G₀(mod kQ) of the classes with their grading forgotten.