Projective/simple q-Euler duality over a nonsplitting field

Let A be a nonnegatively graded finite-dimensional algebra over a field k whose degree-zero part is a semisimple ring, and let eᵢ be a complete family of orthogonal primitive idempotents of degree zero whose left ideals A eᵢ are pairwise nonisomorphic. The classes [A eᵢ] form a Laurent basis of K₀^gr(proj A) (TauCeti.gradedIdempotentProjectiveBasisOfPrimitive) and the classes of the heads Sᵢ = A eᵢ / A₊eᵢ form a Laurent basis of G₀^gr(mod A) (TauCeti.gradedIdempotentHeadBasisOfPrimitive).

In these bases the projective/module q-Euler pairing is diagonal with entries the constants dim_k End_A(Sᵢ), the dimensions of the Schur division algebras. Consequently both of its radicals vanish, but it identifies either Grothendieck group with the dual of the other exactly when every Sᵢ has dim_k End_A(Sᵢ) = 1, that is, when k splits the simple modules. This is the graded form of the general-field correction to the duality between projective covers and simple modules: nondegeneracy survives over every field, while perfectness of the raw q-Euler pairing does not.

Main results

• TauCeti.gradedProjectiveExtEuler_nondegenerate_of_isPrimitiveIdempotent: both radicals of the q-Euler pairing vanish.
• TauCeti.gradedProjectiveExtEuler_bijective_iff_finrank_end_eq_one and TauCeti.gradedProjectiveExtEuler_flip_bijective_iff_finrank_end_eq_one: the q-Euler pairing identifies either side with the dual of the other exactly when dim_k End_A(Sᵢ) = 1 for every i.

References

• Peter Webb, A Course in Finite Group Representation Theory, Chapter 7, Proposition 7.4.1, for the factor dim_k End_A(Sᵢ) in the pairing of projective covers with simples.
• Zsuzsanna Dancso and Anthony Licata, "Koszul algebras and flow lattices", Sections 2.2 and 3.1--3.2, for the q-Euler form and its duals.

theorem TauCeti.gradedProjectiveExtEuler_nondegenerate_of_isPrimitiveIdempotent {k : Type uk} [Field k] {A : Type uA} [Ring A] [Algebra k A] [Module.Finite k A] {𝒜 : ℤ → Submodule k A} [GradedAlgebra 𝒜] [IsSemisimpleRing ↥(𝒜 0)] [CategoryTheory.HasExt (GradedModuleCat 𝒜)] {I : Type uI} [Fintype I] {e : I → A} (hneg : ∀ p < 0, 𝒜 p = ⊥) (he : CompleteOrthogonalIdempotents e) (he₀ : ∀ (i : I), e i ∈ 𝒜 0) (hprim : ∀ (i : I), IsPrimitiveIdempotent (e i)) (hnoniso : Pairwise fun (i j : I) => IsEmpty (↥(Ideal.span {e i}) ≃ₗ[A] ↥(Ideal.span {e j}))) : (gradedProjectiveExtEuler 𝒜).Nondegenerate

The projective/module q-Euler pairing is nondegenerate over every field for a nonnegatively graded finite-dimensional algebra with semisimple degree-zero part and a complete family of orthogonal primitive idempotents of degree zero with pairwise nonisomorphic left ideals: the only projective class pairing to zero with every finite graded module is zero, and dually.

theorem TauCeti.gradedProjectiveExtEuler_bijective_iff_finrank_end_eq_one {k : Type uk} [Field k] {A : Type uA} [Ring A] [Algebra k A] [Module.Finite k A] {𝒜 : ℤ → Submodule k A} [GradedAlgebra 𝒜] [IsSemisimpleRing ↥(𝒜 0)] [CategoryTheory.HasExt (GradedModuleCat 𝒜)] {I : Type uI} [Fintype I] {e : I → A} (hneg : ∀ p < 0, 𝒜 p = ⊥) (he : CompleteOrthogonalIdempotents e) (he₀ : ∀ (i : I), e i ∈ 𝒜 0) (hprim : ∀ (i : I), IsPrimitiveIdempotent (e i)) (hnoniso : Pairwise fun (i j : I) => IsEmpty (↥(Ideal.span {e i}) ≃ₗ[A] ↥(Ideal.span {e j}))) : Function.Bijective ⇑(gradedProjectiveExtEuler 𝒜) ↔ ∀ (i : I), Module.finrank k (Module.End A (gradedPositiveMulQuotient 𝒜 ⋯).carrier) = 1

The q-Euler pairing identifies the projective group with the dual of the module group exactly over a splitting field. For a nonnegatively graded finite-dimensional algebra with semisimple degree-zero part and a complete family of orthogonal primitive idempotents eᵢ of degree zero with pairwise nonisomorphic left ideals, every linear functional on G₀^gr(mod A) is χ_q(x, -) for exactly one projective class x if and only if every head Sᵢ = A eᵢ / A₊eᵢ has dim_k End_A(Sᵢ) = 1.

theorem TauCeti.gradedProjectiveExtEuler_flip_bijective_iff_finrank_end_eq_one {k : Type uk} [Field k] {A : Type uA} [Ring A] [Algebra k A] [Module.Finite k A] {𝒜 : ℤ → Submodule k A} [GradedAlgebra 𝒜] [IsSemisimpleRing ↥(𝒜 0)] [CategoryTheory.HasExt (GradedModuleCat 𝒜)] {I : Type uI} [Fintype I] {e : I → A} (hneg : ∀ p < 0, 𝒜 p = ⊥) (he : CompleteOrthogonalIdempotents e) (he₀ : ∀ (i : I), e i ∈ 𝒜 0) (hprim : ∀ (i : I), IsPrimitiveIdempotent (e i)) (hnoniso : Pairwise fun (i j : I) => IsEmpty (↥(Ideal.span {e i}) ≃ₗ[A] ↥(Ideal.span {e j}))) : Function.Bijective ⇑(gradedProjectiveExtEuler 𝒜).flip ↔ ∀ (i : I), Module.finrank k (Module.End A (gradedPositiveMulQuotient 𝒜 ⋯).carrier) = 1

The q-Euler pairing identifies the module group with the semilinear dual of the projective group exactly over a splitting field. For a nonnegatively graded finite-dimensional algebra with semisimple degree-zero part and a complete family of orthogonal primitive idempotents eᵢ of degree zero with pairwise nonisomorphic left ideals, every q ↦ q⁻¹ semilinear functional on K₀^gr(proj A) is χ_q(-, y) for exactly one module class y if and only if every head Sᵢ = A eᵢ / A₊eᵢ has dim_k End_A(Sᵢ) = 1.