Idempotent coordinates on the graded Grothendieck group of finite graded modules

Let A be a finite-dimensional algebra over a field k, with homogeneous pieces 𝒜 : ℤ → Submodule k A, and let e ∈ 𝒜 0 be an idempotent of degree zero. For a finitely generated graded A-module M, the subspaces e • Mₚ are finite-dimensional and vanish for all but finitely many degrees p, so they have a graded dimension

gdim_e(M) = ∑ₚ dim_k(e • Mₚ) qᵖ ∈ ℤ[q,q⁻¹].

Because e is an idempotent of degree zero, M ↦ e • Mₚ is exact in each degree, so gdim_e is additive on short exact sequences of graded modules. Shifting the grading multiplies it by a power of q, so it descends to a ℤ[q,q⁻¹]-linear coordinate

TauCeti.gradedIdempotentCoordinate : G₀^gr(mod A) ⟶ ℤ[q,q⁻¹]

on the Laurent Grothendieck group of finite graded modules, the target of the graded Cartan map TauCeti.gradedCartanMap.

For e = 1 this is the graded dimension of M. In general e • Mₚ is the degree-p piece of the summand e • M. Suppose the graded simple modules are the shifts Sᵢ{d} of modules Sᵢ concentrated in degree zero, and the idempotents eᵢ satisfy dim_k(eᵢ • Sⱼ) = δᵢⱼ, as for the vertex idempotents of a basic algebra whose simple modules are one-dimensional. Then additivity along a graded composition series shows that dim_k(eᵢ • Mₚ) counts the composition factors of M isomorphic to Sᵢ{p}, so these coordinates read graded composition multiplicities, and in particular the entries of the graded Cartan matrix. Kronecker-delta coordinates give linear independence of classes over ℤ[q,q⁻¹].

On the graded projective Af generated by an idempotent f, the e-coordinate of the image under the graded Cartan map is the graded dimension ∑ₚ dim_k(eAf ∩ 𝒜 p) qᵖ of the corner eAf. Consequently, in a module basis dual to the idempotent coordinates of e₁, …, eₙ and a projective basis containing the classes [A fⱼ], the graded Cartan matrix has entries ∑ₚ dim_k(eᵢ A fⱼ ∩ 𝒜 p) qᵖ.

Main definitions

• TauCeti.gradedIdempotentCoordinate: the induced ℤ[q,q⁻¹]-linear coordinate on the graded Grothendieck group of finite graded modules.

Main results

• TauCeti.gradedIdempotentCoordinate_of: the coordinate of the class of M is gdim_e(M).
• TauCeti.linearIndependent_laurentK0_of_smulGradedDimension: classes with Kronecker-delta idempotent coordinates are linearly independent over ℤ[q,q⁻¹].
• TauCeti.coeff_gradedIdempotentCoordinate_gradedCartanMap_ofIdeal_span_singleton: the idempotent coordinates of the graded Cartan map at [Af] are the graded dimensions of the corners eAf.
• TauCeti.coeff_gradedCartanMatrix_of_ofIdeal_span_singleton: the entries of the graded Cartan matrix in a module basis dual to idempotent coordinates, at a column given by [A fⱼ].

References

• C. Năstăsescu and F. Van Oystaeyen, Methods of Graded Rings, Section 2.3, for graded modules and their degree shifts.
• Z. Dancso and A. Licata, "Koszul algebras and flow lattices", Section 2.2, for the graded Grothendieck group as a ℤ[q,q⁻¹]-module and graded dimensions.
• Ibrahim Assem, Daniel Simson and Andrzej Skowroński, Elements of the Representation Theory of Associative Algebras I, Chapter III, Section 3, for the ungraded coordinates dim_k(e M) on the Grothendieck group and the Cartan matrix.

The coordinate on the graded Grothendieck group

noncomputable def TauCeti.gradedIdempotentCoordinate {k : Type uk} [Field k] {A : Type uA} [Ring A] [Algebra k A] {𝒜 : ℤ → Submodule k A} [Module.Finite k A] {e : A} (he : IsIdempotentElem e) (he₀ : e ∈ 𝒜 0) : LaurentK0 (gradedFiniteModulesExactStructure 𝒜) →ₗ[LaurentPolynomial ℤ] LaurentPolynomial ℤ

The idempotent coordinate on the graded Grothendieck group of finite graded modules over a finite-dimensional algebra: the ℤ[q,q⁻¹]-linear map sending the class of M to the graded dimension ∑ₚ dim_k(e • Mₚ) qᵖ of e • M, for an idempotent e of degree zero.

Equations

• TauCeti.gradedIdempotentCoordinate he he₀ = TauCeti.LaurentK0.lift (TauCeti.gradedFiniteModulesExactStructure 𝒜) (TauCeti.smulGradedDimensionInvariant✝ he he₀)

@[simp]

theorem TauCeti.gradedIdempotentCoordinate_of {k : Type uk} [Field k] {A : Type uA} [Ring A] [Algebra k A] {𝒜 : ℤ → Submodule k A} [Module.Finite k A] {e : A} (he : IsIdempotentElem e) (he₀ : e ∈ 𝒜 0) (M : (gradedFiniteModules 𝒜).FullSubcategory) : (gradedIdempotentCoordinate he he₀) (LaurentK0.of (gradedFiniteModulesExactStructure 𝒜) M) = GradedModuleCat.smulGradedDimension e M.obj

The idempotent coordinate of a class. Evaluating gradedIdempotentCoordinate on the class [M] of a finite graded module recovers the graded dimension ∑ₚ dim_k(e • Mₚ) qᵖ of e • M.

theorem TauCeti.linearIndependent_laurentK0_of_smulGradedDimension {k : Type uk} [Field k] {A : Type uA} [Ring A] [Algebra k A] {𝒜 : ℤ → Submodule k A} [Module.Finite k A] {I : Type u_1} {e : I → A} (he : ∀ (i : I), IsIdempotentElem (e i)) (he₀ : ∀ (i : I), e i ∈ 𝒜 0) (S : I → (gradedFiniteModules 𝒜).FullSubcategory) (hne : Pairwise fun (i j : I) => GradedModuleCat.smulGradedDimension (e i) (S j).obj = 0) (hself : ∀ (i : I), GradedModuleCat.smulGradedDimension (e i) (S i).obj = 1) : LinearIndependent (LaurentPolynomial ℤ) fun (i : I) => LaurentK0.of (gradedFiniteModulesExactStructure 𝒜) (S i)

Classes with Kronecker-delta idempotent coordinates are linearly independent. If the graded dimension of eᵢ • Sⱼ is 1 for i = j and 0 otherwise, the classes of the finite graded modules Sⱼ are linearly independent over ℤ[q,q⁻¹].

The columns of the graded Cartan map at the projectives Af

theorem TauCeti.coeff_gradedIdempotentCoordinate_gradedCartanMap_ofIdeal_span_singleton {k : Type uk} [Field k] {A : Type uA} [Ring A] [Algebra k A] {𝒜 : ℤ → Submodule k A} [Module.Finite k A] [GradedAlgebra 𝒜] {e f : A} (he : IsIdempotentElem e) (he₀ : e ∈ 𝒜 0) (hf : IsIdempotentElem f) (hI : Ideal.IsHomogeneous 𝒜 (Ideal.span {f})) (p : ℤ) : ((gradedIdempotentCoordinate he he₀) ((gradedCartanMap 𝒜) (LaurentK0.of (gradedFiniteProjectiveModulesExactStructure 𝒜) { obj := GradedModuleCat.ofIdeal 𝒜 (Ideal.span {f}) hI, property := ⋯ }))).coeff p = ↑(Module.finrank k ↥(cornerSubmodule k e f ⊓ 𝒜 p))

The idempotent coordinates of the graded Cartan map at Af are graded corner dimensions. For idempotents e of degree zero and f, the coefficient of qᵖ in the e-coordinate of the image of [Af] in G₀^gr(mod A) is dim_k(eAf ∩ 𝒜 p).

theorem TauCeti.coeff_gradedCartanMatrix_of_ofIdeal_span_singleton {k : Type uk} [Field k] {A : Type uA} [Ring A] [Algebra k A] {𝒜 : ℤ → Submodule k A} [Module.Finite k A] [GradedAlgebra 𝒜] {f : A} {I : Type u_1} {J : Type u_2} [Fintype I] [Finite J] (bP : Module.Basis I (LaurentPolynomial ℤ) (LaurentK0 (gradedFiniteProjectiveModulesExactStructure 𝒜))) (bM : Module.Basis J (LaurentPolynomial ℤ) (LaurentK0 (gradedFiniteModulesExactStructure 𝒜))) {e : J → A} (he : ∀ (i : J), IsIdempotentElem (e i)) (he₀ : ∀ (i : J), e i ∈ 𝒜 0) (hne : Pairwise fun (i i' : J) => (gradedIdempotentCoordinate ⋯ ⋯) (bM i') = 0) (hself : ∀ (i : J), (gradedIdempotentCoordinate ⋯ ⋯) (bM i) = 1) (j : I) (hf : IsIdempotentElem f) (hI : Ideal.IsHomogeneous 𝒜 (Ideal.span {f})) (hj : bP j = LaurentK0.of (gradedFiniteProjectiveModulesExactStructure 𝒜) { obj := GradedModuleCat.ofIdeal 𝒜 (Ideal.span {f}) hI, property := ⋯ }) (i : J) (p : ℤ) : (gradedCartanMatrix 𝒜 bP bM i j).coeff p = ↑(Module.finrank k ↥(cornerSubmodule k (e i) f ⊓ 𝒜 p))

The graded Cartan matrix in idempotent coordinates. Let bM be a basis of G₀^gr(mod A) dual to the idempotent coordinates of degree-zero idempotents eᵢ, in the sense that the eᵢ-coordinate of bM i' is 1 for i' = i and 0 otherwise, and let the j-th projective basis vector be the class of A fⱼ for an idempotent fⱼ. Then the coefficient of qᵖ in the (i, j) entry of the graded Cartan matrix is dim_k(eᵢ A fⱼ ∩ 𝒜 p).