Graded Euler-admissibility and the graded Cartan isomorphism

Let A be a finite-dimensional algebra over a field k, nonnegatively graded with semisimple degree-zero part, and let S be an exhaustive family of graded simples up to internal shift, indexed by a finite type. If every pair (Sᵢ, Sⱼ) is graded Euler-admissible, then every finite graded module has finite projective dimension, so it has a finite resolution by finite graded projectives. Admissibility of (M, Sⱼ) gives a cohomological bound on Extⁿ(M, Sⱼ{d}) that is uniform in the internal degree d, and every finite simple graded module is some Sⱼ{d}; hence Extⁿ(M, -) vanishes on all finite simples from some degree on, and this bounds the projective dimension of M (TauCeti.GradedProjectiveResolution.hasProjectiveDimensionLT_of_subsingleton_ext_simple).

Consequently the graded Cartan map is an isomorphism of ℤ[q,q⁻¹]-modules, and the graded Cartan matrix in any projective and module bases indexed by the same finite type has unit determinant. Nothing is assumed about the field splitting the simple modules, so this applies to the bases TauCeti.gradedIdempotentProjectiveBasisOfPrimitive and TauCeti.gradedIdempotentHeadBasisOfPrimitive of a family of primitive idempotents (TauCeti.isUnit_det_gradedCartanMatrix_gradedIdempotentProjectiveBasisOfPrimitive).

Admissibility of the simples is a genuine hypothesis: for the dual numbers k[ε]/(ε²) with ε in degree two, the graded Cartan matrix is [1 + q²], whose determinant is not a unit of ℤ[q,q⁻¹].

Main results

• TauCeti.gradedFiniteModules_le_admitsFiniteResolution_of_isGradedEulerAdmissible: every finite graded module has a finite resolution by finite graded projectives.
• TauCeti.gradedCartanMap_bijective_of_isGradedEulerAdmissible: the graded Cartan map is bijective.
• TauCeti.isUnit_det_gradedCartanMatrix_of_isGradedEulerAdmissible: graded Cartan unimodularity in any two bases indexed by the same finite type.

References

• Charles A. Weibel, An Introduction to Homological Algebra, Section 4.1, for projective dimension, and The K-book, Chapter II, Theorem 7.6, for the resolution theorem.
• A. Beilinson, V. Ginzburg and W. Soergel, Koszul duality patterns in representation theory, Section 1.2, for minimal resolutions over a positively graded ring with semisimple degree-zero part.
• Z. Dancso and A. Licata, "Koszul algebras and flow lattices", Section 2.2, for the graded Cartan matrix over ℤ[q,q⁻¹].

theorem TauCeti.gradedFiniteModules_le_admitsFiniteResolution_of_isGradedEulerAdmissible {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)] (hneg : ∀ p < 0, 𝒜 p = ⊥) {I : Type uI} [Finite I] {S : I → (gradedFiniteModules 𝒜).FullSubcategory} (hS : IsExhaustiveGradedSimpleFamily S) (h : ∀ (i j : I), IsGradedEulerAdmissible k (GradedModuleCat.shift 𝒜) (S i).obj (S j).obj) : gradedFiniteModules 𝒜 ≤ (gradedModuleCanonicalExactStructure 𝒜).admitsFiniteResolution (gradedFiniteProjectiveModules 𝒜)

Euler-admissible simples force finite resolutions. Over a finite-dimensional nonnegatively graded algebra with semisimple degree-zero part, if every pair of members of an exhaustive family of graded simples is graded Euler-admissible, then every finite graded module has a finite resolution by finite graded projectives.

theorem TauCeti.gradedCartanMap_bijective_of_isGradedEulerAdmissible {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)] (hneg : ∀ p < 0, 𝒜 p = ⊥) {I : Type uI} [Finite I] {S : I → (gradedFiniteModules 𝒜).FullSubcategory} (hS : IsExhaustiveGradedSimpleFamily S) (h : ∀ (i j : I), IsGradedEulerAdmissible k (GradedModuleCat.shift 𝒜) (S i).obj (S j).obj) : Function.Bijective ⇑(gradedCartanMap 𝒜)

The graded Cartan map is an isomorphism when the simples are Euler-admissible. Over a finite-dimensional nonnegatively graded algebra with semisimple degree-zero part, if every pair of members of an exhaustive family of graded simples is graded Euler-admissible, then the graded Cartan map K₀^gr(proj A) ⟶ G₀^gr(mod A) is bijective.

theorem TauCeti.isUnit_det_gradedCartanMatrix_of_isGradedEulerAdmissible {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)] (hneg : ∀ p < 0, 𝒜 p = ⊥) {I : Type uI} [Finite I] {S : I → (gradedFiniteModules 𝒜).FullSubcategory} (hS : IsExhaustiveGradedSimpleFamily S) (h : ∀ (i j : I), IsGradedEulerAdmissible k (GradedModuleCat.shift 𝒜) (S i).obj (S j).obj) {J : Type u_1} [Fintype J] [DecidableEq J] (bP : Module.Basis J (LaurentPolynomial ℤ) (LaurentK0 (gradedFiniteProjectiveModulesExactStructure 𝒜))) (bM : Module.Basis J (LaurentPolynomial ℤ) (LaurentK0 (gradedFiniteModulesExactStructure 𝒜))) : IsUnit (gradedCartanMatrix 𝒜 bP bM).det

Graded Cartan unimodularity from Euler-admissible simples. Over a finite-dimensional nonnegatively graded algebra with semisimple degree-zero part, if every pair of members of an exhaustive family of graded simples is graded Euler-admissible, then the graded Cartan matrix in a projective basis and a module basis indexed by the same finite type has determinant a unit of ℤ[q,q⁻¹]. No splitting hypothesis on the field is needed.