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TauCeti.RepresentationTheory.Quiver.Zigzag.Projective.LinearResolution

Linear resolutions over a zigzag algebra #

Let Z be the zigzag relation quotient of a finite simple graph G without isolated vertices, graded by path length, with vertex idempotents e_u, arrows a_d and volume classes x_u.

Graded projectives generated in one degree #

Let P be a graded Z-module whose underlying module is projective and which is generated by its homogeneous piece P_n. Then P is determined degreewise by the vertex parts e_u • P_n:

e_u • P_n ≅ e_u • P_{n+2},                 x ↦ x_u • x,
⨁_{w ∼ u} e_w • P_n ≅ e_u • P_{n+1},         (x_w)_w ↦ ∑_w a_{w → u} • x_w,

and P vanishes in all other degrees. The maps are onto because Z is spanned by idempotents, arrows and volume classes. The volume map is injective because P is a direct summand of a free module: an element x of e_u • P_n with x_u • x = 0 has all its coefficients in the radical, so it lies in Z₊ P, which has no component in the lowest degree n. Multiplying by reverse arrows reduces the injectivity of the arrow map to that of the volume maps.

Linear resolutions #

Let M be a finite-dimensional graded Z-module concentrated in degree 0, and let ⋯ ⟶ X₁ ⟶ X₀ ⟶ M be a linear graded projective resolution, so X_n is generated in degree n. Write m_n(u) = dim_k (e_u • (X_n)_n), the multiplicity of the shifted projective P_u{n} in X_n. In internal degree p only X_p, X_{p-1} and X_{p-2} are nonzero, so the resolution restricts to short exact sequences 0 ⟶ e_u (X_p)_p ⟶ e_u (X_{p-1})_p ⟶ e_u (X_{p-2})_p ⟶ 0. Counting dimensions with the description above gives the recursion

m_{p+2}(u) + m_p(u) = ∑_{w ∼ u} m_{p+1}(w),     m_1(u) = ∑_{w ∼ u} m_0(w),     m_0(u) = dim e_u M,

that is, C_G(-X) m(X) = dim M for the graded Cartan matrix C_G(q) = (1 + q²) I + q A_G (TauCeti.zigzagGradedCartanMatrix_map_aeval_neg_X_mulVec_eq). For the graded simple module S_v this says that the multiplicities of any linear resolution of S_v are the entries of the column v of C_G(-X)⁻¹ (TauCeti.coeff_inv_zigzagGradedCartanMatrix_map_aeval_neg_X_eq). In particular, if S_v has a linear resolution then every coefficient in that column is nonnegative: nonnegativity of C_G(-X)⁻¹ is a necessary condition for the zigzag algebra to be Koszul.

Main results #

References #

Graded projectives generated in one degree #

The volume map #

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theorem TauCeti.finrank_zigzagVertexIdempotent_smul_piece_add_two {k : Type w} [Field k] {V : Type u} {G : SimpleGraph V} [Finite V] {P : GradedModuleCat (zigzagIntegerGrade k G)} {n : ℤ} [Module.Projective (nonisolatedZigzagQuotient k G) P.carrier] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) (hP : P.grading.IsGeneratedInDegree (nonisolatedZigzagQuotient k G) n) (u : V) :
Module.finrank k ↥(zigzagVertexIdempotent k G u • P.grading.piece ℤ (n + 2)) = Module.finrank k ↥(zigzagVertexIdempotent k G u • P.grading.piece ℤ n)

Degree n + 2 of a graded projective generated in degree n. If P is a graded projective over the zigzag algebra generated in degree n, then multiplication by the volume class identifies e_u • P_n with e_u • P_{n+2}.

The arrow map #

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theorem TauCeti.finrank_zigzagVertexIdempotent_smul_piece_add_one {k : Type w} [Field k] {V : Type u} {G : SimpleGraph V} [Finite V] {P : GradedModuleCat (zigzagIntegerGrade k G)} {n : ℤ} [Fintype V] [DecidableRel G.Adj] [Module.Projective (nonisolatedZigzagQuotient k G) P.carrier] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) (hP : P.grading.IsGeneratedInDegree (nonisolatedZigzagQuotient k G) n) [∀ (w : V), FiniteDimensional k ↥(zigzagVertexIdempotent k G w • P.grading.piece ℤ n)] (u : V) :
Module.finrank k ↥(zigzagVertexIdempotent k G u • P.grading.piece ℤ (n + 1)) = ∑ w ∈ G.neighborFinset u, Module.finrank k ↥(zigzagVertexIdempotent k G w • P.grading.piece ℤ n)

Degree n + 1 of a graded projective generated in degree n. If P is a graded projective over the zigzag algebra generated in degree n, with finite-dimensional vertex parts e_w • P_n, then the arrows into u identify e_u • P_{n+1} with ⨁_{w ∼ u} e_w • P_n.

Linear resolutions #

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theorem TauCeti.finrank_zigzagVertexIdempotent_smul_piece_zero_of_isLinear {k : Type w} [Field k] {V : Type u} {G : SimpleGraph V} [Finite V] {M : GradedModuleCat (zigzagIntegerGrade k G)} {r : GradedProjectiveResolution (zigzagIntegerGrade k G) M.grading} (hr : r.IsLinear) (u : V) :
Module.finrank k ↥(zigzagVertexIdempotent k G u • (r.grading 0).piece ℤ 0) = Module.finrank k ↥(zigzagVertexIdempotent k G u • M.grading.piece ℤ 0)

Degree zero of a linear resolution: the augmentation identifies the lowest vertex parts of X₀ with those of the resolved module.

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theorem TauCeti.finrank_zigzagVertexIdempotent_smul_piece_one_of_isLinear {k : Type w} [Field k] {V : Type u} {G : SimpleGraph V} [Finite V] {M : GradedModuleCat (zigzagIntegerGrade k G)} {r : GradedProjectiveResolution (zigzagIntegerGrade k G) M.grading} [Module.Finite k M.carrier] [Fintype V] [DecidableRel G.Adj] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) (hr : r.IsLinear) {u : V} (hM : zigzagVertexIdempotent k G u • M.grading.piece ℤ 1 = ⊥) :
Module.finrank k ↥(zigzagVertexIdempotent k G u • (r.grading 1).piece ℤ 1) = ∑ w ∈ G.neighborFinset u, Module.finrank k ↥(zigzagVertexIdempotent k G w • (r.grading 0).piece ℤ 0)

Degree one of a linear resolution of a module M with e_u • M₁ = 0: the multiplicity of P_u{1} in X₁ is the sum of the multiplicities in X₀ over the neighbours of u.

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theorem TauCeti.finrank_zigzagVertexIdempotent_smul_piece_add_two_add_finrank_of_isLinear {k : Type w} [Field k] {V : Type u} {G : SimpleGraph V} [Finite V] {M : GradedModuleCat (zigzagIntegerGrade k G)} {r : GradedProjectiveResolution (zigzagIntegerGrade k G) M.grading} [Module.Finite k M.carrier] [Fintype V] [DecidableRel G.Adj] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) (hr : r.IsLinear) (p : ℕ) {u : V} (hM : zigzagVertexIdempotent k G u • M.grading.piece ℤ 2 = ⊥) :
Module.finrank k ↥(zigzagVertexIdempotent k G u • (r.grading (p + 2)).piece ℤ ↑(p + 2)) + Module.finrank k ↥(zigzagVertexIdempotent k G u • (r.grading p).piece ℤ ↑p) = ∑ w ∈ G.neighborFinset u, Module.finrank k ↥(zigzagVertexIdempotent k G w • (r.grading (p + 1)).piece ℤ ↑(p + 1))

The multiplicity recursion of a linear resolution. Let X be a linear graded projective resolution of a finite-dimensional graded module M over the zigzag algebra of a finite graph without isolated vertices, and let u be a vertex with e_u • M₂ = 0. Then the multiplicities m_n(u) = dim_k (e_u • (X_n)_n) of the shifted projectives P_u{n} in X_n satisfy m_{p+2}(u) + m_p(u) = ∑_{w ∼ u} m_{p+1}(w).

The inverse graded Cartan matrix #

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theorem TauCeti.zigzagGradedCartanMatrix_map_aeval_neg_X_mulVec_eq {k : Type w} [Field k] {V : Type u} {G : SimpleGraph V} [Finite V] {M : GradedModuleCat (zigzagIntegerGrade k G)} {r : GradedProjectiveResolution (zigzagIntegerGrade k G) M.grading} [Module.Finite k M.carrier] [Fintype V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) (hM : ∀ (u : V) (p : ℤ), p ≠ 0 → zigzagVertexIdempotent k G u • M.grading.piece ℤ p = ⊥) (hr : r.IsLinear) :
(((zigzagGradedCartanMatrix k G).map ⇑(Polynomial.aeval (-PowerSeries.X))).mulVec fun (u : V) => PowerSeries.mk fun (n : ℕ) => ↑(Module.finrank k ↥(zigzagVertexIdempotent k G u • (r.grading n).piece ℤ ↑n))) = fun (u : V) => PowerSeries.C ↑(Module.finrank k ↥(zigzagVertexIdempotent k G u • M.grading.piece ℤ 0))

The multiplicities of a linear resolution invert the graded Cartan matrix. Let X be a linear graded projective resolution of a finite-dimensional graded module M concentrated in degree 0 over the zigzag algebra of a finite graph without isolated vertices, and let m_u(X) = ∑_n dim_k (e_u • (X_n)_n) Xⁿ record the multiplicities of the shifted projectives P_u{n} in X_n. Then C_G(-X) m(X) is the constant vector (dim_k (e_u • M))_u, for the graded Cartan matrix C_G(q) = (1 + q²) I + q A_G.

Linear resolutions of the graded simples #

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theorem TauCeti.coeff_inv_zigzagGradedCartanMatrix_map_aeval_neg_X_eq {k : Type w} [Field k] {V : Type u} {G : SimpleGraph V} [Finite V] [DecidableEq V] [Fintype V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) (v : V) {r : GradedProjectiveResolution (zigzagIntegerGrade k G) (zigzagGradedSimple k G v).grading} (hr : r.IsLinear) (n : ℕ) (u : V) :
(PowerSeries.coeff n) (((zigzagGradedCartanMatrix k G).map ⇑(Polynomial.aeval (-PowerSeries.X)))⁻¹ u v) = ↑(Module.finrank k ↥(zigzagVertexIdempotent k G u • (r.grading n).piece ℤ ↑n))

The multiplicities of a linear resolution of a graded simple. Let G be a finite graph without isolated vertices and S_v the graded simple module of its zigzag algebra at v. For any linear graded projective resolution X of S_v, the multiplicity dim_k (e_u • (X_n)_n) of the shifted projective P_u{n} in X_n is the coefficient of Xⁿ in the entry (u, v) of the inverse C_G(-X)⁻¹ of the graded Cartan matrix C_G(q) = (1 + q²) I + q A_G.

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theorem TauCeti.coeff_inv_zigzagGradedCartanMatrix_map_aeval_neg_X_nonneg {k : Type w} [Field k] {V : Type u} {G : SimpleGraph V} [Finite V] [DecidableEq V] [Fintype V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) (v : V) (h : ∃ (r : GradedProjectiveResolution (zigzagIntegerGrade k G) (zigzagGradedSimple k G v).grading), r.IsLinear) (n : ℕ) (u : V) :
0 ≤ (PowerSeries.coeff n) (((zigzagGradedCartanMatrix k G).map ⇑(Polynomial.aeval (-PowerSeries.X)))⁻¹ u v)

Linearly resolved simples give nonnegative inverse Cartan coefficients. If the graded simple module S_v of the zigzag algebra of a finite graph without isolated vertices has a linear graded projective resolution, then every coefficient of the column v of the inverse C_G(-X)⁻¹ of the graded Cartan matrix is nonnegative.