Path algebras with one quadratic relation per vertex are large

Let R be a finite quiver with finitely many arrows, k a field, and let every vertex v carry a relator r_v which is homogeneous of path length two and lies in the corner e_v kR e_v. This file bounds the growth of A = kR / (r_v) from below, in the spirit of the Golod--Shafarevich inequality, and deduces that A is infinite-dimensional as soon as the vertices carry a nonzero nonnegative weight δ with

2 δ_i ≤ ∑_j #(i ⟶ j) δ_j.

The preprojective algebra of a quiver is of this form, with the doubled quiver for R and the local preprojective relators for the r_v; this is its intended application.

Write P_n(j) for the span of the paths of length n ending at j, which is the degree-n part of the corner e_j kR, and K_n(j) for its intersection with the relation ideal I. Since kR is generated by its vertex idempotents and arrows, I = ∑_v r_v kR + ∑_b b I, and cutting this down to degree n + 2 at the vertex j gives

K_{n+2}(j) = r_j P_n(j) + ∑_{b : i ⟶ j} b K_{n+1}(i).

Moreover r_j = ∑_b b c_b for some c_b, so r_j K_n(j) is already contained in the second summand. Counting dimensions, the dimensions d_n(j) = dim P_n(j) - dim K_n(j) of the images of the P_n(j) in A satisfy d_0(j) = 1, d_1(j) = ∑_{b : i ⟶ j} d_0(i) and

d_{n+2}(j) + d_n(j) ≥ ∑_{b : i ⟶ j} d_{n+1}(i),

which is the vertexwise form of the coefficientwise inequality H_A(t) (1 - C t + t²) ≥ 1 for the matrix Hilbert series of A, C being the arrow-count matrix of R. Weighting by δ, the sums s_n = ∑_j δ_j d_n(j) then satisfy s_{n+2} - s_{n+1} ≥ s_{n+1} - s_n ≥ ⋯ ≥ s_1 - s_0 ≥ s_0, so s_n ≥ (n + 1) s_0 ≥ (n + 1) ∑_j δ_j (TauCeti.add_one_mul_sum_mul_le_sum_mul). If A had finite dimension N, then d_N(j) ≤ N would give s_N ≤ N ∑_j δ_j, contradicting this at n = N since ∑_j δ_j > 0.

Main results

• TauCeti.PathAlgebra.sum_card_mul_finrank_map_pathsInto_le_add: the Anick-type inequality ∑_{b : i ⟶ j} d_{m+1}(i) ≤ d_{m+2}(j) + d_m(j) for the dimensions of the images of these spans in the quotient.
• TauCeti.PathAlgebra.not_module_finite_quotient_span_range_of_two_mul_le_sum: a path algebra with one quadratic corner relation per vertex, whose arrow counts admit a nonzero nonnegative weight δ with 2 δ_i ≤ ∑_j #(i ⟶ j) δ_j, is infinite-dimensional.

Implementation notes

The general path-span API is in TauCeti.RepresentationTheory.Quiver.PathAlgebra.PathsInto, and the decomposition I = ∑ᵥ r_v kR + ∑_b b I of the relation ideal is TauCeti.PathAlgebra.exists_eq_sum_mul_add_sum_ofArrow_mul_of_mem_span in TauCeti.RepresentationTheory.Quiver.PathAlgebra.RelationIdeal. All the dimension counting takes place among subspaces of kR itself; the quotient enters only through the images of the spaces P_n(j), so no grading of the quotient is needed. The intersections K_n(j) with the relation ideal are private to this file.

References

• E. S. Golod and I. R. Shafarevich, On the class field tower, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), for the inequality dim A_n ≥ d dim A_{n-1} - r dim A_{n-2} for algebras with d generators and r quadratic relations.
• D. J. Anick, Non-commutative graded algebras and their Hilbert series, J. Algebra 78 (1982), for the Hilbert-series form of that inequality.
• P. Etingof and C.-H. Eu, Koszulity and the Hilbert series of preprojective algebras, Math. Res. Lett. 14 (2007), where the matrix Hilbert series of a preprojective algebra is compared with (1 - C t + t²)⁻¹.

theorem TauCeti.PathAlgebra.sum_card_mul_finrank_map_pathsInto_le_add {R : Type u} [Quiver R] {k : Type w} [Field k] [Fintype R] [(a b : R) → Fintype (a ⟶ b)] (r : R → pathAlgebra k R) (h2 : ∀ (v : R), r v ∈ grade k R 2) (hl : ∀ (v : R), vertexIdempotent k v * r v = r v) (hr : ∀ (v : R), r v * vertexIdempotent k v = r v) (m : ℕ) (j : R) : ∑ i : R, Fintype.card (i ⟶ j) * Module.finrank k ↥(Submodule.map (Ideal.Quotient.mkₐ k (TwoSidedIdeal.asIdeal (TwoSidedIdeal.span (Set.range r)))).toLinearMap (pathsInto k (m + 1) i)) ≤ Module.finrank k ↥(Submodule.map (Ideal.Quotient.mkₐ k (TwoSidedIdeal.asIdeal (TwoSidedIdeal.span (Set.range r)))).toLinearMap (pathsInto k (m + 2) j)) + Module.finrank k ↥(Submodule.map (Ideal.Quotient.mkₐ k (TwoSidedIdeal.asIdeal (TwoSidedIdeal.span (Set.range r)))).toLinearMap (pathsInto k m j))

The Anick-type inequality for one quadratic corner relation per vertex. Let each vertex v of R carry a relator r_v of path length two in the corner e_v kR e_v, and write d_n(j) for the dimension of the image in kR / (r_v) of the span of the paths of length n into j. Then ∑_{b : i ⟶ j} d_{m+1}(i) ≤ d_{m+2}(j) + d_m(j), the vertexwise form of the coefficientwise inequality H_A(t) (1 - C t + t²) ≥ 1 for the matrix Hilbert series of the quotient.

theorem TauCeti.PathAlgebra.not_module_finite_quotient_span_range_of_two_mul_le_sum {R : Type u} [Quiver R] {k : Type w} [Field k] [Fintype R] [(a b : R) → Fintype (a ⟶ b)] {S : Type u_1} [CommRing S] [LinearOrder S] [IsStrictOrderedRing S] (r : R → pathAlgebra k R) (h2 : ∀ (v : R), r v ∈ grade k R 2) (hl : ∀ (v : R), vertexIdempotent k v * r v = r v) (hr : ∀ (v : R), r v * vertexIdempotent k v = r v) {δ : R → S} (hδ0 : 0 ≤ δ) (hδ : δ ≠ 0) (hδle : ∀ (i : R), 2 * δ i ≤ ∑ j : R, ↑(Fintype.card (i ⟶ j)) * δ j) : ¬Module.Finite k (pathAlgebra k R ⧸ TwoSidedIdeal.asIdeal (TwoSidedIdeal.span (Set.range r)))

A path algebra with one quadratic corner relation per vertex is infinite-dimensional when its arrow counts admit a suitable weight. Let R be a finite quiver with finitely many arrows and let each vertex v carry a relator r_v of path length two in the corner e_v kR e_v. Suppose a nonzero nonnegative weight δ on the vertices satisfies 2 δ_i ≤ ∑_j #(i ⟶ j) δ_j at every vertex. Then kR / (r_v) is not a finite-dimensional k-vector space.

For the doubled quiver of a graph, #(i ⟶ j) is the adjacency matrix A and the hypothesis reads (2I - A) δ ≤ 0.