Koszulity of preprojective algebras over quivers with superadditive weights

Let Π = Π_k(Q) be the preprojective algebra of a finite quiver Q over a field k. The Koszul complex of the vertex module S_v,

0 ⟶ e_v Π ⟶ ⨁_{b : w ⟶ v} e_w Π ⟶ e_v Π ⟶ S_v ⟶ 0,

is exact except possibly at its left end, whose defect consists of the y ∈ e_v Π killed by every arrow b* of the doubled quiver leaving v. This file proves that the defect vanishes as soon as Q carries a positive superadditive weight, a weight δ on its vertices with δ_i > 0 and 2 δ_i ≤ ∑_j (#(i ⟶ j) + #(j ⟶ i)) δ_j at every vertex. No assumption on the sinks of Q or on its shape is needed: the condition lives at quiver level. For orientations of a connected simple graph, a connected graph which is not a Dynkin diagram produces such a weight (TauCeti.exists_pos_two_mul_le_sum_orientedQuiver_of_not_exists_dynkinType_iso, in Koszul/NonDynkin.lean), so there the Koszul complexes are projective resolutions and the matrix Hilbert series of Π is ((1 + t²) I - t A)⁻¹ (TauCeti.preprojectiveHilbertSeries_eq_inv_of_not_exists_dynkinType_iso, Koszul/NonDynkin.lean).

Main results

• TauCeti.forall_preprojectiveMk_ofArrow_reverse_mul_eq_zero_iff_of_two_mul_le_sum: the Koszul complex of every vertex module is exact at its left end when Q carries a positive superadditive weight.
• TauCeti.preprojectiveKoszulLeft_injective_of_two_mul_le_sum: injectivity of the bundled map.
• TauCeti.mul_preprojectiveHilbertSeries_eq_one_of_two_mul_le_sum and TauCeti.preprojectiveHilbertSeries_eq_inv_of_two_mul_le_sum: for such a quiver, with A the arrow-count matrix of the doubled quiver, ((1 + t²) I - t A) H = I and H = ((1 + t²) I - t A)⁻¹ for the matrix Hilbert series H of Π, with no hypothesis on sinks.

References

• P. Etingof and C.-H. Eu, Koszulity and the Hilbert series of preprojective algebras, Math. Res. Lett. 14 (2007), for the superadditive weight argument in the non-Dynkin case.
• I. N. Bernstein, I. M. Gelfand and V. A. Ponomarev, Coxeter functors and Gabriel's theorem, Russian Math. Surveys 28 (1973), for reflection at the sources of a quiver and the splitting of simple summands from representations on which a reflection is not faithful.
• I. M. Gelfand and V. A. Ponomarev, Model algebras and representations of graphs, Funct. Anal. Appl. 13 (1979), where the preprojective algebra is introduced through these reflections.

theorem TauCeti.forall_preprojectiveMk_ofArrow_reverse_mul_eq_zero_iff_of_two_mul_le_sum (k : Type w) {Q : Type u} [Field k] [Quiver Q] [Fintype Q] [(i j : Q) → Fintype (i ⟶ j)] {S : Type u_1} [CommRing S] [LinearOrder S] [IsStrictOrderedRing S] {δ : Q → S} (hδ : ∀ (i : Q), 0 < δ i) (hδle : ∀ (i : Q), 2 * δ i ≤ ∑ j : Q, ↑(Fintype.card (i ⟶ j) + Fintype.card (j ⟶ i)) * δ j) (v : Q) {y : preprojectiveAlgebra k Q} (hy : (preprojectiveMk k Q) (doubledVertexIdempotent k v) * y = y) : (∀ (w : Quiver.Symmetrify Q) (b : w ⟶ Quiver.Symmetrify.of.obj v), (preprojectiveMk k Q) (PathAlgebra.ofArrow (Quiver.reverse b)) * y = 0) ↔ y = 0

Exactness of the Koszul complex at its left end, for a quiver with a positive superadditive weight. Let δ be a weight on the vertices of a finite quiver Q with δ_i > 0 and 2 δ_i ≤ ∑_j (#(i ⟶ j) + #(j ⟶ i)) δ_j at every vertex, that is (2I - A) δ ≤ 0 for the adjacency matrix A of the underlying multigraph. If y = e_v y in the preprojective algebra of Q over a field k, then b* y = 0 for every arrow b of the doubled quiver into v exactly when y = 0.

theorem TauCeti.preprojectiveKoszulLeft_injective_of_two_mul_le_sum (k : Type w) {Q : Type u} [Field k] [Quiver Q] [Fintype Q] [(i j : Q) → Fintype (i ⟶ j)] {S : Type u_1} [CommRing S] [LinearOrder S] [IsStrictOrderedRing S] {δ : Q → S} (hδ : ∀ (i : Q), 0 < δ i) (hδle : ∀ (i : Q), 2 * δ i ≤ ∑ j : Q, ↑(Fintype.card (i ⟶ j) + Fintype.card (j ⟶ i)) * δ j) (v : Q) : Function.Injective ⇑(preprojectiveKoszulLeft k v)

For a quiver with a positive superadditive weight, the left Koszul differential at each vertex is injective.

theorem TauCeti.mul_preprojectiveHilbertSeries_eq_one_of_two_mul_le_sum (k : Type w) {Q : Type u} [Field k] [Quiver Q] [Fintype Q] [(i j : Q) → Fintype (i ⟶ j)] [DecidableEq Q] {S : Type u_1} [CommRing S] [LinearOrder S] [IsStrictOrderedRing S] {δ : Q → S} (hδ : ∀ (i : Q), 0 < δ i) (hδle : ∀ (i : Q), 2 * δ i ≤ ∑ j : Q, ↑(Fintype.card (i ⟶ j) + Fintype.card (j ⟶ i)) * δ j) : ((1 + PowerSeries.X ^ 2) • 1 - PowerSeries.X • Matrix.of fun (v w : Q) => ↑(Fintype.card (Quiver.Symmetrify.of.obj w ⟶ Quiver.Symmetrify.of.obj v))) * preprojectiveHilbertSeries k Q = 1

The Hilbert series identity for a quiver with a superadditive weight. Let δ be a weight on the vertices of a finite quiver Q over a strictly ordered commutative ring, positive with

2 δ_i ≤ ∑_j (#(i ⟶ j) + #(j ⟶ i)) δ_j

at every vertex. No hypothesis on the sinks of Q is needed: the matrix Hilbert series H of Π satisfies ((1 + X²) I - X A) H = I for A_{v,w} = #(w ⟶ v) the arrow-count matrix of the doubled quiver.

theorem TauCeti.preprojectiveHilbertSeries_eq_inv_of_two_mul_le_sum (k : Type w) {Q : Type u} [Field k] [Quiver Q] [Fintype Q] [(i j : Q) → Fintype (i ⟶ j)] [DecidableEq Q] {S : Type u_1} [CommRing S] [LinearOrder S] [IsStrictOrderedRing S] {δ : Q → S} (hδ : ∀ (i : Q), 0 < δ i) (hδle : ∀ (i : Q), 2 * δ i ≤ ∑ j : Q, ↑(Fintype.card (i ⟶ j) + Fintype.card (j ⟶ i)) * δ j) : preprojectiveHilbertSeries k Q = ((1 + PowerSeries.X ^ 2) • 1 - PowerSeries.X • Matrix.of fun (v w : Q) => ↑(Fintype.card (Quiver.Symmetrify.of.obj w ⟶ Quiver.Symmetrify.of.obj v)))⁻¹

The Hilbert series of a preprojective algebra whose quiver carries a superadditive weight is the inverse matrix H = ((1 + X²) I - X A)⁻¹, under the same hypothesis on δ.