Preprojective algebras of quivers without sinks

Let Q be a finite quiver without sinks, that is, every vertex is the tail of an arrow of Q, and let Π = Π_k(Q) be its preprojective algebra over a commutative ring k. This file proves that for every arrow a : v ⟶ w of Q, left multiplication by a is injective on e_v Π.

Consequently the left-hand map y ↦ (ε_b b* y)_b of the Koszul complex

0 ⟶ e_v Π ⟶ ⨁_{b : i ⟶ v} e_i Π ⟶ e_v Π ⟶ S_v ⟶ 0

of TauCeti.RepresentationTheory.Quiver.Preprojective.Koszul.Complex is injective. Together with the exactness at the other terms proved there, the complex is a projective resolution of the vertex module S_v, for every vertex v and every commutative ring k.

The hypothesis is on the orientation, while Π does not depend on the orientation up to isomorphism (TauCeti.reorientPreprojectiveAlgebraEquiv). A connected graph admits an orientation without sinks exactly when it is not a tree, so the non-Dynkin trees, such as D~ₙ and E₆~, E₇~, E₈~, are not covered here.

Main results

• TauCeti.preprojectiveMk_ofArrow_mul_eq_zero_iff: for y ∈ e_v Π and an arrow a of Q out of v, a y = 0 exactly when y = 0.
• TauCeti.forall_preprojectiveMk_ofArrow_reverse_mul_eq_zero_iff: exactness of the Koszul complex at its left end: b* y = 0 for every arrow b of the doubled quiver into v exactly when y = 0.

Implementation notes

The local preprojective relators are weighted backtrack relators with the signs TauCeti.doubledArrowSign, and a choice of an arrow of Q out of every vertex is non-backtracking in the doubled quiver, since no arrow of Q is the reverse of one. Both results are therefore instances of TauCeti.PathAlgebra.map_ofArrow_mul_eq_zero_iff_of_ne_reverse, which builds a faithful module from the rewriting rules solving each local relation for one backtrack.

References

• G. M. Bergman, The diamond lemma for ring theory, Adv. Math. 29 (1978), for the construction of a module from a rewriting system without ambiguities.
• P. Etingof and C.-H. Eu, Koszulity and the Hilbert series of preprojective algebras, Math. Res. Lett. 14 (2007), Sections 2 and 3, for the Koszul complex and the Koszulity of the preprojective algebras of non-Dynkin quivers.

theorem TauCeti.preprojectiveMk_ofArrow_mul_eq_zero_iff (k : Type w) {Q : Type u} [CommRing k] [Quiver Q] [Fintype Q] [(i j : Q) → Fintype (i ⟶ j)] (hQ : ∀ (u : Q), ∃ (w : Q), Nonempty (u ⟶ w)) {v w : Q} (a : v ⟶ w) {y : preprojectiveAlgebra k Q} (hy : (preprojectiveMk k Q) (doubledVertexIdempotent k v) * y = y) : (preprojectiveMk k Q) (PathAlgebra.ofArrow (Quiver.Symmetrify.of.map a)) * y = 0 ↔ y = 0

An arrow of a quiver without sinks acts injectively on e_v Π. If every vertex of Q is the tail of an arrow, a : v ⟶ w is an arrow of Q and y = e_v y, then a y = 0 exactly when y = 0.

theorem TauCeti.forall_preprojectiveMk_ofArrow_reverse_mul_eq_zero_iff (k : Type w) {Q : Type u} [CommRing k] [Quiver Q] [Fintype Q] [(i j : Q) → Fintype (i ⟶ j)] (hQ : ∀ (u : Q), ∃ (w : Q), Nonempty (u ⟶ w)) (v : Q) {y : preprojectiveAlgebra k Q} (hy : (preprojectiveMk k Q) (doubledVertexIdempotent k v) * y = y) : (∀ (i : Quiver.Symmetrify Q) (b : i ⟶ 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 without sinks. If every vertex of Q is the tail of an arrow and y = e_v y, then b* y = 0 for every arrow b of the doubled quiver into v exactly when y = 0: the map y ↦ (ε_b b* y)_b of TauCeti.sum_preprojectiveMk_ofArrow_mul_eq_zero_iff is injective.