Koszul complexes of preprojective algebras of graphs which are not trees

Let Π = Π_k(Q) be the preprojective algebra of a finite quiver over a commutative ring k. 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 exact at its left end as soon as Q has no sinks (TauCeti.forall_preprojectiveMk_ofArrow_reverse_mul_eq_zero_iff). This file removes the dependence on the orientation. Since Π does not depend on the orientation up to isomorphism (TauCeti.reorientPreprojectiveAlgebraEquiv), it is enough that some reorientation of Q has no sinks; the isomorphism carries every doubled arrow to a doubled arrow up to sign, so it carries the left-hand map of the Koszul complex of Q to that of the reorientation.

A preconnected simple graph which is not acyclic has an orientation without sinks (SimpleGraph.Preconnected.exists_forall_adj_and_apply_apply_ne_of_not_isAcyclic). Hence for every orientation of such a graph, over every commutative ring, the Koszul complex of every vertex module is exact at its left end, so it is a projective resolution of the vertex module. This covers the cycles A~ₙ (n ≥ 2) and every connected graph with more edges than a spanning tree, all of which are non-Dynkin; the non-Dynkin trees, such as D~ₙ and E₆~, E₇~, E₈~, are not covered. Over a field, every connected non-Dynkin graph, trees included, is covered by TauCeti.forall_preprojectiveMk_ofArrow_reverse_mul_eq_zero_iff_of_not_exists_dynkinType_iso. The shifted graded Koszul complex is consequently a linear graded projective resolution of each right vertex module, over the ordinary opposite algebra.

Main results

• TauCeti.forall_preprojectiveMk_ofArrow_reverse_mul_eq_zero_iff_of_reorient: the Koszul complex of Π_k(Q) is exact at its left end if some reorientation of Q has no sinks.
• TauCeti.forall_preprojectiveMk_ofArrow_reverse_mul_eq_zero_iff_of_not_isAcyclic: the Koszul complex of the preprojective algebra of any orientation of a connected graph which is not a tree (more generally, of a preconnected graph which is not acyclic) is exact at its left end.
• TauCeti.preprojectiveKoszulLeft_injective_of_not_isAcyclic: injectivity of the bundled map.
• TauCeti.isLinear_preprojectiveGradedKoszulResolution_of_not_isAcyclic: linear graded projective resolutions for every orientation, without a hypothesis on its sinks.

References

• 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.
• W. Crawley-Boevey, Quiver algebras, weighted projective lines, and the Deligne--Simpson problem, Section 1, for the independence of the orientation.

theorem TauCeti.forall_preprojectiveMk_ofArrow_reverse_mul_eq_zero_iff_of_reorient (k : Type w) {Q : Type u} [CommRing k] [Quiver Q] [Fintype Q] [(i j : Q) → Fintype (i ⟶ j)] (σ : ⦃i j : Q⦄ → (i ⟶ j) → Bool) (hσ : ∀ (u : Q), ∃ (w : Q), Nonempty (reorientVertex σ u ⟶ reorientVertex σ 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 with a reorientation without sinks. If turning around the arrows of Q which σ labels true leaves no vertex without an outgoing arrow, and y = e_v y in Π_k(Q), then b* y = 0 for every arrow b of the doubled quiver into v exactly when y = 0.

theorem TauCeti.forall_preprojectiveMk_ofArrow_reverse_mul_eq_zero_iff_of_not_isAcyclic {V : Type u} {G : SimpleGraph V} (k : Type w) [CommRing k] [Finite V] (hG : G.Preconnected) (hc : ¬G.IsAcyclic) (o : DoubledQuiver.Orientation G) (v : DoubledQuiver.OrientedQuiver G o) {y : preprojectiveAlgebra k (DoubledQuiver.OrientedQuiver G o)} (hy : (preprojectiveMk k (DoubledQuiver.OrientedQuiver G o)) (doubledVertexIdempotent k v) * y = y) : (∀ (i : Quiver.Symmetrify (DoubledQuiver.OrientedQuiver G o)) (b : i ⟶ Quiver.Symmetrify.of.obj v), (preprojectiveMk k (DoubledQuiver.OrientedQuiver G o)) (PathAlgebra.ofArrow (Quiver.reverse b)) * y = 0) ↔ y = 0

Exactness of the Koszul complex at its left end, for a graph which is not a tree. Let G be a finite preconnected simple graph which is not acyclic, and o any orientation of G. If y = e_v y in the preprojective algebra of o over a commutative ring k, then b* y = 0 for every arrow b of the doubled quiver into v exactly when y = 0: the left-hand map of the Koszul complex of TauCeti.sum_preprojectiveMk_ofArrow_mul_eq_zero_iff is injective.

theorem TauCeti.preprojectiveKoszulLeft_injective_of_not_isAcyclic {V : Type u} {G : SimpleGraph V} (k : Type w) [CommRing k] [Finite V] (hG : G.Preconnected) (hc : ¬G.IsAcyclic) (o : DoubledQuiver.Orientation G) (v : DoubledQuiver.OrientedQuiver G o) : Function.Injective ⇑(preprojectiveKoszulLeft k v)

For every orientation of a finite preconnected graph containing a cycle, the left Koszul differential at each vertex is injective. The chosen orientation may have sinks.

theorem TauCeti.isLinear_preprojectiveGradedKoszulResolution_of_not_isAcyclic {V : Type u} {G : SimpleGraph V} (k : Type w) [CommRing k] [Finite V] (hG : G.Preconnected) (hc : ¬G.IsAcyclic) (o : DoubledQuiver.Orientation G) (v : DoubledQuiver.OrientedQuiver G o) : (preprojectiveGradedKoszulResolution k v ⋯).IsLinear

The right vertex module of every orientation of a finite preconnected graph containing a cycle has a linear graded Koszul resolution. Its projective terms are generated in internal degrees zero, one and two, respectively, and all later terms vanish.