The Koszul complex of a vertex module of a signless preprojective algebra #
Let R be a finite quiver with an involutive reversal of arrows, such as the doubled quiver of a
simple graph, and let Π = Π^s_k(R) be its signless preprojective algebra over a commutative ring
k, the quotient of the path algebra by the signless relators s_v = ∑_{b : i ⟶ v} b b*. For a
vertex v the Koszul complex of the vertex module S_v is
0 ⟶ e_v Π ⟶ ⨁_{b : i ⟶ v} e_i Π ⟶ e_v Π ⟶ S_v ⟶ 0,
y ↦ (b* y)_b, (z_b) ↦ ∑_b b z_b,
the sum running over the arrows b of R into v. It differs from the Koszul complex of the
preprojective algebra (TauCeti.RepresentationTheory.Quiver.Preprojective.Koszul.Complex) only in
carrying no signs. For the doubled quiver of a finite simple graph G the signless algebra is the
quadratic dual of the quadratic zigzag presentation
(TauCeti.quadraticDualQuadraticZigzagEquivSignless), and so, when G is connected with at least
three vertices and that presentation is the zigzag algebra
(TauCeti.quadraticZigzagPresentationEquivZigzagQuotient), the quadratic dual of the zigzag
algebra of G. When G is bipartite the signless algebra is isomorphic to the
preprojective algebra of a source--sink orientation
(TauCeti.symmetrifySignlessPreprojectiveAlgebraEquiv), but when G is not bipartite and 2 ≠ 0
in k it is a different algebra, and its Koszul complex must be treated on its own.
This file proves that the two maps compose to zero, that the complex is exact at its middle term,
and that the left-hand map is injective whenever R admits a non-backtracking choice of arrows:
an arrow α_u out of every vertex u such that the arrow chosen at the head of α_u is never the
reverse of α_u. For the doubled quiver of a graph G such a choice is a map f sending every
vertex to a neighbour with f (f v) ≠ v. Following f from any vertex eventually runs around a
cycle of G, so such an f exists only when every connected component of G contains a cycle;
conversely every such graph admits one. For the symmetrification of a bipartite quiver Q, the
comparison with Π_k(Q) carries vertex idempotents to vertex idempotents and arrows to signed
arrows, so left-end exactness of the preprojective Koszul complex gives that of the signless one.
Exactness at e_v Π itself, the statement that the classes of positive degree are the sums
∑_b b z_b, is not addressed here.
Main results #
TauCeti.signlessPreprojectiveRelator_eq_sum_ofArrow_mul: the signless relator atvis∑_{b : i ⟶ v} b b*, decomposed along the last arrow of its paths.TauCeti.sum_signlessPreprojectiveMk_ofArrow_mul_ofArrow_reverse_mul_eq_zero: the two maps compose to zero.TauCeti.sum_signlessPreprojectiveMk_ofArrow_mul_eq_zero_iff: exactness at the middle term.TauCeti.signlessPreprojectiveMk_ofArrow_mul_eq_zero_iff_of_ne_reverse: for a non-backtracking choice of arrowsα, left multiplication byα_vis injective one_v Π.TauCeti.forall_signlessPreprojectiveMk_ofArrow_reverse_mul_eq_zero_iff_of_ne_reverse: exactness at the left end for a non-backtracking choice of arrows.TauCeti.forall_signlessPreprojectiveMk_ofArrow_reverse_mul_eq_zero_iff_of_adj: the same for the doubled quiver of a simple graph, given a neighbour mapfwithf (f v) ≠ v.TauCeti.forall_signlessPreprojectiveMk_ofArrow_reverse_mul_eq_zero_iff_of_bipartite: the same for the symmetrification of a bipartite quiverQ, given left-end exactness forΠ_k(Q).
References #
- S. Huerfano and M. Khovanov, A category for the adjoint representation, J. Algebra 246 (2001), Section 3, for the signless relation of the quadratic dual of the zigzag algebra.
- 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 of a vertex module.
The signless relator along the last arrow of its paths: in the later-factor-first
convention s_v = ∑_{b : i ⟶ v} b b*, the sum over the arrows of R into v.
The Koszul complex is a complex: the composite y ↦ ∑_b b (b* y) is left multiplication by
the signless relator s_v, which vanishes in the signless algebra.
Exactness of the Koszul complex at its middle term. Let z_b ∈ e_i Π for the arrows
b : i ⟶ v of R. Then ∑_b b z_b = 0 exactly when there is one y ∈ e_v Π with z_b = b* y for
every b.
An arrow of a non-backtracking choice acts injectively. Let σ choose an arrow α_u out
of every vertex u of R such that the arrow chosen at the head of α_u is never the reverse of
α_u. Then for y ∈ e_v Π, α_v y = 0 exactly when y = 0.
Exactness of the Koszul complex at its left end, for a non-backtracking choice of arrows.
Let σ choose an arrow α_u out of every vertex u of R such that the arrow chosen at the head
of α_u is never the reverse of α_u. Then for y ∈ e_v Π, b* y = 0 for every arrow b into
v exactly when y = 0: the map y ↦ (b* y)_b is injective.
Exactness of the Koszul complex at its left end, for a graph. Let G be a finite simple
graph and f a map sending every vertex to a neighbour, with f (f v) ≠ v for every v. Then in
the signless algebra of the doubled quiver of G, for y ∈ e_v Π, b* y = 0 for every arrow b
into v exactly when y = 0.
Exactness of the Koszul complex at its left end, for a bipartite quiver. Let c colour
the vertices of Q so that the two ends of every arrow have different colours. If the Koszul
complex of the preprojective algebra Π_k(Q) is exact at its left end at v, then so is the Koszul
complex of the signless algebra of Quiver.Symmetrify Q: for y ∈ e_v Π, b* y = 0 for every
arrow b into v exactly when y = 0. The comparison isomorphism
TauCeti.symmetrifySignlessPreprojectiveAlgebraEquiv fixes the vertex idempotents and multiplies
every doubled arrow by a sign.