The Koszul complex of a vertex module of a preprojective algebra #
Let Q be a finite quiver and Π = Π_k(Q) its preprojective algebra over a commutative ring k.
For a vertex v, the right ideal e_v Π is the projective right Π-module at v, spanned by the
classes of the paths of the doubled quiver ending at v. The vertex augmentation module S_v at
v is its quotient by its part of positive degree, a copy of k on which every arrow acts by zero;
it is the simple right module at v when k is a field. The Koszul complex of S_v is
0 ⟶ e_v Π ⟶ ⨁_{b : i ⟶ v} e_i Π ⟶ e_v Π ⟶ S_v ⟶ 0,
y ↦ (ε_b b* y)_b, (z_b) ↦ ∑_b b z_b,
the sum running over the arrows b of the doubled quiver Quiver.Symmetrify Q into v, with
b* the formal reverse of b and ε_b = 1 when b is an arrow of Q and -1 when it is the
reverse of one (TauCeti.doubledArrowSign). Both maps are left multiplications, so they are maps of
right modules. The signs are those of the local relator: in Tau Ceti's later-factor-first
convention
ρ_v = ∑_{b : i ⟶ v} ε_b b b*
(TauCeti.localPreprojectiveRelator_eq_sum_ofArrow_mul), which makes the two maps compose to zero.
This file proves that the complex is exact at its two right-hand terms:
TauCeti.mem_iSup_preprojectiveGrade_add_one_iff_exists_eq_sum: an element ofe_v Πhas positive degree exactly when it is a sum∑_b b z_b, so the right-hand map has image the kernel ofe_v Π ⟶ S_v;TauCeti.sum_preprojectiveMk_ofArrow_mul_eq_zero_iff: a familyz_b ∈ e_i Πhas∑_b b z_b = 0exactly whenz_b = ε_b b* yfor oney ∈ e_v Π.
These hold for every finite quiver and every commutative ring k. What is left is the left-hand
map: the complex is a linear projective resolution of S_v exactly when that map is injective,
that is, when the only y ∈ e_v Π with b* y = 0 for every arrow b into v is 0. Over a
field, counting dimensions degree by degree, this injectivity in every degree amounts to equality
in the Anick-type inequality TauCeti.PathAlgebra.sum_card_mul_finrank_map_pathsInto_le_add.
It fails whenever Π is finite-dimensional and nonzero, as for a Dynkin quiver: a nonzero element
of e_v Π of top degree is killed by every arrow.
That the two maps compose to zero and exactness at the middle term are the instances, for the signs
ε_b, of the results for weighted backtrack relators in
TauCeti.RepresentationTheory.Quiver.PathAlgebra.BacktrackRelator.
Main results #
TauCeti.sum_preprojectiveMk_ofArrow_mul_doubledArrowSign_smul_eq_zero: the two maps compose to zero.TauCeti.sum_preprojectiveMk_ofArrow_mul_eq_zero_iff: exactness at the middle term.TauCeti.mem_iSup_preprojectiveGrade_add_one_iff_exists_eq_sum: exactness ate_v Π.
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 this complex and the Koszulity of preprojective algebras of non-Dynkin quivers.
- S. Brenner, M. C. R. Butler and A. D. King, Periodic algebras which are almost Koszul, Algebr. Represent. Theory 5 (2002), for the Dynkin case.
The Koszul complex is a complex: the composite y ↦ ∑_b b (ε_b b* y) is left
multiplication by the local relator ρ_v, which vanishes in the preprojective algebra.
Exactness of the Koszul complex at its middle term. Let z_b ∈ e_i Π for the arrows
b : i ⟶ v of the doubled quiver. Then ∑_b b z_b = 0 exactly when there is one y ∈ e_v Π with
z_b = ε_b b* y for every b.
Exactness of the Koszul complex at e_v Π. An element x ∈ e_v Π has positive degree,
so maps to zero in the vertex augmentation module S_v, exactly when x = ∑_b b z_b for some
z_b ∈ e_i Π, the sum over the arrows b : i ⟶ v of the doubled quiver.