The linear-resolution obstruction for finite ADE zigzag algebras

For a finite simple graph without isolated vertices, finite-dimensionality of the preprojective algebra of an orientation forces a negative coefficient in the inverse graded Cartan matrix of the zigzag algebra. A linear projective resolution of every graded vertex simple would make all these coefficients nonnegative. Hence at least one vertex simple has no linear graded projective resolution.

Applied to any finite graph on at least two vertices isomorphic to a valid Dynkin diagram, this proves the non-Koszul half of the finite-ADE classification in the resolution-theoretic form: not every graded vertex simple is linearly resolved. The rank-one zigzag convention is deliberately excluded because its generator has degree two and it is not the quadratic presentation used here.

Main results

• TauCeti.exists_not_exists_isLinear_zigzagGradedSimple_of_finiteDimensional_preprojective: a finite-dimensional preprojective algebra obstructs linear resolutions of zigzag vertex simples.
• TauCeti.exists_not_exists_isLinear_zigzagGradedSimple_of_iso: every finite graph on at least two vertices isomorphic to a valid Dynkin diagram has a vertex simple without a linear graded projective resolution.

References

• A. Beilinson, V. Ginzburg and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), Sections 1.2 and 2.11.
• S. Brenner, M. C. R. Butler and A. D. King, Periodic algebras which are almost Koszul, Algebr. Represent. Theory 5 (2002), Sections 3 and 4.
• P. Etingof and C.-H. Eu, Koszulity and the Hilbert series of preprojective algebras, Math. Res. Lett. 14 (2007), Sections 2 and 3.

theorem TauCeti.exists_not_exists_isLinear_zigzagGradedSimple_of_finiteDimensional_preprojective (k : Type w) [Field k] {V : Type u} {G : SimpleGraph V} [Finite V] [Nonempty V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) (o : DoubledQuiver.Orientation G) [FiniteDimensional k (preprojectiveAlgebra k (DoubledQuiver.OrientedQuiver G o))] : ∃ (v : V), ¬∃ (r : GradedProjectiveResolution (zigzagIntegerGrade k G) (zigzagGradedSimple k G v).grading), r.IsLinear

A finite-dimensional preprojective algebra obstructs linear resolutions of zigzag simples. If some orientation of a nonempty finite graph without isolated vertices has a finite-dimensional preprojective algebra, then at least one graded vertex simple of the zigzag algebra has no linear graded projective resolution.

theorem TauCeti.exists_not_exists_isLinear_zigzagGradedSimple_of_iso (k : Type w) [Field k] {V : Type u} {G : SimpleGraph V} [Finite V] [Nontrivial V] {t : DynkinType} (ht : t.Valid) (e : G ≃g diagramGraph t.cartanMatrix) : ∃ (v : V), ¬∃ (r : GradedProjectiveResolution (zigzagIntegerGrade k G) (zigzagGradedSimple k G v).grading), r.IsLinear

Every finite graph on at least two vertices isomorphic to a valid Dynkin diagram has a zigzag vertex simple without a linear resolution.