The Koszul complex of a zigzag simple module #
Let Z be the zigzag relation quotient TauCeti.nonisolatedZigzagQuotient of a finite graph G
and let Π be the signless preprojective algebra of the doubled quiver of G, the quadratic dual
of the quadratic zigzag presentation (TauCeti.quadraticDualQuadraticZigzagEquivSignless). For a
vertex v, the Koszul complex of the simple Z-module at v has in homological degree n the
left Z-module
K_n = ⨁_u Hom_k(e_u Π_n e_v, Z e_u),
the Hom form of ⨁_u Z e_u ⊗_k (e_u Π_n e_v)^*, and the differential
(d f)_w (y) = ∑_{j ∼ w} f_j (a_{wj} y) · a_{wj},
where a_{wj} is the arrow w → j, acting on the left of Π and on the right of Z. Composing
two differentials multiplies by a length-two zigzag path; only the returning paths survive, all
equal to the volume class, against the signless relator ∑_{j ∼ w} (w → j → w) = 0. So K is a
complex.
The main theorem is the Koszul duality between K and the vertex Koszul complexes
0 → e_w Π → ⨁_{j ∼ w} e_j Π → e_w Π of Π. The vertex, arrow and volume coordinates of the
values of f ∈ K_{n+1} pair with the three terms of these complexes: the vertex coordinates of a
cycle vanish because Π is generated by its arrows, the arrow coordinates lift by exactness of the
signless complex at its middle term, and the volume coordinates lift by exactness at its left end.
Hence, if every signless vertex complex is exact at its left end, K is exact in every positive
homological degree. A non-backtracking choice of neighbours supplies this exactness, so the result
holds for every connected graph which is not a tree, and these graphs are never finite ADE Dynkin
diagrams. The augmentation of K_0 onto the simple module and the internal grading of the terms,
which make K a linear resolution, are in
TauCeti.RepresentationTheory.Quiver.Zigzag.Projective.Koszul.Resolution.
Main definitions #
TauCeti.ZigzagKoszulTerm: the termK_n.TauCeti.zigzagKoszulDifferential: the differentialK_{n+1} → K_n.
Main results #
TauCeti.zigzagKoszulDifferential_comp_zigzagKoszulDifferential:d ∘ d = 0.TauCeti.range_zigzagKoszulDifferential_eq_ker: exactness in positive homological degrees, given exactness of the signless vertex Koszul complexes at their left ends.TauCeti.range_zigzagKoszulDifferential_eq_ker_of_adjandTauCeti.range_zigzagKoszulDifferential_eq_ker_of_not_isAcyclic: exactness for a non-backtracking choice of neighbours, and for connected graphs which are not trees.
References #
- R. S. Huerfano and M. Khovanov, A category for the adjoint representation, J. Algebra 246 (2001), Section 3, for the quadratic dual of the zigzag algebra.
- A. Beilinson, V. Ginzburg and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), Section 2, for the Koszul complex and the exactness of the Koszul complexes of a quadratic algebra and its dual.
- 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 vertex Koszul complexes of
Π.
The term in homological degree n of the Koszul complex of the zigzag simple module at v:
the left module ⨁_u Hom_k(e_u Π_n e_v, Z e_u) over the zigzag relation quotient Z, where Π
is the signless preprojective algebra of the doubled quiver, the quadratic dual of the quadratic
zigzag presentation (TauCeti.quadraticDualQuadraticZigzagEquivSignless).
Instances For
The Koszul differential of the zigzag simple module at v, from homological degree
n + 1 to n. At the vertex w it sends f to y ↦ ∑_{j ∼ w} f_j (a_{wj} y) · a_{wj}: the
signless arrow w → j moves y ∈ e_w Π_n e_v to e_j Π_{n+1} e_v, and right multiplication by
the zigzag arrow w → j moves the value from Z e_j back to Z e_w.
Instances For
The value of the Koszul differential, computed in the zigzag algebra.
The Koszul complex is a complex #
The Koszul complex is a complex: two consecutive Koszul differentials compose to zero.
Two consecutive Koszul differentials compose to zero, pointwise.
Coordinates in the vertex projectives #
The signless Koszul complex, indexed by neighbours #
Exactness #
Koszul duality for the zigzag Koszul complex. Let G be a finite graph. If the Koszul
complex 0 → e_w Π → ⨁_{j ∼ w} e_j Π of the signless preprojective
algebra Π is exact at its left end at every vertex w, then the Koszul complex of every zigzag
simple module is exact in every positive homological degree: the image of the differential out of
degree n + 2 is the kernel of the differential out of degree n + 1.
The zigzag Koszul complex is exact for a non-backtracking choice of neighbours. If f
sends every vertex of G to a neighbour, with f (f u) ≠ u for every u, then the Koszul complex
of every zigzag simple module of G is exact in every positive homological degree. Such an f
exists exactly when every connected component of G contains a cycle.
The zigzag Koszul complex of a connected graph which is not a tree is exact in every positive homological degree, at every vertex. Such a graph is never a finite ADE Dynkin diagram.