Exactness of the fixed-term zigzag bimodule tensor actions

The vertex ideals are cyclic over the zigzag algebra. Its canonical finiteness instance and their tensor factorization make the actual vertex bimodule finite over the coefficient ring and hence over the algebra on the right. The regular bimodule is finite directly through its canonical equivalence with the regular right module. Together with the established right projectivity, the general finite-projective graded tensor theorem proves homology preservation for the actual vertex bimodule, the regular bimodule, and internal shifts of the vertex bimodule.

This is a fixed-term tensor result. Preservation of quasi-isomorphisms by the totalized braid complexes, localization and derived action transfer require further steps.

theorem TauCeti.preservesHomology_zigzagGradedBraidBimoduleTensor (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (i : V) : ((GradedModuleCat.bimoduleModuleTensor (zigzagAlgebraInternalGrading k G) (zigzagAlgebraInternalGrading k G)).obj (zigzagGradedBraidBimodule k G i)).PreservesHomology

Tensoring with the actual vertex bimodule preserves homology of graded modules.

theorem TauCeti.preservesHomology_zigzagGradedBimoduleTensor (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] : ((GradedModuleCat.bimoduleModuleTensor (zigzagAlgebraInternalGrading k G) (zigzagAlgebraInternalGrading k G)).obj (zigzagGradedBimodule k G)).PreservesHomology

Tensoring with the regular graded zigzag bimodule preserves homology.

theorem TauCeti.preservesHomology_zigzagGradedBraidBimoduleTensor_shift (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (i : V) (n : ℤ) : ((GradedModuleCat.bimoduleModuleTensor (zigzagAlgebraInternalGrading k G) (zigzagAlgebraInternalGrading k G)).obj ((GradedModuleCat.shiftFunctor n).obj (zigzagGradedBraidBimodule k G i))).PreservesHomology

An internal shift of the vertex bimodule retains homology preservation of tensoring.