The derived orbit functor in finite global dimension

Let C be an abelian category with enough projectives in which every object has projective dimension < d, and let n be positive. Then the orbit functor TauCeti.PeriodicComplex.orbitDerivedCompressionFunctor n : D^b(C)/[n] ⥤ D_n(C), the descent of compression D^b(C) ⥤ D_n(C) to the orbit category, is fully faithful (TauCeti.PeriodicComplex.fullyFaithfulOrbitDerivedCompressionFunctor):

⨁_{j : ℤ} Hom_{D^b(C)}(X, Y⟦n * j⟧) ≅ Hom_{D_n(C)}(Δₙ X, Δₙ Y).

For finitely generated modules over a finite-dimensional algebra of finite global dimension, this is Stai's Lemma 3.12.

The proof reduces to the full faithfulness of the orbit functor K^b(C)/[n] ⥤ K_n(C) on homotopy categories, which holds without any hypothesis on C (TauCeti.PeriodicComplex.full_orbitCompressionFunctor). In finite global dimension:

• every object of D^b(C) is the image of a bounded complex of projectives (DerivedCategory.Bounded.exists_projective_Qh_obj_iso_of_hasProjectiveDimensionLT);
• the localization K^b(C) ⥤ D^b(C) is bijective on morphisms out of such a complex (TauCeti.DerivedCategory.Bounded.Qh_map_bijective_of_projective), hence so is the induced functor K^b(C)/[n] ⥤ D^b(C)/[n] on orbit morphisms (TauCeti.OrbitCategory.map_map_bijective_iff);
• the compression of a bounded complex of projectives is a periodic complex of projectives, so it is homotopically projective (TauCeti.PeriodicComplex.isHomotopicallyProjective_of_projective) and the localization K_n(C) ⥤ D_n(C) is bijective on morphisms out of it.

So, on the full subcategory of K^b(C)/[n] on the bounded complexes of projectives, the functor to D^b(C)/[n] is an equivalence, and the composite with the derived orbit functor is the homotopy orbit functor followed by K_n(C) ⥤ D_n(C) (TauCeti.PeriodicComplex.orbitDerivedCompressionFunctorFactors), which is fully faithful there.

The finite global dimension hypothesis is used twice, for the bounded projective representatives in D^b(C) and for the homotopical projectivity of their compressions. The positivity of n is needed already for the homotopy orbit functor.

Main results

• TauCeti.PeriodicComplex.fullyFaithfulOrbitDerivedCompressionFunctor: in finite global dimension, the orbit functor D^b(C)/[n] ⥤ D_n(C) is fully faithful.

References

• T. Stai, The triangulated hull of periodic complexes, Mathematical Research Letters 25 (2018), 199–236, Section 3.3 and Lemma 3.12.
• B. Keller, On triangulated orbit categories, Documenta Mathematica 10 (2005), 551–581, Section 1.

noncomputable def TauCeti.PeriodicComplex.fullyFaithfulOrbitDerivedCompressionFunctor {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] (n : ℕ) [HasDerivedCategory C] [CategoryTheory.EnoughProjectives C] [NeZero n] (d : ℕ) (hd : ∀ (X : C), CategoryTheory.HasProjectiveDimensionLT X d) : (orbitDerivedCompressionFunctor n).FullyFaithful

The derived orbit functor is fully faithful in finite global dimension (Stai, Lemma 3.12). Let C be an abelian category with enough projectives in which every object has projective dimension < d, and let n be positive. Then the orbit functor D^b(C)/[n] ⥤ D_n(C) induced by compression is fully faithful: ⨁_{j : ℤ} Hom_{D^b(C)}(X, Y⟦n * j⟧) ≅ Hom_{D_n(C)}(Δₙ X, Δₙ Y).