The orbit functor K^b(C)/[n] ⥤ K_n(C) is fully faithful

Let C be a preadditive category with finite coproducts and let n be positive. Compression K^b(C) ⥤ K_n(C) descends to the orbit functor TauCeti.PeriodicComplex.orbitCompressionFunctor n : K^b(C)/[n] ⥤ K_n(C), which sends a homogeneous piece f : X ⟶ Y⟦n * j⟧ of an orbit morphism to the compression of f followed by the j-th power of the periodicity isomorphism. This file proves that the orbit functor is fully faithful:

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

The j-th power of the periodicity isomorphism is, up to a sign, the periodicity isomorphism for the shift by n * j (TauCeti.PeriodicComplex.exists_periodicityZPow_hom_app_eq), because the periodicity isomorphisms compose up to the signs TauCeti.PeriodicComplex.homotopyCompressionFunctorPeriodicityIso_shiftComp_hom_app. Hence the image of an orbit morphism is represented by the sum of the compressions of its homogeneous pieces, transported along the periodicity isomorphisms. Faithfulness follows by extracting the components of degree n * j (TauCeti.PeriodicComplex.compressionComponent), which is compatible with homotopies, and fullness from the decomposition of every morphism between compressions of bounded complexes into its finitely many components. The hypothesis that n is nonzero is needed: for n = 0 an orbit hom group is a direct sum of infinitely many copies of a hom group of K^b(C), all sent to the same hom group of K_0(C).

For finitely generated modules over a finite-dimensional algebra of finite global dimension, bounded complexes of projectives model the bounded derived category and periodic complexes of projectives model the periodic derived category. This computation then underlies the full faithfulness of the derived orbit functor D^b(C)/[n] ⥤ D_n(C) (Stai, Lemma 3.12).

Main results

• TauCeti.PeriodicComplex.exists_periodicityZPow_hom_app_eq: the powers of the periodicity isomorphism are the periodicity isomorphisms for the multiples of n, up to signs.
• TauCeti.PeriodicComplex.faithful_orbitCompressionFunctor: the orbit functor is faithful.
• TauCeti.PeriodicComplex.full_orbitCompressionFunctor: the orbit functor is full.

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.

theorem TauCeti.PeriodicComplex.exists_periodicityZPow_hom_app_eq {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] (n : ℕ) (j : ℤ) : ∃ (ε : ℤˣ), ∀ (X : HomotopyCategory.Bounded C), ((homotopyCompressionFunctorPeriodicityIso n ↑n ⋯).periodicityZPow j).hom.app X = ε • (homotopyCompressionFunctorPeriodicityIso n (↑n * j) ⋯).hom.app X

The j-th power of the periodicity isomorphism of K^b(C) ⥤ K_n(C) for the shift by n is, up to a sign independent of the object, the periodicity isomorphism for the shift by n * j.

instance TauCeti.PeriodicComplex.faithful_orbitCompressionFunctor {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] (n : ℕ) [NeZero n] : (orbitCompressionFunctor n).Faithful

For positive n, the orbit functor K^b(C)/[n] ⥤ K_n(C) is faithful.

instance TauCeti.PeriodicComplex.full_orbitCompressionFunctor {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasFiniteCoproducts C] (n : ℕ) [NeZero n] : (orbitCompressionFunctor n).Full

For positive n, the orbit functor K^b(C)/[n] ⥤ K_n(C) is full.