Periodic complexes of projectives in finite global dimension

In an abelian category with enough projectives in which every object has projective dimension < d, every periodic complex with projective terms is homotopically projective (TauCeti.PeriodicComplex.isHomotopicallyProjective_of_projective). Morphisms out of it into an acyclic periodic complex are null-homotopic, so the periodic Verdier localization is bijective on morphisms out of it, and quasi-isomorphisms between such complexes are homotopy equivalences (TauCeti.PeriodicComplex.IsHomotopicallyProjective.quasiIso_iff).

The bound on projective dimensions cannot be dropped: over the dual numbers k[ε], which have infinite global dimension, multiplication by ε on k[ε] is a periodic complex with free terms which is not homotopically projective (TauCeti.not_isHomotopicallyProjective_dualNumberPeriodicComplex).

The proof inducts on a bound < m + 1 for the projective dimensions of the cycles of the acyclic target L (TauCeti.PeriodicComplex.nonempty_homotopy_zero_of_hasProjectiveDimensionLT). When the cycles of L are projective, L itself is contractible. Otherwise, choose a degreewise epimorphism π : Q ⟶ L from a contractible complex with projective terms. Its kernel N is acyclic, and the cycles of N are syzygies of those of L, so they satisfy the bound < m. Given f : K ⟶ L, lift it degreewise along π to g : Kⁱ ⟶ Qⁱ. The failure of g to commute with the differentials is a chain map K⟦-1⟧ ⟶ N, null-homotopic by induction, and the homotopy corrects g into a chain map K ⟶ Q lifting f. Since Q is contractible, f is null-homotopic (TauCeti.PeriodicComplex.nonempty_homotopy_zero_of_epi). No positivity of the period is needed. At period zero, where ZMod 0 is integer indexing, this is the K-projectivity of unbounded complexes of projectives in finite global dimension.

Main results

• TauCeti.PeriodicComplex.nonempty_homotopy_zero_of_hasProjectiveDimensionLT: a morphism from a periodic complex of projectives to an acyclic periodic complex whose cycles have bounded projective dimension is null-homotopic.
• TauCeti.PeriodicComplex.isHomotopicallyProjective_of_projective: in finite global dimension, periodic complexes of projectives are homotopically projective.

References

• T. Stai, The triangulated hull of periodic complexes, Mathematical Research Letters 25 (2018), 199–236, Section 3, Proposition 3.10.
• N. Spaltenstein, Resolutions of unbounded complexes, Compositio Mathematica 65 (1988), 121–154, for K-projectivity of complexes of projectives in finite global dimension.

theorem TauCeti.PeriodicComplex.nonempty_homotopy_zero_of_hasProjectiveDimensionLT {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {n : ℕ} [CategoryTheory.EnoughProjectives C] {K L : CochainComplex C (ZMod n)} [∀ (i : ZMod n), CategoryTheory.Projective (K.X i)] (hL : HomologicalComplex.Acyclic L) (d : ℕ) (hd : ∀ (i : ZMod n), CategoryTheory.HasProjectiveDimensionLT (HomologicalComplex.cycles L i) d) (f : K ⟶ L) : Nonempty (Homotopy f 0)

A morphism from a periodic complex of projectives to an acyclic periodic complex whose cycles all have projective dimension < d is null-homotopic.

theorem TauCeti.PeriodicComplex.isHomotopicallyProjective_of_projective {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Abelian C] {n : ℕ} [CategoryTheory.EnoughProjectives C] (d : ℕ) (hd : ∀ (X : C), CategoryTheory.HasProjectiveDimensionLT X d) (K : CochainComplex C (ZMod n)) [∀ (i : ZMod n), CategoryTheory.Projective (K.X i)] : IsHomotopicallyProjective K

In an abelian category with enough projectives in which every object has projective dimension < d, every periodic complex of projectives is homotopically projective.