Compression detects acyclicity #
In a preadditive category with homology, a strictly bounded cochain complex K is acyclic exactly
when its n-periodic compression Δₙ K is acyclic
(TauCeti.PeriodicComplex.compression_acyclic_iff). Exactness of the compression at the residue
of i implies exactness of K at i, assuming only zero morphisms, a compression, and homology
of K at i (TauCeti.PeriodicComplex.exactAt_of_compression_exactAt). Reflection of acyclicity
follows degreewise (TauCeti.PeriodicComplex.acyclic_of_compression_acyclic).
For a strictly bounded K, the homology of
Δₙ K in residue r is the direct sum of the homology objects of K in the degrees congruent to
r; we do not construct this decomposition, but only the finite sum decomposition of the identity
of the short complex of Δₙ K at r through the short complexes of K that it rests on. The
summand inclusions and projections are morphisms of short complexes because the differential of
Δₙ K is induced summand by summand by the differential of K, and the identity is the finite
sum of their composites because K has only finitely many nonzero terms. Conversely, each short
complex of K is a retract of a short complex of Δₙ K.
This is the input needed to descend compression from bounded homotopy categories to bounded and periodic derived categories: compression preserves and reflects quasi-isomorphisms of bounded complexes.
References #
- T. Stai, The triangulated hull of periodic complexes, Mathematical Research Letters 25 (2018), 199–236, Section 3.3.
A cochain complex with homology at i is exact there when its n-periodic compression
is exact at the residue of i. This holds in any category with zero morphisms.
A cochain complex with homology in every degree is acyclic when its n-periodic
compression is acyclic. This holds in any category with zero morphisms.
A strictly bounded cochain complex is acyclic exactly when its n-periodic compression is
acyclic.