Periodicity of compression #
The n-periodic compression Δₙ is invariant under shifting by a multiple k of n: there is
a natural isomorphism Δₙ (K⟦k⟧) ≅ Δₙ K. In residue degree r, it sends the summand
(K⟦k⟧)^i = K^(i+k) to the summand of degree i + k of Δₙ K, which lies in the same residue
class, with the sign (-1)^(k * i). The sign compensates for the factor (-1)^k on the
differential of K⟦k⟧, so it is needed exactly when k is odd.
This is not the composite of compressionShiftIso with a periodicity isomorphism of
ZMod n-graded complexes: for odd n, the shift by n of a periodic complex has the negated
differential and need not be isomorphic to it. The sign (-1)^(k * i) depends on the integer
degree i, which the compression still remembers summand by summand.
The isomorphism is natural, so it gives the periodicity isomorphism
[k] ⋙ Δₙ ≅ Δₙ of the compression functor on bounded complexes, and, along the localization
functor to the bounded homotopy category, of the compression functor K^b(C) ⥤ K_n(C). Taking
k = n, Keller's universal property of the orbit category descends compression to the orbit
functor K^b(C)/[n] ⥤ K_n(C) (TauCeti.PeriodicComplex.orbitCompressionFunctor).
The periodicity isomorphisms for shifts by a, b and a + b are compatible with the
identification K⟦a + b⟧ ≅ K⟦a⟧⟦b⟧ only up to the sign (-1)^(a * b)
(TauCeti.PeriodicComplex.homotopyCompressionFunctorPeriodicityIso_shiftComp_hom_app), since the
summand of degree i acquires (-1)^(b * i) * (-1)^(a * (i + b)) on one side and
(-1)^((a + b) * i) on the other. The one for the shift by 0 is the identification X⟦0⟧ ≅ X.
Hence the powers of the periodicity isomorphism for the shift by n, which define the action of
the orbit functor on morphisms, are the periodicity isomorphisms for the shifts by multiples of
n up to signs.
Main definitions #
TauCeti.PeriodicComplex.compressionPeriodicityIso: the isomorphismΔₙ (K⟦k⟧) ≅ Δₙ Kforkdivisible byn.TauCeti.PeriodicComplex.compressionFunctorPeriodicityIso: its version for the compression functor on bounded complexes.TauCeti.PeriodicComplex.homotopyCompressionFunctorPeriodicityIso: its version for the compression functorK^b(C) ⥤ K_n(C)on homotopy categories.TauCeti.PeriodicComplex.orbitCompressionFunctor: the orbit functorK^b(C)/[n] ⥤ K_n(C).
Main results #
TauCeti.PeriodicComplex.compressionPeriodicityIso_add_hom: the periodicity isomorphisms compose up to the sign(-1)^(a * b).TauCeti.PeriodicComplex.homotopyCompressionFunctorPeriodicityIso_shiftComp_hom_app: the same for the compression functorK^b(C) ⥤ K_n(C).
References #
- B. Keller, On triangulated orbit categories, Documenta Mathematica 10 (2005), 551–581, Section 1.
- T. Stai, The triangulated hull of periodic complexes, Mathematical Research Letters 25 (2018), 199–236, Section 3.3.
compressionFunctorPeriodicityIsoon bounded complexes is assembled likeTauCeti.PeriodicComplex.compressionFunctorShiftIso, from their shared comparisonTauCeti.PeriodicComplex.shiftCompCompressionFunctorIso(TauCeti.Algebra.Homology.Periodic.Compression.Shift), and the descent ofhomotopyCompressionFunctorPeriodicityIsoalong the localization functor follows the construction of Mathlib'sCategoryTheory.Functor.commShiftOfLocalization.iso.
The degreewise periodicity isomorphism (Δₙ (K⟦k⟧))^r ≅ (Δₙ K)^r for k divisible by n:
the summand of degree i of K⟦k⟧ goes to the summand of degree i + k of K with the sign
(-1)^(k * i).
Instances For
On a summand, the periodicity isomorphism is the inclusion of the summand whose degree has
been translated by k, with the sign (-1)^(k * i).
On a summand, the periodicity isomorphism is the inclusion of the summand whose degree has
been translated by k, with the sign (-1)^(k * i).
On a summand, the inverse periodicity isomorphism is the inclusion of the summand whose
degree has been translated back by k, with the sign (-1)^(k * (j - k)).
On a summand, the inverse periodicity isomorphism is the inclusion of the summand whose
degree has been translated back by k, with the sign (-1)^(k * (j - k)).
The compression of a complex is invariant under shifting it by a multiple k of n. The
isomorphism reindexes the summands by translation by k and multiplies the summand of degree
i by (-1)^(k * i).
Instances For
The component in residue degree r of the periodicity isomorphism.
The inverse component in residue degree r of the periodicity isomorphism.
The periodicity isomorphism is natural in the complex.
The periodicity isomorphism is natural in the complex.
The periodicity isomorphisms for shifts by a, b and c = a + b, all divisible by n, are
compatible with the identification K⟦c⟧ ≅ K⟦a⟧⟦b⟧ up to the sign (-1)^(a * b).
The periodicity isomorphisms for shifts by a, b and c = a + b, all divisible by n, are
compatible with the identification K⟦c⟧ ≅ K⟦a⟧⟦b⟧ up to the sign (-1)^(a * b).
The compression functor on bounded complexes is invariant under the shift by a multiple k
of n.
Instances For
The component of compressionFunctorPeriodicityIso at a bounded complex is the periodicity
isomorphism of its underlying complex, after identifying the underlying complex of its shift with
the shift of its underlying complex.
The component of the inverse of compressionFunctorPeriodicityIso at a bounded complex is the
inverse periodicity isomorphism of its underlying complex, followed by the identification of the
shift of its underlying complex with the underlying complex of its shift.
The compression of a map f : K ⟶ L⟦k⟧ of bounded complexes followed by the periodicity
isomorphism is, on the compressions of the underlying complexes, the compression of the underlying
chain map followed by the periodicity isomorphism of the underlying complex.
The periodicity isomorphisms of compression of bounded complexes for shifts by a, b and
c = a + b, all divisible by n, compose to the one for c up to the sign (-1)^(a * b).
The compression functor K^b(C) ⥤ K_n(C) is invariant under the shift by a multiple k of
n. It is induced, along the localization functor HomotopyCategory.Bounded.quotient C, by the
periodicity isomorphism of compression of bounded complexes.
Instances For
On the class of a bounded complex, the periodicity isomorphism of K^b(C) ⥤ K_n(C) is the
class of the periodicity isomorphism of compression, conjugated by the identifications of the
descended compression on classes of bounded complexes.
On the class of a bounded complex, the inverse of the periodicity isomorphism of
K^b(C) ⥤ K_n(C) is the class of the inverse periodicity isomorphism of compression, conjugated by
the identifications of the descended compression on classes of bounded complexes.
On the class of a bounded complex, the periodicity isomorphism of K^b(C) ⥤ K_n(C), preceded
by the identification of the class of the shift with the shift of the class, is the class of the
periodicity isomorphism of compression, conjugated by the identifications of the descended
compression on classes of bounded complexes.
On the class of a bounded complex, the periodicity isomorphism of K^b(C) ⥤ K_n(C), preceded
by the identification of the class of the shift with the shift of the class, is the class of the
periodicity isomorphism of compression, conjugated by the identifications of the descended
compression on classes of bounded complexes.
The periodicity isomorphisms of K^b(C) ⥤ K_n(C) for shifts by a, b and c = a + b, all
divisible by n, compose to the one for c up to the sign (-1)^(a * b).
The periodicity isomorphism of K^b(C) ⥤ K_n(C) for the shift by k = 0 is induced by the
identification X⟦0⟧ ≅ X.
The orbit functor K^b(C)/[n] ⥤ K_n(C): the descent of the compression functor
K^b(C) ⥤ K_n(C) to the orbit category, along its periodicity isomorphism for the shift by n.
It restricts to compression along the projection K^b(C) ⥤ K^b(C)/[n]
(TauCeti.OrbitCategory.projectionCompLift).