Group cohomology of the cyclic representation of an automorphism of finite order #
Let T be an additive commutative group, m > 0, and A : T ≃+ T an additive automorphism of
order dividing m, that is A^[m] = id. Then the cyclic group C_m = Multiplicative (ZMod m)
acts on T with its standard generator ofAdd 1 acting by A; this is the integral
representation AddEquiv.cyclicRep A hA : Rep ℤ C_m. Its norm N = ∑ g, ρ g is the ordered sum
of powers ∑ j < m, A^j (AddEquiv.cyclicRepresentation_norm_apply).
The group cohomology of C_m is computed by Mathlib's periodic resolution
(Rep.FiniteCyclicGroup.groupCohomologyπOdd, Rep.FiniteCyclicGroup.groupCohomologyπEven). For
the cyclic representation of A these quotient maps read as follows.
- In odd degree, and so in degree one, the classes are the elements
twith∑ j < m, A^j t = 0, two of them having the same class exactly when they differ byb - A bfor someb : T(AddEquiv.groupCohomologyπOdd_cyclicRep_eq_iff). The image ofA - 1is that of1 - A, by negating the argument. These are the twiststfor which the affine mapx ↦ A x + thas order dividingm, up to conjugation by translations, which changesttot + (b - A b): two norm-zero twists have the same class exactly when their affine maps are conjugate by a translation (AddEquiv.groupCohomologyπOdd_cyclicRep_eq_iff_exists_conj_affineEquiv). - In positive even degree, the classes are the fixed points of
A, two of them having the same class exactly when they differ by a norm∑ j < m, A^j y(AddEquiv.groupCohomologyπEven_cyclicRep_eq_iff).
Every class arises in each case (Rep.FiniteCyclicGroup.groupCohomologyπOdd_surjective,
Rep.FiniteCyclicGroup.groupCohomologyπEven_surjective). For a homomorphism χ : C_n → C_m
sending the generator to its d-th power, an additive map
f intertwining A^d with B induces change-of-group maps on these classes. In degree one,
the class of t maps to the class of f (partialNorm A d t); in degree two, the class of a
fixed point v maps to the class of q • f v, where d * n = q * m
(AddMonoidHom.map_cyclicRepResHom_groupCohomologyπOdd_one,
AddMonoidHom.map_cyclicRepResHom_groupCohomologyπEven_two). Coefficient maps are the case
χ = id and d = q = 1.
The representation itself is constructed in TauCeti.RepresentationTheory.Cyclic.Automorphism.
The second part of the file treats an exact sequence 0 → Λ → E → T → 0 of abelian groups whose
maps i and p intertwine automorphisms AΛ, AE, AT of order dividing m, for instance a
lattice Λ in a real vector space E with quotient torus T. It is a short exact sequence of
cyclic representations (AddMonoidHom.shortExact_cyclicRepShortComplex), and its connecting
homomorphism δ : H¹(C_m, T) ⟶ H²(C_m, Λ) is computed on classes: for t killed by the norm
and a lift s : E of t, the norm ∑ j < m, AE^j s is the image of a fixed point v of AΛ
(AddMonoidHom.exists_apply_eq_sum_range_iterate), and δ sends the class of t to the class
of v (TauCeti.δ_cyclicRepShortComplex_groupCohomologyπOdd_one). Changing s by the
image of l : Λ changes v by the norm of l (AddMonoidHom.sum_range_iterate_add_apply),
which does not change its class. In particular, when multiplication by m is injective on E
and m • s = i v, the class of the image of s is sent to the class of v, that is
δ [v / m] = [v]
(TauCeti.δ_cyclicRepShortComplex_groupCohomologyπOdd_one_of_nsmul_eq). When multiplication
by m is bijective on E, as for a real vector space, the cohomology of E vanishes in positive
degrees and δ is an isomorphism (TauCeti.cyclicConnectingEquiv). The connecting homomorphism
is natural under arbitrary homomorphisms χ : C_l → C_m: maps between two such sequences
which intertwine the appropriate powers of the automorphisms and commute with the sequence
maps commute with δ (TauCeti.map_δ_cyclicRepShortComplex).
The construction is compatible with products. The cyclic representation of the product
automorphism A × B of T × M is the biproduct of those of A and B
(AddEquiv.cyclicRepBinaryBiconeIsBilimit), so the projections identify its cohomology with the
product of the two cohomologies
(CategoryTheory.Limits.BinaryBicone.IsBilimit.groupCohomologyProdEquiv); in degrees one and two
this sends the class of (t, u) to the pair of classes of t and u
(TauCeti.groupCohomologyProdEquiv_groupCohomologyπOdd_one,
TauCeti.groupCohomologyProdEquiv_groupCohomologyπEven_two). The product of two equivariant exact
sequences is again one (AddMonoidHom.shortExact_cyclicRepShortComplex_prodMap), and under these
identifications its connecting homomorphism is the product of the two connecting homomorphisms
(TauCeti.groupCohomologyProdEquiv_δ_cyclicRepShortComplex_prodMap); in particular the connecting
isomorphism of a product of tori is the product of their connecting isomorphisms
(TauCeti.groupCohomologyProdEquiv_cyclicConnectingEquiv).
The last part of the file specializes to the trivial action, given by the identity automorphism.
Then the odd-degree cohomology of C_m is the m-torsion T[m]
(TauCeti.groupCohomologyOddEquivTorsionBy) and its positive even-degree cohomology is T ⧸ m T
(TauCeti.groupCohomologyEvenEquivQuotSMulTop). Through these identifications the connecting
homomorphism of 0 → Λ → E → T → 0 becomes the snake-lemma map TauCeti.torsionByδ of
multiplication by m, which sends t to the coset of v when m • s = i v for a lift s of
t (TauCeti.groupCohomologyEvenEquivQuotSMulTop_δ). When multiplication by m is bijective on
E, the snake-lemma isomorphism T[m] ≃ Λ ⧸ m Λ is therefore the connecting isomorphism
(TauCeti.torsionByδEquiv_eq_trans_cyclicConnectingEquiv). For a torus E ⧸ Λ this is the
description of its m-torsion points as (1 / m) Λ ⧸ Λ ≃ Λ ⧸ m Λ.
Main definitions #
AddMonoidHom.cyclicRepShortComplex: the short complexΛ ⟶ E ⟶ Tof cyclic representations.TauCeti.cyclicConnectingEquiv: the connecting isomorphismH¹(C_m, T) ≃ H²(C_m, Λ)when multiplication bymis bijective onE.TauCeti.groupCohomologyOddEquivTorsionBy: odd-degree cohomology of the trivial action isT[m].TauCeti.groupCohomologyEvenEquivQuotSMulTop: positive even-degree cohomology of the trivial action isT ⧸ m T.
Main results #
AddEquiv.groupCohomologyπOdd_cyclicRep_eq_zero_iff,AddEquiv.groupCohomologyπOdd_cyclicRep_eq_iff: odd-degree classes are norm-zero elements modulo the image of1 - A.AddEquiv.groupCohomologyπOdd_cyclicRep_eq_iff_exists_conj_affineEquiv: two norm-zero twists have the same odd-degree class exactly when their affine maps are conjugate by a translation.AddEquiv.groupCohomologyπEven_cyclicRep_eq_zero_iff,AddEquiv.groupCohomologyπEven_cyclicRep_eq_iff: positive even-degree classes are fixed points ofAmodulo the image of the norm.AddMonoidHom.map_cyclicRepHom_groupCohomologyπOdd_one,AddMonoidHom.map_cyclicRepHom_groupCohomologyπEven_two: an equivariant map acts on degree-1and degree-2classes by acting on their representatives.AddMonoidHom.shortExact_cyclicRepShortComplex_iff: a sequence of abelian groups with equivariant maps is exact exactly when it is a short exact sequence of cyclic representations.AddMonoidHom.shortExact_cyclicRepShortComplex_prodMap: the product of two such short exact sequences is short exact.AddMonoidHom.map_cyclicRepResHom_groupCohomologyπOdd_one,AddMonoidHom.map_cyclicRepResHom_groupCohomologyπEven_two: change-of-group maps act on degree-one twists by partial norms and on degree-two fixed points by the degree factor.AddMonoidHom.shortExact_cyclicRepShortComplex: an exact sequence of abelian groups with equivariant maps is a short exact sequence of cyclic representations.TauCeti.δ_cyclicRepShortComplex_groupCohomologyπOdd_one,TauCeti.δ_cyclicRepShortComplex_groupCohomologyπOdd_one_of_nsmul_eq: the connecting homomorphism sends the class oftto the class of the norm of a lift oft, and the class ofv / mto the class ofv.TauCeti.map_δ_cyclicRepShortComplex: naturality of the connecting homomorphism under maps of equivariant exact sequences.TauCeti.groupCohomologyProdEquiv_groupCohomologyπOdd_one,TauCeti.groupCohomologyProdEquiv_groupCohomologyπEven_two: the cohomology of a product of cyclic representations is the product of the cohomologies, computed on representatives.TauCeti.groupCohomologyProdEquiv_δ_cyclicRepShortComplex_prodMap,TauCeti.groupCohomologyProdEquiv_cyclicConnectingEquiv: the connecting homomorphism and the connecting isomorphism of a product of sequences are the products of those of the factors.TauCeti.map_δ_cyclicRepShortComplex: naturality of the connecting homomorphism under homomorphisms of cyclic groups and compatible maps of exact sequences.TauCeti.groupCohomologyEvenEquivQuotSMulTop_δ: for trivial actions, the connecting homomorphism is the snake-lemma mapT[m] → Λ ⧸ m Λ.TauCeti.torsionByδEquiv_eq_trans_cyclicConnectingEquiv: for trivial actions andmbijective onE, the snake-lemma isomorphismT[m] ≃ Λ ⧸ m Λis the connecting isomorphism.
References #
- K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics 87, Springer (1982), Chapter I, §6 (the periodic resolution of a cyclic group).
- J.-P. Serre, Local Fields, Graduate Texts in Mathematics 67, Springer (1979), Chapter VIII, §4 (the cohomology of cyclic groups).
Odd-degree classes of the cyclic representation. A norm-zero element x has zero class
in odd degree exactly when x = b - A b for some b : T.
Odd-degree classes of the cyclic representation. Two norm-zero elements x and y
have the same class in odd degree exactly when y = x + (b - A b) for some b : T.
Translation conjugacy classes of affine maps. Two norm-zero twists x and y have the
same class in odd degree, in particular in degree one, exactly when the affine maps
z ↦ A z + x and z ↦ A z + y are conjugate by a translation of T.
Even-degree classes of the cyclic representation. A fixed point x of A has zero
class in positive even degree exactly when it is a norm ∑ j < m, A^j y.
Even-degree classes of the cyclic representation. Two fixed points x and y of A
have the same class in positive even degree exactly when y = x + ∑ j < m, A^j z for some
z : T.
Equivariant maps on the cohomology classes #
Pulling back a degree-one class along a homomorphism sending the generator to its
d-th power replaces its twist x by f (partialNorm A d x).
Pulling back a degree-two class multiplies its fixed-point representative by q,
where d * n = q * m. In particular, restriction along the standard inclusion of a cyclic
subgroup (m = d * n) preserves the representative.
The connecting class of an equivariant short exact sequence #
The short complex Λ ⟶ E ⟶ T of cyclic representations of C_m defined by additive maps
i : Λ →+ E and p : E →+ T intertwining automorphisms AΛ, AE, AT of order dividing m,
with p ∘ i = 0. Its body is exposed, so that its terms are the cyclic representations of AΛ,
AE and AT by definition and its connecting homomorphisms act on their cohomology.
Instances For
The short complex Λ ⟶ E ⟶ T of cyclic representations is short exact when
0 → Λ → E → T → 0 is an exact sequence of abelian groups.
The short complex Λ ⟶ E ⟶ T of cyclic representations is short exact exactly when
0 → Λ → E → T → 0 is an exact sequence of abelian groups.
The product of two short exact sequences of cyclic representations is short exact: for
equivariant exact sequences 0 → Λ → E → T → 0 and 0 → Λ' → E' → T' → 0, the sequence
0 → Λ × Λ' → E × E' → T × T' → 0 of cyclic representations of the product automorphisms is
short exact.
The integral connecting class. For the short exact sequence 0 → Λ → E → T → 0 of
cyclic representations, let t be killed by the norm ∑ j < m, AT^j, lift it to s : E, and
write the norm ∑ j < m, AE^j s as the image of a fixed point v of AΛ. Then the connecting
homomorphism sends the degree-1 class of t to the degree-2 class of v.
The connecting class of v / m. If multiplication by m is injective on E and
m • s = i v for a fixed point v of AΛ, then the connecting homomorphism sends the
degree-1 class of the image t of s in T to the degree-2 class of v.
The connecting isomorphism H¹(C_m, T) ≃ H²(C_m, Λ). When multiplication by m is
bijective on E, for instance when E is a real vector space, the cohomology of E vanishes in
positive degrees and the connecting homomorphism of 0 → Λ → E → T → 0 is an isomorphism.
Instances For
The connecting isomorphism H¹(C_m, T) ≃ H²(C_m, Λ) is the connecting homomorphism.
Naturality under homomorphisms of cyclic groups. Suppose χ : C_l → C_m sends the
selected generator to its d-th power. Maps of exact sequences intertwining the d-th
iterates of the source automorphisms with the target automorphisms commute with the
connecting homomorphisms. This includes restriction, inflation, and coefficient maps.
Products #
Degree-one classes of a product. The identification of H¹(C_m, T × M) with
H¹(C_m, T) × H¹(C_m, M) by the projections sends the class of a norm-zero element (t, u) of
T × M to the pair of the classes of t and u.
Degree-two classes of a product. The identification of H²(C_m, T × M) with
H²(C_m, T) × H²(C_m, M) by the projections sends the class of a fixed point (t, u) of A × B
to the pair of the classes of t and u.
The connecting homomorphism of a product. For equivariant exact sequences
0 → Λ → E → T → 0 and 0 → Λ' → E' → T' → 0, the connecting homomorphism
Hⁿ(C_m, T × T') ⟶ H^q(C_m, Λ × Λ') of the product sequence (short exact by
AddMonoidHom.shortExact_cyclicRepShortComplex_prodMap) is the product of the two connecting
homomorphisms, once the cohomology of each product is identified with the product of the
cohomologies by the projections.
The connecting isomorphism of a product. When multiplication by m is bijective on E
and on E', the connecting isomorphism H¹(C_m, T × T') ≃ H²(C_m, Λ × Λ') of the product
sequence is the product of the connecting isomorphisms H¹(C_m, T) ≃ H²(C_m, Λ) and
H¹(C_m, T') ≃ H²(C_m, Λ'), once the cohomology of each product is identified with the product of
the cohomologies by the projections.
Trivial actions #
Odd-degree cohomology of the trivial action. When C_m acts trivially on T, its
cohomology in odd degree is the m-torsion T[m]: the class of a norm-zero element x
corresponds to x itself.
Instances For
The odd-degree class corresponding to x : T[m] is the class of x.
The odd-degree class of a norm-zero element x corresponds to x in T[m].
Even-degree cohomology of the trivial action. When C_m acts trivially on T, its
cohomology in positive even degree is T ⧸ m T: the class of x corresponds to the coset of
x.
Instances For
The even-degree class corresponding to the coset of x is the class of x.
The even-degree class of x corresponds to the coset of x in T ⧸ m T.
The connecting map of trivial actions is the snake map. For the trivial actions of C_m
on an exact sequence 0 → Λ → E → T → 0, identify H¹(C_m, T) with T[m] and H²(C_m, Λ)
with Λ ⧸ m Λ. Then the connecting homomorphism δ becomes the snake-lemma connecting map
TauCeti.torsionByδ, which sends t to the coset of v when m • s = i v for a lift s of
t.
T[m] ≃ Λ ⧸ m Λ is the connecting isomorphism. When multiplication by m is bijective on
E, the snake-lemma isomorphism T[m] ≃ Λ ⧸ m Λ is the connecting isomorphism
H¹(C_m, T) ≃ H²(C_m, Λ) of the trivial actions, read through H¹(C_m, T) ≃ T[m] and
H²(C_m, Λ) ≃ Λ ⧸ m Λ.