Shapiro's isomorphism and the cup product #
Let U be a subgroup of a compact group G and let μ : A →+ B →+ C be a U-equivariant
pairing of discrete U-modules. The pointwise pairing
TauCeti.DiscreteCoind.pointwisePairing of the coinduced modules is a G-equivariant pairing
Coind_U^G A × Coind_U^G B → Coind_U^G C, and the explicit Shapiro maps are multiplicative for
it:
sh (a ⌣ b) = sh a ⌣ sh b,
where the left cup is taken over G along the pointwise pairing and the right cup over U along
μ. The forward Shapiro map is the compatible-pair pullback along U ↪ G and evaluation at 1,
and evaluation at 1 intertwines the two pairings, so each of the six low-degree statements is an
instance of the naturality of the cup product in compatible pairs
(TauCeti/RepresentationTheory/Homological/ContCohomology/Cup/Naturality.lean).
When U is open in a profinite group G, A and B are discrete U-modules and the target C
is a discrete G-module restricted to U, corestriction is inverse Shapiro followed by the
coefficient map of the trace, and the multiplicativity turns into the corestriction of a cup
product of U:
cor (a ⌣_U b) = tr_* (sh⁻¹ a ⌣_G sh⁻¹ b).
The right-hand side is the cup over G of the inverse Shapiro images along the pointwise pairing
of coinduced modules, pushed to P by the trace Coind_U^G P → P. For 𝔽_p-coefficients and
μ the multiplication, tr ∘ pointwisePairing is the standard G-invariant symmetric form on the
permutation module 𝔽_p[G ⧸ U], and the formula reads the cup product of U on the cohomology of
G with coefficients in that module. This is how the duality of an open subgroup of a Demushkin
group is read from the duality of the group itself.
Main results #
TauCeti.ContCohomology.explicitShapiro0_explicitCup00,explicitShapiroMap1_explicitCup01,explicitShapiroMap1_explicitCup10,explicitShapiroMap2_explicitCup02,explicitShapiroMap2_explicitCup11andexplicitShapiroMap2_explicitCup20: the Shapiro maps are multiplicative, in each bidegree(p, q)withp + q ≤ 2.TauCeti.ContCohomology.explicitCor0_explicitCup00,explicitCor1_explicitCup01,explicitCor1_explicitCup10,explicitCor2_explicitCup02,explicitCor2_explicitCup11andexplicitCor2_explicitCup20: the corestriction of a cup product of an open subgroup is the trace of the cup product overGof the inverse Shapiro images.
References #
- J.-P. Serre, Structure de certains pro-p-groupes (d'après Demuškin), Séminaire Bourbaki 8 (1962/63), exposé 252, §9.2: the duality of an open subgroup is deduced from that of the group through Shapiro's lemma.
- J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., Springer (2008), (1.4.2), (1.5.3) and (1.6.4): naturality of the cup product, its behaviour under the change of group, and Shapiro's lemma.
The Shapiro maps are multiplicative #
The degree-zero Shapiro isomorphism is multiplicative for the (0,0) cup product.
The Shapiro maps are multiplicative for the (0,1) cup product.
The Shapiro maps are multiplicative for the (1,0) cup product.
The Shapiro maps are multiplicative for the (0,2) cup product.
The Shapiro maps are multiplicative for the (1,1) cup product: sh (a ⌣ b) = sh a ⌣ sh b,
with the cup over G taken along the pointwise pairing of the coinduced modules.
The Shapiro maps are multiplicative for the (2,0) cup product.
The corestriction of a cup product of an open subgroup #
Here M and N are discrete U-modules, P is a discrete G-module restricted to the open
subgroup U, and μ is a U-equivariant pairing M × N → P; only the target P needs the
ambient G-action, for the corestriction and the trace. Corestriction is inverse Shapiro followed
by the trace
(TauCeti.ContCohomology.explicitCor2_eq_explicitCoeff2_trace and its lower-degree companions),
so the multiplicativity above computes the corestriction of a cup product of U.
The corestriction of a (0,0) cup product of U is the trace of the (0,0) cup product
over G of the inverse Shapiro images, along the pointwise pairing of the coinduced modules.
The corestriction of a (0,1) cup product of U is the trace of the (0,1) cup product
over G of the inverse Shapiro images, along the pointwise pairing of the coinduced modules.
The corestriction of a (1,0) cup product of U is the trace of the (1,0) cup product
over G of the inverse Shapiro images, along the pointwise pairing of the coinduced modules.
The corestriction of a (0,2) cup product of U is the trace of the (0,2) cup product
over G of the inverse Shapiro images, along the pointwise pairing of the coinduced modules.
The corestriction of a (1,1) cup product of U is the trace of the (1,1) cup product
over G of the inverse Shapiro images, along the pointwise pairing of the coinduced modules:
cor (a ⌣_U b) = tr_* (sh⁻¹ a ⌣_G sh⁻¹ b).
The corestriction of a (2,0) cup product of U is the trace of the (2,0) cup product
over G of the inverse Shapiro images, along the pointwise pairing of the coinduced modules.