The Euler formulas for the dimensions of Hⁱ(-, 𝔽_p) #
Let G be a topologically finitely generated pro-p group and U an open subgroup. This file
states the two-term and the three-term Euler formulas as identities between the 𝔽_p-dimensions
of the cohomology groups Hⁱ(-, 𝔽_p) = cohomFp p - i with trivial coefficients, with the
alternating sums taken in ℤ:
dim H⁰(U) - dim H¹(U) = [G : U] * (dim H⁰(G) - dim H¹(G)) (cd_p G ≤ 1),
dim H⁰(U) - dim H¹(U) + dim H²(U) = [G : U] * (dim H⁰(G) - dim H¹(G) + dim H²(G)) (cd_p G ≤ 2).
All the spaces involved are finite-dimensional: H⁰(-, 𝔽_p) ≅ 𝔽_p has dimension one
(TauCeti.finrank_cohomFp_zero); H¹(-, 𝔽_p) is finite-dimensional of dimension the topological
generator rank (TauCeti.IsProP.finite_cohomFp_one_iff, TauCeti.IsProP.finrank_cohomFp_one),
and U is topologically finitely generated with G; and in the three-term formula H²(G, 𝔽_p)
is finite by hypothesis, which makes H²(U, 𝔽_p) finite
(TauCeti.IsProP.finite_cohomFp_openSubgroup). The hypothesis on H²(G, 𝔽_p) is not
redundant: topological finite generation bounds H⁰ and H¹ only, and dim H²(G, 𝔽_p) is the
relation rank of G.
The formulas are the generator-rank identities of
TauCeti.Topology.Algebra.Group.Profinite.ProP.EulerCharacteristic.Basic and
TauCeti.Topology.Algebra.Group.Profinite.ProP.EulerCharacteristic.ThreeTerm, read through these
dimension counts. For a free pro-p group of finite rank the two-term formula is the Schreier
index formula; for a Demushkin group, where dim H⁰ = dim H² = 1, the three-term formula is the
rank formula d(U) - 2 = [G : U] * (d(G) - 2) of an open subgroup.
Main results #
TauCeti.IsProP.finrank_cohomFp_zero_sub_finrank_cohomFp_one_openSubgroupand its formTauCeti.CohomologicalDimensionLE.finrank_cohomFp_zero_sub_finrank_cohomFp_one_openSubgroupundercd_p G ≤ 1: the two-term Euler formula.finrank_cohomFp_zero_sub_finrank_cohomFp_one_add_finrank_cohomFp_two_openSubgroup, in the namespaceTauCeti.IsProPand, undercd_p G ≤ 2, inTauCeti.CohomologicalDimensionLE: the three-term Euler formula.
References #
- J.-P. Serre, Galois Cohomology, Ch. I, §4.1.
- H. Koch, Galois Theory of
p-Extensions, §5.4. - J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., Ch. III §3.
The two-term Euler formula. Let G be a topologically finitely generated profinite pro-p
group whose explicit H² vanishes on the discrete modules of order p with trivial action, and
let U be an open subgroup. Then, in ℤ,
dim H⁰(U, 𝔽_p) - dim H¹(U, 𝔽_p) = [G : U] * (dim H⁰(G, 𝔽_p) - dim H¹(G, 𝔽_p)).
The four spaces are finite-dimensional: H⁰ always, and H¹ because G and U are
topologically finitely generated (TauCeti.IsProP.finite_cohomFp_one_iff).
The three-term Euler formula. Let G be a topologically finitely generated profinite
pro-p group whose canonical H³ vanishes on the discrete modules of order p with trivial
action, with H²(G, 𝔽_p) finite, and let U be an open subgroup. Then, in ℤ,
dim H⁰(U) - dim H¹(U) + dim H²(U) = [G : U] * (dim H⁰(G) - dim H¹(G) + dim H²(G)),
for the cohomology with trivial coefficients 𝔽_p. The six spaces are finite-dimensional: H⁰
always, H¹ because G and U are topologically finitely generated, and H²(U, 𝔽_p) by
TauCeti.IsProP.finite_cohomFp_openSubgroup.
The two-term Euler formula under cd_p G ≤ 1. For a topologically finitely generated
profinite pro-p group G with cd_p G ≤ 1 and an open subgroup U, in ℤ:
dim H⁰(U, 𝔽_p) - dim H¹(U, 𝔽_p) = [G : U] * (dim H⁰(G, 𝔽_p) - dim H¹(G, 𝔽_p)).
The three-term Euler formula under cd_p G ≤ 2. For a topologically finitely generated
profinite pro-p group G with cd_p G ≤ 2 and H²(G, 𝔽_p) finite, and an open
subgroup U, in ℤ:
dim H⁰(U) - dim H¹(U) + dim H²(U) = [G : U] * (dim H⁰(G) - dim H¹(G) + dim H²(G)) for the
cohomology with trivial coefficients 𝔽_p.