Ext¹(-, ℤ_p) of a presented module is the dual of its p-power torsion #
Let 0 → P₁ → P₀ → M → 0 be an exact sequence of ℤ_p-modules, with first map f. Applying
Hom_{ℤ_p}(-, ℤ_p) gives Hom(P₀, ℤ_p) → Hom(P₁, ℤ_p), whose cokernel is Ext¹_{ℤ_p}(M, ℤ_p) when
P₀ is projective. For M finitely generated this cokernel is the Pontryagin dual
Hom_{ℤ_p}(M[p^∞], ℚ_p / ℤ_p) of the p-power torsion M[p^∞] of M. More generally,
the comparison holds whenever M / M[p^∞] is projective over ℤ_p; finite generation guarantees
this because the quotient is torsion-free.
The comparison is the connecting map of the long exact sequence for
0 → ℤ_p → ℚ_p → ℚ_p / ℤ_p → 0, written out by hand: a torsion element t lifts to x ∈ P₀ with
p ^ n • x = f k for some k ∈ P₁, and a functional ψ on P₁ sends t to the class of
ψ k / p ^ n. Its kernel consists of the functionals extending along f, because M[p^∞] is a
direct summand of M over ℤ_p. Surjectivity needs only projectivity of P₀, since
ℚ_p / ℤ_p is injective over ℤ_p and every character of the torsion extends to M.
The construction and surjectivity allow an action of a semiring A, compatible with that of
ℤ_p, for which the maps are A-linear. The kernel and cokernel statements assume A is a ring.
The connecting map is compatible with the contragredient actions of A:
precomposing a functional on P₁ with the action of a ∈ A corresponds to precomposing a character
of M[p^∞] with the action of a. Applied to the group algebra A = ℤ_p[G] of a finite group,
this identifies E¹(M) = Ext¹_{ℤ_p[G]}(M, ℤ_p[G]) of a module of projective dimension one with the
Pontryagin dual of its p-power torsion.
Main definitions #
TauCeti.torsionDualMap: the connecting mapHom(P₁, ℤ_p) → Hom(M[p^∞], ℚ_p / ℤ_p).TauCeti.torsionDualEquiv: the induced isomorphism of its cokernel with the dual of the torsion.
Main results #
TauCeti.torsionDualMap_apply: the connecting map on representatives.TauCeti.torsionDualMap_apply_of_extension: the connecting map at a functional extending alongfto aℚ_p-valued functionalΦsendsπ xto the class ofΦ x.TauCeti.ker_torsionDualMap: its kernel is the range ofHom(P₀, ℤ_p) → Hom(P₁, ℤ_p).TauCeti.torsionDualMap_surjective: it is surjective whenP₀is projective overℤ_p.TauCeti.torsionDualMap_comp_toLinearMap_apply: it is compatible with the contragredient actions ofAon functionals and on characters.
References #
- J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., Grundlehren 323, Springer (2008), (5.4.11) and (5.6.9).
The connecting map Hom(P₁, ℤ_p) → Hom(M[p^∞], ℚ_p / ℤ_p) of a presentation
0 → P₁ → P₀ → M → 0: for t in the p-power torsion of M, choose x ∈ P₀ over t and
p ^ n • x = f k; then ψ sends t to the class of ψ k / p ^ n. When M is finitely generated
over ℤ_p, its kernel consists of the functionals extending along f
(TauCeti.ker_torsionDualMap). It is surjective whenever P₀ is projective, without assuming
finite generation (TauCeti.torsionDualMap_surjective). Together these identify
Ext¹_{ℤ_p}(M, ℤ_p) with the Pontryagin dual of the p-power torsion of M
(TauCeti.torsionDualEquiv).
Instances For
The connecting map on a p-power torsion element t: if π x = t and p ^ n • x = f k,
then ψ sends t to the class of ψ k / p ^ n.
The connecting map at a functional ψ that extends along f to a ℚ_p-valued functional Φ
on P₀: it sends π x to the class of Φ x.
The connecting map is compatible with the contragredient actions of A: precomposing a
functional with the action of a ∈ A on P₁ corresponds to evaluating its image at a • t.
The connecting map vanishes on the functionals that extend along f.
The connecting map is surjective when P₀ is projective over ℤ_p: every character of the
p-power torsion of M comes from a functional on P₁, without a finite-generation assumption
on M.
Kernel and cokernel of the connecting map #
Throughout this section, the quotient M / M[p^∞] is assumed projective over ℤ_p. Under this
hypothesis, the kernel consists of the functionals extending along f. When P₀ is also
projective, the connecting map induces the cokernel equivalence. Finite generation of M
implies the common quotient-projectivity hypothesis.
If the quotient of M by its p-power torsion is projective over ℤ_p, the kernel of the
connecting map consists exactly of the functionals that extend along f. In particular, this holds
when M is finitely generated over ℤ_p.
Ext¹_{ℤ_p}(M, ℤ_p) is the Pontryagin dual of the p-power torsion of M. For an exact
sequence 0 → P₁ → P₀ → M → 0 with P₀ and M / M[p^∞] projective over ℤ_p, the cokernel
of Hom(P₀, ℤ_p) → Hom(P₁, ℤ_p) is isomorphic to
Hom_{ℤ_p}(M[p^∞], ℚ_p / ℤ_p), through the connecting map TauCeti.torsionDualMap. The
projectivity of M / M[p^∞] holds in particular when M is finitely generated over ℤ_p.
Instances For
TauCeti.torsionDualEquiv sends the class of a functional to its image under the connecting
map.