The transpose of a ℤ_p[G]-module of projective dimension one #
Let G be a finite group and 0 → P₁ → P₀ → M → 0 an exact sequence of ℤ_p[G]-modules with
P₀, P₁ finitely generated projective, so that M has projective dimension at most one. Its
Auslander–Reiten transpose Tr f, the cokernel of
Hom_{ℤ_p[G]}(P₀, ℤ_p[G]) → Hom_{ℤ_p[G]}(P₁, ℤ_p[G]), is E¹(M) = Ext¹_{ℤ_p[G]}(M, ℤ_p[G]).
This file identifies it with the Pontryagin dual Hom_{ℤ_p}(M[p^∞], ℚ_p / ℤ_p) of the p-power
torsion of M, compatibly with the action of ℤ_p[G]ᵐᵒᵖ. The identification composes the
symmetric Frobenius duality of ℤ_p[G] (TauCeti.AuslanderReitenTranspose.groupAlgebraDualEquiv),
which rewrites the transpose through ℤ_p-linear duals, with the connecting map
TauCeti.torsionDualMap over ℤ_p.
Since the transpose determines a finitely presented module up to projective summands
(TauCeti.AuslanderReitenTranspose.nonempty_linearEquiv_prod_of_linearEquiv), two such modules
with isomorphic p-power torsion are stably isomorphic. This is how Neukirch–Schmidt–Wingberg
compare modules of projective dimension one over ℤ_p[G] before cancelling projective summands.
Main definitions #
TauCeti.instModulePadicCharacterContragredient: the contragredient action ofℤ_p[G]ᵐᵒᵖon theℚ_p / ℤ_p-valued characters of aℤ_p[G]-module.TauCeti.AuslanderReitenTranspose.torsionDualLinearEquiv: the transposeTr fisℤ_p[G]ᵐᵒᵖ-linearly isomorphic to the Pontryagin dual of thep-power torsion ofM.
Main results #
TauCeti.nonempty_linearEquiv_prod_of_pPowerTorsion_linearEquiv: modules of projective dimension one with isomorphicp-power torsion satisfyM ⊕ P₁ ⊕ Q₀ ≃ N ⊕ P₀ ⊕ Q₁.TauCeti.exists_projective_prod_linearEquiv_of_torsion: the same, asM ⊕ P ≃ N ⊕ Qfor some finitely generated projectivePandQ.
References #
- J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., Grundlehren 323, Springer (2008), (5.4.11) and (5.6.9).
- M. Auslander, M. Bridger, Stable module theory, Mem. Amer. Math. Soc. 94 (1969), Section 2.1.
The contragredient action of ℤ_p[G] on the ℚ_p / ℤ_p-valued characters of a
ℤ_p[G]-module: op a sends χ to x ↦ χ (a • x).
The transpose of a ℤ_p[G]-module of projective dimension one is the Pontryagin dual of
its p-power torsion. For a finite group G and an exact sequence 0 → P₁ → P₀ → M → 0 of
ℤ_p[G]-modules with P₀ projective over ℤ_p and M finitely generated over ℤ_p, the
transpose Tr f (which is E¹(M) when P₀ and P₁ are projective over ℤ_p[G]) is isomorphic
to Hom_{ℤ_p}(M[p^∞], ℚ_p / ℤ_p), compatibly with the action of ℤ_p[G]ᵐᵒᵖ on the transpose and
the contragredient action on characters.
Instances For
TauCeti.AuslanderReitenTranspose.torsionDualLinearEquiv sends the class of a group-algebra
functional to the image of its coefficient-at-one functional under the connecting map.
Modules of projective dimension one with isomorphic p-power torsion are stably
isomorphic (NSW (5.4.11) with (5.6.9)). Let G be a finite group and let
0 → P₁ → P₀ → M → 0 and 0 → Q₁ → Q₀ → N → 0 be exact sequences of ℤ_p[G]-modules with
finitely generated projective P₀, P₁, Q₀, Q₁. If the p-power torsion submodules of M and N
are isomorphic as ℤ_p[G]-modules, then M ⊕ P₁ ⊕ Q₀ ≃ N ⊕ P₀ ⊕ Q₁ as ℤ_p[G]-modules.
The stable class of a module of projective dimension one is determined by its torsion
(NSW (5.4.11) with (5.6.9)). For a finite group G, two ℤ_p[G]-modules presented as quotients
of finitely generated projectives by projective submodules, with isomorphic p-power torsion
submodules, become isomorphic after adding finitely generated projective ℤ_p[G]-modules:
M ⊕ P ≃ N ⊕ Q. The summands are explicit in
TauCeti.nonempty_linearEquiv_prod_of_pPowerTorsion_linearEquiv.