The lattice defect of a lattice

Let k be a commutative ring of characteristic ℓ and G a monoid. For a G-module V which is finitely generated and has no ℓ-torsion, such as a G-stable lattice, the lattice defect [k ⊗_ℤ (V ⧸ ℓV)] - [k ⊗_ℤ V[ℓ]] is the reduction class [k ⊗_ℤ V] (TauCeti.latticeDefect_eq_reductionK0): the torsion term vanishes, and since ℓ = 0 in k, the reduction of the quotient map V → V ⧸ ℓV is an isomorphism k ⊗_ℤ V ≅ k ⊗_ℤ (V ⧸ ℓV).

For the permutation lattice ℤ[X] = X →₀ ℤ of a finite G-set X, whose reduction is the permutation representation k[X] (TauCeti.baseChangeComapEquiv), so its reduction class is the permutation class (TauCeti.reductionK0_finsupp_int), and this gives latticeDefect ℤ[X] = [k[X]] (TauCeti.latticeDefect_finsupp_int).

Two lattices V and W with equivalent rationalizations ℚ ⊗_ℤ V ≅ ℚ ⊗_ℤ W have the same reduction class over every field k (TauCeti.reductionK0_eq_of_nonempty_equiv_baseChange_rat), although their reductions need not be isomorphic: for G of order 2, the permutation lattice ℤ[G] and the lattice ℤ ⊕ ℤ(-1) have isomorphic rationalizations, but modulo 2 the first reduces to the indecomposable 𝔽₂[G] and the second to two trivial lines. Clearing denominators in a rational equivalence gives equivariant maps f : V → W and f' : W → V whose composites are multiplication by a nonzero integer s (Representation.Equiv.exists_intertwiningMap_comp_eq_smul). In characteristic zero s is invertible in k, so f reduces to an isomorphism. In characteristic ℓ, f is injective with finite cokernel, so the two lattices have the same lattice defect (TauCeti.latticeDefect_eq_of_finiteIndex), which is their reduction class.

Main results

• TauCeti.latticeDefect_eq_reductionK0: the defect of a lattice is its reduction class.
• TauCeti.reductionK0_finsupp_int: the reduction class of the permutation lattice ℤ[X] is the permutation class [k[X]].
• TauCeti.latticeDefect_finsupp_int: the defect of the permutation lattice ℤ[X] is the permutation class [k[X]].
• TauCeti.reductionK0_eq_of_nonempty_equiv_baseChange_rat_of_charP and TauCeti.reductionK0_eq_of_nonempty_equiv_baseChange_rat: lattices with equivalent rationalizations have the same reduction class, in characteristic ℓ over any commutative ring, and over every field.

References

• J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., Springer (2008), §VII.3, (7.3.3).
• J. S. Milne, Arithmetic Duality Theorems, 2nd ed. (2006), Chapter I, Lemma 2.12, the same statement for finitely generated ℤ_p[G]-modules with isomorphic ℚ_p-rationalizations.

theorem TauCeti.latticeDefect_eq_reductionK0 (k G : Type u) [CommRing k] [Monoid G] (ℓ : ℕ) [CharP k ℓ] (V : Type u) [AddCommGroup V] [DistribMulAction G V] [Finite (QuotSMulTop (↑ℓ) V)] [Subsingleton ↥(Submodule.torsionBy ℤ V ↑ℓ)] : latticeDefect k G ℓ V = reductionK0 k (Representation.ofDistribMulAction ℤ G V)

The lattice defect of a lattice is its reduction class: in characteristic ℓ, a G-module V with V ⧸ ℓV finite and without ℓ-torsion has defect [k ⊗_ℤ V]. Since ℓ = 0 in k, the reduction of V ⧸ ℓV is that of V (TauCeti.reductionK0_quotSMulTop).

theorem TauCeti.reductionK0_eq_of_nonempty_equiv_baseChange_rat_of_charP (k G : Type u) [CommRing k] [Monoid G] (ℓ : ℕ) [Fact (Nat.Prime ℓ)] [CharP k ℓ] (V W : Type u) [AddCommGroup V] [DistribMulAction G V] [Module.Finite ℤ V] [HasUniqueDiv V] [AddCommGroup W] [DistribMulAction G W] [Module.Finite ℤ W] [HasUniqueDiv W] (h : Nonempty ((Representation.baseChange ℚ (Representation.ofDistribMulAction ℤ G V)).Equiv (Representation.baseChange ℚ (Representation.ofDistribMulAction ℤ G W)))) : reductionK0 k (Representation.ofDistribMulAction ℤ G V) = reductionK0 k (Representation.ofDistribMulAction ℤ G W)

Lattices with equivalent rationalizations have the same reduction class in characteristic ℓ: for finitely generated torsion-free G-modules V and W with ℚ ⊗_ℤ V ≅ ℚ ⊗_ℤ W, the classes [k ⊗_ℤ V] and [k ⊗_ℤ W] agree. The reductions themselves need not be isomorphic.

theorem TauCeti.reductionK0_finsupp_int (k G : Type u) [CommRing k] [Monoid G] (X : Type u) [MulAction G X] [Finite X] : reductionK0 k (Representation.ofDistribMulAction ℤ G (X →₀ ℤ)) = permK0 k G X

The reduction class of a permutation lattice ℤ[X] = X →₀ ℤ, on which G acts by pushing the support forward (Finsupp.comapDistribMulAction), is the permutation class [k[X]].

theorem TauCeti.latticeDefect_finsupp_int (k G : Type u) [CommRing k] [Monoid G] (ℓ : ℕ) [CharP k ℓ] [NeZero ℓ] (X : Type u) [MulAction G X] [Finite X] : latticeDefect k G ℓ (X →₀ ℤ) = permK0 k G X

The lattice defect of a permutation lattice ℤ[X] = X →₀ ℤ, on which G acts by pushing the support forward (Finsupp.comapDistribMulAction), is the permutation class [k[X]] in characteristic ℓ ≠ 0.

theorem TauCeti.reductionK0_eq_of_nonempty_equiv_baseChange_rat (k G : Type u) [Field k] [Monoid G] (V W : Type u) [AddCommGroup V] [DistribMulAction G V] [Module.Finite ℤ V] [HasUniqueDiv V] [AddCommGroup W] [DistribMulAction G W] [Module.Finite ℤ W] [HasUniqueDiv W] (h : Nonempty ((Representation.baseChange ℚ (Representation.ofDistribMulAction ℤ G V)).Equiv (Representation.baseChange ℚ (Representation.ofDistribMulAction ℤ G W)))) : reductionK0 k (Representation.ofDistribMulAction ℤ G V) = reductionK0 k (Representation.ofDistribMulAction ℤ G W)

Lattices with equivalent rationalizations have the same reduction class, over every field k: for finitely generated torsion-free G-modules V and W with ℚ ⊗_ℤ V ≅ ℚ ⊗_ℤ W, the classes [k ⊗_ℤ V] and [k ⊗_ℤ W] in G₀(k[G]) agree. In characteristic zero the reductions are isomorphic; in characteristic ℓ they need not be.