Row-compact idempotents have finite projective range

Let R be a complete ultrametric normed ring with Tate scalars and I a discrete index type. If e is a row-compact idempotent endomorphism of the model space C₀(I, R), its range is a finitely generated projective R-module. The finite spectral projectors of the Riesz theory of compact operators over coefficient rings are such idempotents; their finite projective ranges replace the finite-dimensional generalised eigenspaces of the field case.

The range is compared with a sufficiently large finite coordinate projection π_S = ZeroAtInftyContinuousMap.indicatorCLM R S: row compactness gives ‖e - e ∘ π_S ∘ e‖ < 1 for some finite S, and π_S factors through the coordinates C₀(I, R) → (S → R). Hence the range of e is a direct summand of the finite free module S → R. No Noetherian hypothesis is needed and the scalars may have nilpotents. The range need not be free.

Main results

• ContinuousLinearMap.IsRowCompact.exists_ringOpNorm_sub_mul_indicatorCLM_mul_lt_one: a row-compact idempotent e satisfies ‖e - e ∘ π_S ∘ e‖ < 1 for some finite S.
• ContinuousLinearMap.IsRowCompact.exists_comp_eq_id_of_isIdempotentElem: the range of a row-compact idempotent is a direct summand of S → R for a finite set S of coordinates.
• ContinuousLinearMap.IsRowCompact.finite_range_of_isIdempotentElem, ContinuousLinearMap.IsRowCompact.projective_range_of_isIdempotentElem: the range of a row-compact idempotent is finitely generated and projective.

References

• J.-P. Serre, Endomorphismes complètement continus des espaces de Banach p-adiques, Publ. Math. IHÉS 12 (1962), 69–85, §7.
• R. Coleman, p-adic Banach spaces and families of modular forms, Invent. Math. 127 (1997), 417–479, Appendix A.
• C. Johansson and J. Newton, Extended eigenvarieties for overconvergent cohomology, arXiv:1604.07739, §2.2.

theorem ContinuousLinearMap.IsRowCompact.exists_ringOpNorm_sub_mul_indicatorCLM_mul_lt_one {I : Type u_1} {R : Type u_2} [TopologicalSpace I] [DiscreteTopology I] [DecidableEq I] [NormedRing R] [IsUltrametricDist R] [TauCeti.NormedRing.IsTate R] {e : ZeroAtInftyContinuousMap I R →L[R] ZeroAtInftyContinuousMap I R} (hu : e.IsRowCompact) (he : IsIdempotentElem e) : ∃ (S : Finset I), ‖e - e * ZeroAtInftyContinuousMap.indicatorCLM R ↑S * e‖ < 1

A row-compact idempotent endomorphism e of C₀(I, R) is close to its compression by a finite coordinate projection: ‖e - e ∘ π_S ∘ e‖ < 1 for some finite set S of coordinates.

theorem ContinuousLinearMap.IsRowCompact.exists_comp_eq_id_of_isIdempotentElem {I : Type u_1} {R : Type u_2} [TopologicalSpace I] [DiscreteTopology I] [DecidableEq I] [NormedRing R] [IsUltrametricDist R] [TauCeti.NormedRing.IsTate R] [CompleteSpace R] {e : ZeroAtInftyContinuousMap I R →L[R] ZeroAtInftyContinuousMap I R} (hu : e.IsRowCompact) (he : IsIdempotentElem e) : ∃ (S : Finset I) (f : (↥S → R) →ₗ[R] ↥(↑e).range) (g : ↥(↑e).range →ₗ[R] ↥S → R), f ∘ₗ g = LinearMap.id

The range of a row-compact idempotent is a direct summand of a finite free module. For a row-compact idempotent endomorphism e of C₀(I, R) there are a finite set S of coordinates and linear maps f : (S → R) → range e, g : range e → (S → R) with f ∘ g = id.

theorem ContinuousLinearMap.IsRowCompact.finite_range_of_isIdempotentElem {I : Type u_1} {R : Type u_2} [TopologicalSpace I] [DiscreteTopology I] [DecidableEq I] [NormedRing R] [IsUltrametricDist R] [TauCeti.NormedRing.IsTate R] [CompleteSpace R] {e : ZeroAtInftyContinuousMap I R →L[R] ZeroAtInftyContinuousMap I R} (hu : e.IsRowCompact) (he : IsIdempotentElem e) : Module.Finite R ↥(↑e).range

A row-compact idempotent has finitely generated range.

theorem ContinuousLinearMap.IsRowCompact.projective_range_of_isIdempotentElem {I : Type u_1} {R : Type u_2} [TopologicalSpace I] [DiscreteTopology I] [DecidableEq I] [NormedRing R] [IsUltrametricDist R] [TauCeti.NormedRing.IsTate R] [CompleteSpace R] {e : ZeroAtInftyContinuousMap I R →L[R] ZeroAtInftyContinuousMap I R} (hu : e.IsRowCompact) (he : IsIdempotentElem e) : Module.Projective R ↥(↑e).range

A row-compact idempotent has projective range.