Common complements for finite subspace chains

Abstract

A finite increasing chain of subspaces has one ambient complement that splits every layer.

Theorem 1.1 (One complement for all layers).

Lean statement: D5/S3/HomologicalAlgebra/FilteredVectorSpaceComplement.exists_common_complement_of_monotone

Proof. Machine-checked in Lean as D5/S3/HomologicalAlgebra/FilteredVectorSpaceComplement.exists_common_complement_of_monotone (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Gunnar Carlsson; Vin de Silva (2008). Zigzag Persistence. URL: https://arxiv.org/abs/0812.0197v1.

Commentary.

Let k be a division ring and V a left k-module with an additive group. Let F be an increasing family of submodules indexed by Fin n, and let K be any submodule of V. There exists a submodule L with K and L complementary in V such that every F(i) is the sum of its intersections with K and L. These intersections are disjoint because K and L are disjoint. Thus the same ambient splitting induces a direct sum in every filtration layer.

The construction proceeds along the finite chain. Inside the next layer, the previous partial complement is still disjoint from the intersection with K. It can therefore be enlarged to a complement of that intersection inside the next layer. The enlarged submodule contains the previous one, so all earlier splitting identities persist. After the last layer, an ambient enlargement gives a complement of K in V; containment again preserves every layer identity.

The empty chain, repeated layers, zero layers, and whole-space layers are included. Neither a zero first layer nor a whole-space last layer is required. The ambient dimension is unrestricted, and multiplication in k need not be commutative.

Carlsson and de Silva’s Proposition 3.11 in arXiv:0812.0197v1, PDF pages 13 and 14, gives the recursive complementary-summand construction for induced subspaces of filtered vector spaces in their finite-dimensional field setting. This theorem proves the division-ring, arbitrary-dimension version with arbitrary chain endpoints and an ambient complement. It does not assert an interval classification or a zigzag decomposition.

References

• Truth anchor: D5/S3/HomologicalAlgebra/FilteredVectorSpaceComplement.exists_common_complement_of_monotone