Manuscript XV-B@2 · XV-B.tex · proposition · 4d3f12e8

Proposition: Domain-sensitive transport of descent

Contact support and domain-sensitive transfer › A transfer lemma with explicit domains · lines 669–707 · proof included or linked

XV-B@2#prop:transfer, XV-B@2#eq:transport

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For $i = 1, 2$ , let $Rec_{i} : P_{i} \to K_{i}$ be linear and let $D_{i} \subset P_{i}$ be vector subspaces. Suppose $F_{1} : D_{1} \to V_{1}$ annihilates $D_{1} \cap ker Rec_{1}$ . Let

α : D_{2} \to D_{1}, β : K_{2} \to K_{1}, τ : V_{1} \to V_{2}

be linear maps satisfying

Rec_{1} α = β Rec_{2} ∣_{D_{2}} .

Then

F_{2} = τ F_{1} α : D_{2} \to V_{2}

annihilates $D_{2} \cap ker Rec_{2}$ and descends onto $Rec_{2} (D_{2})$ .

No assertion is made outside $D_{2}$ . If all maps are continuous and the reconstructed image carries the appropriate quotient topology, the corresponding induced map is continuous.

Proof. For $A \in D_{2} \cap ker Rec_{2}$ ,

Rec_{1} (α A) = β (Rec_{2} A) = 0.

Since $α A \in D_{1}$ , the assumed null condition gives $F_{1} (α A) = 0$ , hence $F_{2} (A) = 0$ . Apply Proposition prop:descent to $Rec_{2} ∣_{D_{2}} : D_{2} \to Rec_{2} (D_{2})$ . The continuity assertion is the quotient-topology statement.

Displayed formulas in this passage

XV-B@2#eq:transport · line 681
$Rec_{1} α = β Rec_{2} ∣_{D_{2}} .$
unlabeled · line 675
$α : D_{2} \to D_{1}, β : K_{2} \to K_{1}, τ : V_{1} \to V_{2}$
unlabeled · line 686
$F_{2} = τ F_{1} α : D_{2} \to V_{2}$
unlabeled · line 699
$Rec_{1} (α A) = β (Rec_{2} A) = 0.$

Nearby in this section

section: A transfer lemma with explicit domains