Manuscript Oai212-A@1 · Oai212-A.tex · proposition · 79323841

Proposition: Relative-lifting criterion

The separate characteristic-composition programme › A conditional affine correction is not a supplied map · lines 1635–1659 · proof included or linked

Oai212-A@1#prop:relative

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A descending member of $F_{0} + A$ exists if and only if

[F_{0} ∣_{H}] = 0 in Hom (H, Y_{0}) / {a ∣_{H} : a \in A} .

An individual candidate descends precisely when

F_{0} ∣_{H} + a ∣_{H} = 0.

If $F_{*}$ is one descending candidate, all such candidates are

F_{*} + {a \in A : a ∣_{H} = 0} .

Proof. Apply Proposition prop:descent to $F_{0} + a$ . The required cancellation equation has a solution exactly when $F_{0} ∣_{H}$ belongs to the restriction image, since that image is a vector space. Subtract one solution to obtain the ambiguity.

Displayed formulas in this passage

unlabeled · line 1638
$[F_{0} ∣_{H}] = 0 in Hom (H, Y_{0}) / {a ∣_{H} : a \in A} .$
unlabeled · line 1645
$F_{0} ∣_{H} + a ∣_{H} = 0.$
unlabeled · line 1649
$F_{*} + {a \in A : a ∣_{H} = 0} .$

Proposition: Relative-lifting criterion

Displayed formulas in this passage

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