Manuscript XV-C@1 · XV-C.tex · section · 5afd124d

Readiness of the full relative-lifting calculation

Readiness of the full relative-lifting calculation · lines 1169–1209

XV-C@1#eq:relative

Once the actual reconstruction and canonical contraction are specified, write

H = ker Rec, V = D^{'} (W; E_{res}), r_{O} : V ⟶ D^{'} (O; E_{res}),

and

Y = ker r_{O} = {t \in V : supp t \subset W ∖ O} .

No equality between $W ∖ O$ and the characteristic locus is assumed.

A preliminary lift must first be defined:

F_{0} : P_{int} ⟶ V, r_{O} F_{0} = m^{\circ} Rec .

Only then is its null defect

d_{0} = F_{0} ∣_{H}

an evaluable object.

If the actual permitted candidate family is proved to be $F_{0} + A$ , with $A \subset Hom (P_{int}, Y)$ a specified vector space, XV-B gives the exact descent criterion

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

It uses membership in the actual correction image, not merely its closure XVB.

Displayed formulas in this passage

XV-C@1#eq:relative · line 1201
$[d_{0}] = 0 in Hom (H, Y) / {a ∣_{H} : a \in A} .$
unlabeled · line 1173
$H = ker Rec, V = D^{'} (W; E_{res}), r_{O} : V ⟶ D^{'} (O; E_{res}),$
unlabeled · line 1179
$Y = ker r_{O} = {t \in V : supp t \subset W ∖ O} .$
unlabeled · line 1187
$F_{0} : P_{int} ⟶ V, r_{O} F_{0} = m^{\circ} Rec .$
unlabeled · line 1192
$d_{0} = F_{0} ∣_{H}$

Nearby in this section

proposition: Proposition: The obstruction has not been evaluated