Reading the figure
A native parameter field and a matrix over it feed an actual ambient matrix obtained by adding the scalar parameter to a lifted matrix. The diagram keeps the derived field separate from the final parameter field.
Mathematical labels
\(E=L(\alpha)\)
\(C\in M_m(E)\)
\(B=\alpha I_m+\mathrm{Lift}_E(C)\)
Selectable labels from the original PDF
Native PDF text retains the figure’s extraction order. Equations and data tables above provide the reading order.
Native parameter field E = L(α) Derived parameter C ∈ Mm(E) Actual ambient matrix B = αIm + LiftE(C) The derived field and the final field are not identified.
Original figure caption
The recursive lift is an actual filtered right inverse of normalized corestriction.
Start over a coefficient field \(L\) with uniformizer \(t_L\) and form
At a recursive stage, \(C\) is an actual matrix over \(E\). The outer matrix is
The lift has prescribed corestriction and preserves its exact filtration valuation. The field and simple-stratum proof uses the actual displayed matrix, together with the cited type-theory derived-stratum criterion.
Scope
The derived field \(E[C]\) is not identified with the final field \(L[B]\). The recursion proves a literal defining sequence without silently declaring intermediate fields to be nested inside the final field.