Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 138 / 6 / a / iv

Table of contents
- Solution iv

Solution

Under (ii), tensor the direct-product decomposition with the residue field

k

. Since

W

is free over

O

,

k \otimes_{O} End_{O} (W) ≅ End_{k} (\overline{W}) ≅ M_{n} (k) .

Hence

k G ≅ End_{k} (\overline{W}) \times (k \otimes_{O} B),

and the reduced action map is the projection onto the first factor. This proves

(ii) ⟹ (iv) .

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