Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 138 / 5 / a

Solution

A block idempotent is a primitive central idempotent

e \in R

M

lies in the corresponding block of a group algebra when

e M = M,

equivalently when every other block idempotent annihilates

M

V

lies in

e

, then the centrality of

e

makes both its submodule

U

W

lie in

e

. Conversely, suppose

e U = U

and

e W = W

. Then

(1 - e) V

maps to zero in

W

, so

(1 - e) V \subseteq U

. But

(1 - e) U = 0

, and applying the idempotent

1 - e

once more gives

(1 - e) V = (1 - e)^{2} V = 0.

Thus

e V = V

, proving

V lies in e ⟺ U and W lie in e .

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