What does the cryptic crossword clue "Supporter at home greeting stroke of genius (9)" mean?

The wordplay uses a charade (building blocks). BRA (synonym="”Supporter”") + IN (synonym="”at home”") + WAVE (synonym="”greeting”") = BRAINWAVE The answer is BRAINWAVE.

Supporter at home greeting stroke of genius (9)

Guardian Cryptic #29659 | 9 Across | 2025-04-03 flower

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