I've put parts of your post in quotes in order to make sense of it. I hope I've guessed right.
There's no grammatical reason for this. We just don't use 'forget' that way in English. Similarly we don't say, "I remembered my umbrella at home." Do you say, "Ho dimenticato il mio ombrello a casa" and "Mi sono ricordato il mio ombrello a casa"? That's interesting.
While it's true that you forgot your umbrella while you were at home, leaving it there is a different action. Your sentence tries to conflate the two verbs.
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