"purchased a rental property, did improvements which were depreciated, rented out the property for 9 months, then sold it."

Ah, mind reading. What a great skill.

Yes, the sequence of events makes a difference.

And depreciation would start when the improvements are in use, but with this tight timeframe, if all of this happened in 2020, there would not be both depreciation and recapture and sale.

Yes, details matter.