The depreciation continues on the *old* basis. If you 'buy up' you may have additional basis to depreciate and there's an election to be made as to how to handle.
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If you are asking how to find your basis after the 1031 here is an example:
your selling price $200,000
your adj basis $125,000
new property price $250,000
now you take your adj basis and add it to the difference in price of your old property and new, and that is what you use for new basis for depreciation. $125,000 + $50,000 = $175,000.
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