If you are on the title of the property, and make the payments, then you can deduct the property tax. If you are a California resident, you do not have to exclude the Ohio property tax from your California return.
In order to be able to deduct mortgage interest, you should be listed as a borrower on the loan, be legally obligated to pay the debt, and actually make the payments.
Please see this article and this one for more information on home mortgages from TurboTax.
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