For any tax year that began after July 2, 2008, tax law was changed relating to the definition of custodial parent for those parents who are divorced, separated or never married. Your former spouse is correct that the parent who has physical custody of the child for the greater part of the year is the custodial parent, as long as the child is not emancipated under state law. If you separated during the year, the custodial parent is the one who had physical custody for the greater part of the rest of the year. Under these circumstances, the child in question would be a qualifying child for the custodial parent if the child lived with that parent for more than half the year and the child did not provide more than half of his or her own support.
For tax years starting after July 2, 2008, the noncustodial parent can claim the child as a dependent only if the custodial parent formally releases that exemption via IRS Form 8332 or with a signed, written statement that conforms to the language in Form 8332.