the IRS says the custodial parent CAN claim them provided all the tests for qualifying child or relative are met. however, by claiming that 1 child you could possibly find yourself held in contempt of court. if you don't want that furnish your ex with signed form 8332.
The IRS cares about who the custodial parent is--not divorce papers.
Are you the custodial parent? Do you have an agreement with the other parent to allow the other parent to claim them--due to divorce or that you live apart and share custody? Did one of you sign a Form 8332?
If there is a signed 8332 then the custodial parent retains the right to file as Head of Household, get earned income credit and the childcare credit + education credits if the child is a full-time college student. The non-custodial parent gets the child tax credit for children under the age of 17.
As far as the IRS is concerned, the custodial parent is the one with whom the child spent the most nights during the tax year--at least 183 nights.