You can't claim a dependency exemption for someone who files a joint tax return. The only exception is for someone who files a joint return for the sole purpose of claiming a refund of income tax withholding and who wouldn't be required to file a return if he filed separately from his spouse. A dependent must live in your household for the entire year, unless he is closely related to you. Special rules apply when determining the dependency status for minor children of divorced or separated parents. The general rule for children of divorce assigns the dependent exemption to the parent who has custody of the child for the greater part of the year.
Support provided by each parent is irrelevant.
The most recent divorce decree or separation agreement determines custody. The parent who has actual physical custody of the child for the greater portion of the year is deemed to have custody. There is one major exception. If the custodial parent consents to give up the exemption, the noncustodial parent can claim the child. The exemption can be released for one year, several years, or all future years.