Old Cat is right. Theoretically if the child lives with his mother she should be the one claiming him as a dependant, even if you pay child support. In another word, only the custodial parent can claim the child as dependant. There are exceptions. The custodial parent can sign a release form 8332 so that the non-custodial parent can claim tax credit for the child. You must do it every year. The other possibility is that the if both of you fail to provide more than 50% of the support, either can claim the child as dependant but not both.
Bottom line is that only one parent can claim dependant and normally is the custodial parent. I have seen judges in state court to order parents to alternate in claiming dependant. This is dangerous since a state court decision will be pre-empted by federal tax laws.