From the information you gave concerning your particular situation, unless you could come to an agreement with your wife, your wife would be considered the custodial parent, and the child would be considered as the dependent to your wife and you would not be entitled to any of the credits relating to the child.
According to the IRS:
"Custodial parent and noncustodial parent.
The custodial parent is the parent with whom the child lived for the greater number of nights during the year. The other parent is the noncustodial parent.
If the parents divorced or separated during the year and the child lived with both parents before the separation, the custodial parent is the one with whom the child lived for the greater number of nights during the rest of the year.
A child is treated as living with a parent for a night if the child sleeps:
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At that parent's home, whether or not the parent is present; or
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In the company of the parent, when the child doesn't sleep at a parent's home (for example, the parent and child are on vacation together)."
"Children of divorced or separated parents (or parents who live apart).
In most cases, because of the residency test, a child of divorced or separated parents is the qualifying child of the custodial parent..."
Even if you do arrange an agreement with your wife, you would be limited as to which credits you would be eligible for:
"Children of divorced or separated parents (or parents who live apart).
In most cases, because of the residency test, a child of divorced or separated parents is the qualifying child of the custodial parent. However, the child will be treated as the qualifying child of the noncustodial parent if all four of the following statements are true.
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The parents:
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Are divorced or legally separated under a decree of divorce or separate maintenance,
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Are separated under a written separation agreement, or
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Lived apart at all times during the last 6 months of the year, whether or not they are or were married.
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The child received over half of his or her support for the year from the parents.
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The child is in the custody of one or both parents for more than half of the year.
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Either of the following statements is true.
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The custodial parent signs a written declaration, discussed later, that he or she won't claim the child as a dependent for the year, and the noncustodial parent attaches this written declaration to his or her return. (If the decree or agreement went into effect after 1984 and before 2009, see Post-1984 and pre-2009 divorce decree or separation agreement, later. If the decree or agreement went into effect after 2008, see Post-2008 divorce decree or separation agreement, later.)
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A pre-1985 decree of divorce or separate maintenance or written separation agreement that applies to 2021 states that the noncustodial parent can claim the child as a dependent, the decree or agreement wasn't changed after 1984 to say the noncustodial parent can't claim the child as a dependent, and the noncustodial parent provides at least $600 for the child's support during the year.
If statements (1) through (4) are all true, only the noncustodial parent can:
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Claim the child as a dependent, and
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Claim the child as a qualifying child for the nonrefundable child tax credit, the credit for other dependents, the refundable child tax credit, or the additional child tax credit.
However, this doesn’t allow the noncustodial parent to claim head of household filing status, the credit for child and dependent care expenses, the exclusion for dependent care benefits, the earned income credit, or the health coverage tax credit. See Applying the tiebreaker rules to divorced or separated parents (or parents who live apart), later."
IRS PUB 501
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