The homicide is the first in Chapel Hill in more than two years.
At 1:18 p.m. Monday, the Orange County 911 Center received a call of shots fired in the area of West Rosemary Street and Lindsey Street.
When officers responded, they found Thompson, 34, of Knolls Street, lying dead on Mitchell Lane, just two blocks from Franklin Street.
Tate was arrested at the scene without incident.
According to the Orange County jail, Tate made bond before he was brought into custody. Bond was $25,000.
Thompson reportedly was the boyfriend of Tate's daughter.
The two men began arguing after Tate locked Thompson out of his house. The confrontation ended with the shooting outside the family's realty business.
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