SUTTON, Chief Judge. An Ohio jury convicted Clarence Mack of aggravated murder arising from a 1991 carjacking. It recommended the death penalty. The judge agreed and imposed a capital sentence. After years of litigation in state courts, Mack petitioned for a writ of habeas corpus in federal court, alleging several claims relevant to today’s dispute: that prosecutors suppressed material evidence, that they introduced false testimony, that his trial counsel was ineffective, and that the state court denied him a fair trial when it denied the admission of certain testimony. The district court denied his petition. We affirm.