A prosecution document unsealed on Tuesday provides the government’s most detailed public account of a shooting last year of an unarmed black man by the police in a Brooklyn stairwell.

The officer who fired the fatal shot, Peter Liang, has been charged with second-degree manslaughter and other offenses. In the court document, prosecutors say he and his partner failed to aid the man who was shot, Akai Gurley, 28.

Prosecutors have not said Officer Liang intended to shoot Mr. Gurley, who had been visiting his girlfriend in a housing project and was heading down a stairwell with her when he was struck by a ricocheting bullet. But they have said that Officer Liang acted recklessly in his handling of his weapon and that he and his partner, Officer Shaun Landau, did not assist Mr. Gurley after they realized that the shot had struck him.