Spade awoke because he received a phone call telling him that his partner has been killed. When he got to the where the crime was commited, he talked to the police and said that it was possible that Thursby was the one who killed his partner.
Since the cops reveal that Thursby was also killed, about 30 minutes after Spade left the crime scene, it is made clear that the police believes that Spade is the killer.
Most likely, since Spade said that Thursby might have been the one to kill Archer, the police thinks that Spade went for revenge.
