Sam Spade used to be a policeman himself. In fact, it sounds as if he has been a private detective for only a short time, although he has had considerable experience as a cop. His firm is called Spade and Archer until Archer is killed in the alley. Spade tells Brigid towards the end of the novel:
"Miles was a son of a bitch. I found that out the first week we were in business together and I meant to kick him out as soon as the year was up. You didn't do me a damned bit of harm by killing him."
So Miles Archer and Sam Spade are both former cops who have only been in the private detective business for less than a year. Spade and Polhaus are friends and must have worked together before Spade went private. In fact, Polhaus might have been Spade's partner and Lieutenant Dundy is his new partner.
Tom Paulhaus is important to the plot because he maintains friendly relations with Spade and is willing to give him important information. Most importantly, Spade and Polhaus are having lunch at the States Hof Brau in Chapter 15 when Polhaus tells Spade practically everything the police have found out about Floyd Thursby, including the fact that the Webley-Fosbery automatic revolver found at the scene of Archer's murder belonged to Thursby and the bullet taken from the body came from the Webley-Fosbery. That seems to clear Spade of suspicion of killing Archer, although not of killing Thursby later that same night.
Polhaus also gives Spade a sketch of Thursby's arrests and convictions. This information is especially useful to Spade because it helps him trap Brigid in one of her many lies at the end. She claims that she came to Spade's office with her fabricated story about her sister in order to get a detective to follow Thursby. She wanted to be sure it was a detective she would recognize so that she could point him out to Thursby. It could have been Spade himself, but Archer volunteered because he was strongly attracted to the beautiful young client. According to her story:
"And I was afraid Gutman would find me--or find Floyd and buy him over. That's why I came to you and asked you to watch him for--"
"That's a lie," Spade said. "You had Thursby hooked and you knew it. He was a sucker for women. His record shows that--the only falls he took were over women. And once a chump, always a chump."
Spade is using the vital information he got from Tom Polhaus at the States Hof Brau. He knows Brigid wanted to get rid of Thursby so that she wouldn't have to share the proceeds from the Maltese falcon with him when she got it from Captain Jacobi. Spade tells her bluntly:
"You thought Floyd would tackle him and one or the other of them would go down. If Thursby was the one then you were rid of him. If Miles was, then you could see that Floyd was caught and you'd be rid of him. . . . And when you found that Thursby didn't mean to tackle him you borrowed the gun and did it yourself. Right?"
"Yes--though not exactly."
"But exact enough. And you had that plan up your sleeve from the first. You thought Floyd would be nailed for the killing."
Brigid's thinking was correct. The police naturally assumed that Thursby had killed Archer because the Webley-Fosbery found at the murder scene belonged to Thursby, and Spade had told Polhaus that Archer was following Thursby that night. Her scheme was ingenious, though thoroughly vicious. She didn't care whether Archer or Thursby or both of them got killed as long as she was rid of Thursby.
Spade is deeply indebted to his friend Tom Polhaus for the information he uses to clear himself of suspicion of murder and to force Brigid to confess that she not only killed Archer but was responsible for the entire affair involving the Maltese falcon, a conflict that ultimately led to the deaths of Archer, Thursby, Captain Jacobi, Caspar Gutman, and eventually Wilmer Cook, who was sure to be executed for killing Thursby, Jacobi, and Gutman. Brigid stole the statuette for Gutman but then decided not to give it to him. She had Captain Jacobi bring it to San Francisco, and her visit to Spade's office was what got him involved with the whole tangled and lethal melee. Spade had to clear himself of suspicion of two murders and avenge his partner's death, as a matter of honor, by unmasking Archer's killer.