In The Tempest, what did Prospero do to the good spirits upon his arrival at the island?Why did he owe Prospero a grudge?
By "good spirits" I assume you are refering to Ariel and the way that his history became intertwined in that of Prospero's. The dynamic between Ariel and Prospero is fascinating as the play progresses, and many directors have portrayed their relationship in very different ways, however, if we look at Act I scene 2 we are clearly told how Ariel came to willingly serve Prospero and the conditions that he agreed to. Prospero, in response to Ariel's stubbornness to do what he is commanded, reminds Ariel of what he saved him from. Sycorax, because of the way that Ariel refused to serve her and comply with her abhorrent orders, imprisoned Ariel in the middle of a "cloven pine" where he remained for 12 years. Note how Prospero freed him:
Thou best knowst
What torment I did find thee in. Thy groans
Did make wolves howl and penetrate the breasts
Of ever-angry bears. It was a torment
To lay upon the damned, which Sycorax
Could not again undo. It was mine art,
When I arrived and heard thee, that made gape
The pine, and let thee out.
So, Prospero, upon his arrival on the island that has been his home since his exile, freed Ariel from the dark magic of Sycorax, for which Ariel agreed to serve Prospero for a given time.