In The Crucible, who called Elizabeth out and why?

According to Cheever, Abigail Williams made the claim that Elizabeth's spirit stuck needles into her (Abigail's) stomach. Cheever is then summoned to the Proctor house to question Elizabeth and look for evidence of this type of witchcraft. He finds a poppet (doll) with needles stuck in it. This is what Abigail claimed. She concluded that Elizabeth stuck needles in the doll and used her (voodoo) powers of witchcraft to inflict that violence on Abigail herself.
Of course, this is untrue. Abigail is pretending, and she had a role in using Mary Warren to plant the doll (poppet) in Elizabeth's home. As a result of Abigail's accusation and finding the poppet in her home, Cheever is forced to arrest Elizabeth.
Abigail accuses Elizabeth out of jealousy. She and John Proctor, Elizabeth's husband, had an affair, but John ended it. Abigail, still in love with John, did not want their relationship to end. When he abandons her, she takes her revenge upon Elizabeth. At this point, John Proctor's only hope is to convince Mary Warren to admit it was her poppet and the accusations of witchcraft are all false. John is outraged that his wife has been charged, but he must know his affair with Abigail indirectly contributed to his wife's arrest.