Man on a Ledge

In New York City, Nick Cassidy (Sam Worthington) checks in to the Roosevelt Hotel under the false name of Walker, goes to his hotel room on the 21st floor, and climbs on the ledge, apparently ready to commit suicide. The crowd below calls the police, with Dante Marcus (Titus Welliver) controlling the crowd, while Jack Dougherty (Edward Burns) tries to talk with Nick. However, Nick will only speak to negotiator Lydia Mercer (Elizabeth Banks), who is on a leave of absence, after failing to save a suicidal policeman. Lydia arrives at the hotel room and manages to acquire Nick's fingerprints from a cigarette they share. Dougherty has them analyzed and discovers that Nick is an ex-policeman, arrested for stealing the $40 million Monarch diamond from businessman David Englander (Ed Harris). Nick was given a 25-year sentence but escaped from the Sing Sing Correctional Facility one month earlier, after being allowed to attend his father's funeral. Nick, however, maintains his innocence and accuses Englander of framing him for the theft of the diamond, as Englander lost his fortune and was too proud to sell the diamond.
Unknown to the police, Nick is merely distracting them while his brother, Joey (Jamie Bell), and Joey's girlfriend, Angie (GÃ©nesis RodrÃ­guez), break into Englander's vault across the street, to steal the diamond and prove Nick's innocence. Meanwhile, Dougherty informs Marcus of Nick's identity, and Marcus orders the jewelry store's security to check the vault. Joey and Angie are able to evade them but do not find the diamond. They set off the alarms, tricking Englander into retrieving the diamond and ambush him, stealing the diamond at gunpoint. Meanwhile, Nick's ex-partner, Mike Ackerman (Anthony Mackie), arrives at the hotel with evidence that Nick is planning something and demands to be allowed into the hotel room. Lydia does not trust him, and Dougherty backs her up. Ackerman claims he has found bomb schematics in a storage unit Nick rented and is convinced that he will detonate an explosive somewhere. While the crowd is evacuated by the bomb squad, Lydia, believing in Nick's innocence, calls Internal Affairs and discovers that three of the cops employed by Englander were suspected of being corrupt: Ackerman, Marcus and a deceased officer called Walker.
Joey and Angie enter the hotel and hand the bag containing the diamond to the hotel concierge (William Sadler). The concierge pockets the diamond. Englander calls Marcus, one of the men who helped him frame Nick, and has him capture Joey and Angie. Nick begins to make his escape through the hotel, at one point being aided by the concierge who tells him "everyone is rooting for you, kid," while handing him a disguise (and the diamond). Marcus chases Nick to the roof, where he has Lydia arrested for obstruction. Englander brings Joey and Angie, threatening to throw Joey off the roof, if Nick does not give him the diamond. Nick turns it over, and Englander leaves. Meanwhile, Lydia escapes custody and rushes back to the roof. Marcus attempts to force Nick to jump off the roof, but Ackerman arrives and shoots Marcus, who wounds Ackerman. Nick rushes to Ackerman's side, who apologizes and reveals that he had no idea that Englander would frame Nick for stealing the diamond. Marcus survives, as he is wearing a bulletproof vest, and is preparing to kill Nick, when Lydia shoots him, but does not kill him. Nick jumps from the roof onto an airbag set up earlier by the police, catches up to Englander, beats him and pulls the diamond he had only just recently given him from his jacket, proving Nick's innocence in front of both the police and public, resulting in Englander being arrested.
Later, Nick is cleared of all charges and released from custody and meets Joey, Angie, and Lydia at a bar. There it is revealed that the hotel concierge is in fact Nick and Joey's father, Frank Cassidy, who had faked his own death in order to help Nick break out of prison and prove his innocence. Joey proposes to Angie with a diamond ring presumably stolen from Englander's vault. She accepts and they all celebrate together.

Question: What do Nick and Lydia share that allows Lydia to obtain Nick's fingerprints?
Answer: Cigarette