Bancroft killed himself… sort of
Bancroft was both right and wrong about this one. Though he did, technically, pull the trigger that ended his life, it was an act of desperation after he was corrupted by two women: his wife, Miriam, and Kovacs' sister, Reileen (Dichen Lachman).
In his original body, Kovacs was an Envoy, or a form of government soldier outfitted with special abilities that made them useful to the governmental Protectorate. As the Protectorate grew egomaniacal with their stack technology, the Envoys turned on them, attempting to overthrow their leaders and corrupt stack coding so that it would short circuit after 100 years. The new code was created by Quellcrist Falconer (Renée Elise Goldsberry), Kovacs' lover who also originated the stack technology before watching it grow into her own Frankenstein monster. In their attempt to flee the chaos of the battle between the Envoys and the Protectorate, Quell and Reileen were killed in an explosion.
Or so Kovacs thought. In reality, Reileen cut a deal with the Protectorate, selling out the Envoys so she could survive. They were destroyed, and Kovacs' stack placed "on ice" as punishment. Reileen eventually worked her way to a place of status in Bay City, though she took several moral leaps backward to get there; she winds up running a brothel called Head in the Clouds, where Meths come to abuse sex workers whose stack codes were manipulated with the Catholic "DNR" programming; no chance of re-sleeving means there would be no witnesses to crimes committed against Catholics. Essentially, at Head in the Clouds, Meths could torture and kill women with zero consequence.
Reileen was always desperate to have her brother freed from his mind prison, so she concocted an elaborate plan to have Bancroft kill himself so that she could suggest he re-sleeve Kovacs to help solve his murder. The plan worked, but not without some manipulation. Bancroft was having an affair with a sex worker named Lizzie (Hayley Law), and Reileen used this to get Miriam on her side. Miriam learned that not only was Lizzie sleeping with her husband, but she was also pregnant with his child; out of rage, she killed Lizzie and the baby, and administered a drug into Bancroft's system that made him delusional and aggressive. Under the influence, Bancroft killed another sex worker at Head in the Clouds, and then killed himself out of shame. This set in motion Reileen's plan, and Kovacs was brought back.