At the same time, such disputes are not uncommon. There is an ingrained tension between the police and prosecutors that often centers on what charges to bring and, at times, whether there is sufficient evidence to make an arrest. For the police, in some measure, the job ends with handcuffs, while prosecutors are left with proving a case beyond a reasonable doubt or finding some other resolution. But such arguments do not often became public at all, let alone so early in a new administration.Mr. Adams has been complimentary about Mr. Bragg when asked about him in recent interviews, calling him a “great prosecutor” and declining to criticize the memo. The mayor’s office did not immediately respond to a request for comment. A police spokesman said the memo “speaks for itself.”Mr. Bragg and Mr. Adams, both Democrats, have significant histories in law enforcement, and both have pledged some measure of reform. Mr. Bragg, a former federal prosecutor, stood out in a competitive primary vowing to balance safety with justice. Mr. Adams, a former police captain, has spoken out against police brutality and, while serving, pushed for changes within the department.Mr. Bragg is the first Black person to lead the district attorney’s office, Mr. Adams is the second Black mayor in the city’s history, and Commissioner Sewell is the first woman and third Black person to lead the Police Department.In his memo, Mr. Bragg instructed his prosecutors to ask judges for jail or prison time only for those who had committed serious offenses, including murder, sexual assault and major economic crimes, unless required to do otherwise by law. Others, he has said, would be directed to programs better equipped to deal with the issues that had led them to commit the crimes.The new district attorney also instructed his prosecutors not to charge a number of misdemeanors. Many of the crimes on his list already were not being prosecuted by his predecessor, Cyrus R. Vance Jr. But Mr. Bragg directed his staff to avoid charging several misdemeanors which previously had been charged, including resisting arrest.