The Development Team manages how the development work is planned, performed and controlled.
Whereas, the people aspects of the Scrum Team like hiring, firing, compensation and other aspects are handled by HR authority in the organization.
It is only the Development Team that decides how to get the development work done to meet the Sprint Goal.
The Scrum Master coaches the team on Scrum essentials and acts as the guardian of the Scrum implementation.
Keep in mind: there is no exclusive manager for a Scrum Team. In Scrum, there is no role called Agile Coach.