How to upgrade Drupal core manually or via composer?
When upgrading Drupal Core, likewise I am upgrading Drupal core 9.0.7 to Drupal core 9.1.2, to do so I simply ran composer update that will eventually upgrade all the packages along with drupal/core.
In case you encounter the permission issue while running composer update, for example in image blow.
To resolve this problem make sure you have proper permissions on sites/default by running following command.
chmod u+w sites/default
Once permissions are set correct, Drupal Core will be upgraded successfully.