TroubleShoot Alternator

Hi All.
I have a CL9 accord euro.
Which does not charge the battery, When I hook up a multimeter to the battery leads i get a reading of 12.10-12.56. When I start the car this drops slightly and continues to drop slowly..
I am fairly certain that my alternator has gone bad. Is there any fuses I should check before I get the alternator replaced ?

BTW the Check Battery light is also on...

not really. when engine running, should see over 14v

If you're handy, can test certain individual parts inside the alternator.