If you suspect that a frozen pulley is preventing the motor from turning, remove the serpentine belt and try to spin all of the pulleys by hand. A frozen or tight one will be obvious.
Make sure that your battery and alternator cables are clean, tight and not corroded (even underneath the insulating boots on the ends).
Check or have the battery checked with a load tester. If a new battery is not holding charge, the likely culprit is the alternator or the connections to it. Remove the alternator and have it tested. If the battery is fully charged and load tests OK, and the connections are clean and tight....
Put a test light on the starter solenoid wire at the starter and turn the key to start. If it lights up, but you hear no click from the solenoid engaging the starter drive, replace the starter and solenoid.