problem with alternator light coming on?

Hi i have vauxhall vivaro van 1.9 cdti 54 plate, recently had alternator light with seat belt tension light and service light all coming on at same time. garage did diagnostic test and found battery was needing renewed, continued to happen now said alternator done and needed replaced. started up engine tonite and guess what yeh - alternator light coming on again. Not had time to drive round in it yet but would hazard a guess that once im driving the other two lights will come back on. can anyone help. milage on van ia 105,000. Can the problem be coming from the air condition unit aa i have noticed that the other two lights seem to come on once van has heated up with heating on. maybe a coincidence not sure