Vauxhall Calibra Running Rough, Starting & Power Loss Problems

Connected battery and started car and let it run for a little while. When i turned off ignition and removed the key i noticed that the battery lightwas still on. Would this be an altinator problem?

Hi, hope someone can help with this problem, my calibra runs ok for a while, then refuses to start which then results in a dead battery, after recharge it seems ok, then the same thing again (have new battery on)today noticed it was hot aft...