Alternator/belt/running

The seller told me that the truck had been sitting and the only problem is that if you leave the battery connected it drains out. I figured sense there is no aftermarket stuff installed it probably was the alternator that was bad and discharging the battery when turned off. Seller tried to start it the day before and said it would not stay running, they sprayed some gas in the intake and it would run for a few seconds. I get there the next day and that is exactly what happens, spray some gas and it runs for five seconds. As it shut off I noticed the first light to come on is the battery light. I look around and noticed the belt is broke and the alternator is seized. So, I live in New England, and it is freezing cold this time of year. This truck has been sitting for a few months with a quarter tank of gas. And also had to jump it to get it started to begin with. My question is, will the truck not stay running because the alternator is bad, not putting voltage out? Remember the car is being jumped so it should think the alternator is okay because the other car is providing the power. Or is there a circuit that the truck knows the alternator is not working and shutting down the vehicle. I mentioned the cold because I do not know if the fuel lines might have froze. In your opinion, would just replacing the alternator fix the stay running problem?