Can dead batteries cause a derate?

I believe it does. I have the same issue. My alternator puts out 13.1 to 13.5 which I think it should be higher. My truck would ask for regen and would give me derate message every time after I shut the truck of for couple of hours or overnight. If I don't stop and keep going after 45 min or so regen and derate messages would disappear. ( I found that out after a while) The other day while I was driving messages poped up again. By complete luck I saw the alternator output which it was 12.1. So now I am convinced it is an alternator problem and all this time I thought it was aftertreatment problem which I serviced everything including dpf cleaning.