If you get a list, he gets a list. It is being fair and everyone pitches in on keeping the household running. My hubby works full time (40-50 hrs a week) and I usually work part time (8-15 hrs a week) sometimes I take on another job that is 2-3 months long so then working about 30 hours a week plus taking care of our daughter.
When I am working less I pick up the majority of the chores because I am around the house more (cooking, cleaning, laundry, grocery shopping) while hubby usually does the garbage,shoveling, homerepair and sometimes the laundry. When I am busier outside of the home my hubby, even though he works full time, does what needs to be done to have clean clothes and a somewhat clean house. I never leave him a list (unless it is the grocery list which he asks for) and he never leaves me a list. If anything he will ask if I am going to the store and if so I could I pick up something for him which I reply write it down please otherwise I will forget to pick it up.
You will need to talk with him about this, and the two of you figure it out. This is part of living together and there will be much harder things to work out in the future. It may be uncomfortable, he or you may not want to sit down and figure out a fair game plan about the chores but hopefully it helps you both stop disagreeing on this point.