The study is being conducted to understand how the medicine, semaglutide, affects the immune system and other biological processes in people with Alzheimer's disease. Semaglutide is a medicine that doctors can prescribe in some countries for the treatment of type 2 diabetes and excess body weight. This study will help us understand whether semaglutide can also be used for the treatment of Alzheimer's disease. The study will last for about 77 weeks. In the first 12 weeks of treatment, participants will either get semaglutide (active medicine) or placebo (inactive dummy medicine). Which treatment participants get is decided by chance. In the following 52 weeks of treatment, all participants taking part in the study will get semaglutide. Participants must have a study partner, who is willing to take part in the study. Participants will get study medicine in a pen injector. The study partner will need to inject the study medicine into the skin of participant's stomach, thigh or upper arm once every week.