In the corona vaccine, why don't they put antibodies in it, instead of prompting your body to make its own antibodies?

Antibodies only last several months if they are injected and then they disappear and then you are no longer immune.

The purpose of vaccines is to make your body immune from disease and infection. It can do that by having your own body make the antibodies for the rest of your life.  You have to have the body get exposed to parts of the virus which creates cells dedicated to making antibodies. The cells remain with you which gives you the ability to form antibodies against the virus for the rest of your life hopefully.