Question: If I download the eBook Library software to a computer (say, my work computer that is replaced every three years), log in as myself

and make a purchase, will it show up for download when I open up the library on my computer at home? (Assuming connection to the internet, of course.)
I'm wondering if I can browse and purchase from here, and then have it show up and download it from home, without having to connect and register my Reader to this computer. I don't want to have to mess with registering and unregistering every three years when I get a new computer, especially when I'm already all hooked up at home.
Thanks!