I simply began skimmed within the Facebook API and OpenSocial API. In OpenSocial there's the construct of the gadget that's located around the container. You simply write the code inside some XML structure and upload in some way towards the container. Therefore the code ist located through the OpenSocial provider. I could not look for a pendent within the Facebook API. It appears that applications on Facebook should always be located on the designers system (and this is not on Facebook). Is the fact that correct? Or did I miss something?

Yes, that's correct. Facebook is busy enough running its very own stuff without your stuff keeping their servers busy!