Does it still show it when you're not logged in or just logged in as a general user? I've run into situations where something was not displaying as I thought it should only to realize that it only showed for the Admin account. I can't remember if it was the same situation that you're having or not.

Thanks for your answer.  Now that you say that it is obvious.

Is there a "No Container" container available?  This would only show the down arrow in edit mode, but it would make things a little easier during construction of the site.  I know that I saw this type of container in the past when working on a site quite some time ago.