tabindex in CSS

Is it possible to control tabindex with CSS and if yes, which browsers support it and on which elements?

EDIT

I should say, my goal is to catch keydown events on a div. I saw this page http://www.quirksmode.org/js/events/keys.html# that tests keyboard events and it shows that the only way keydown fires on anything other than document and body or some kind of form element or link is to have tabindex declared on it. But I read on W3C site:

The following elements support the атрибут tabindex: A, AREA, BUTTON, INPUT, OBJECT, SELECT, and TEXTAREA.

So I am a little confused, what to do in order to be standarts compliant and make my use case work?

EDIT2

My whole use case is a div with a lot of content with an artificial scroll bar. I am able to scroll it with mouse events but no luck with the keyboard so far.