Ok. I have a question about guitars if anyone can help.?

I know that if I am playing the open "G" string and I want to sharp it. I have to fret the "G" string at the first fret. I also know that if I want to flat it I have to fret the fourth fret of the "D" string. Here's my question. If I'm playing the open "G" string, and I want to flat that "G" an octave higher. Don't I flat the "G" string at the 11th fret? Just curious.