There are so many things called cohomology and so many ways of writing the letter “H”. And I’m not even talking about the distinction between de Rham cohomology and singular cohomology and absolute Hodge cohomology: I really just mean the difference between writing \mathbb{H}^q(X;\Omega^p) and \mathrm{H}^q(X,\Omega^p). Why does one use a semicolon and the other a comma, and one an \mathbb{H} and the other an \mathrm{H}? And what about that one time that I saw somebody write \mathscr{H}?

I remember being so confused by this during my PhD, and as a result all my notes from those days are a nightmare to read. Normally this wouldn’t matter at all, but for various reasons I’m interested in revisiting some of my old calculations. So here we go: I’m finally going to set my notation straight, and clear up as many of my past confusions as possible.