I'm sure I am missing something quite simple, but why are divisors on an algebraic variety actually homological cycles? I can see them as chains, but in order to take homology (as one always does), or to integrate along them, I would like to see them as singular cycles ($\partial D=0$). (Not just algebraic cycles, of course, which they are by definition.)
Perhaps I am just a bit confused about definitions. Thank you in advance for any clarifying suggestion.