I'm currently reading Olivier Debarre's book “Higher-Dimensional Algebraic Geometry” and I'm stuck on a small remark made in the definition of Cartier divisors (it's on page 3 if you have the book). Note that here, all schemes are Noetherian and separated (this is an assumption made earlier). To be precise, I can't prove the passage in bold :
A Cartier divisor on a scheme $X$ is a collection of pairs $(U_i, f_i)$, where $(U_i)$ is an open cover of $X$ and $f_i$ an invertible element of $\mathcal K_X(U_i)$ (the sheaf of total quotient rings of $\mathcal O_X$), such that $f_i / f_j$ is in $\mathcal O_X^*(U_i \cap U_j)$. When $X$ is reduced, we may take integral open sets $U_i$ (...).
I am not able to build these integral open sets. I've tried to “play” with the finitely many irreducible components of X and the assumption of $X$ being reduced, but it doesn't seem obvious to me. All $U_i$ are reduced and Noetherian too, so I tried to look at the irreducible components of these open sets to cobble something together, but it came to nothing. Do you have any ideas?
