I have a probably silly question about proof of Proposition 6.6.5(b) from Hartshorne (Algebraic Geometry) on page 133:
Proposition 6.5. Let $X$ satisfy (*), let $Z$ be a proper closed subset of $X$, and let $U = X - Z$. Then:
(a) there is a surjective homomorphism $\operatorname{Cl} X \to \operatorname{Cl} U$ defined by $D = \sum n_i Y_i \mapsto \sum n_i (Y_i \cap U)$, where we ignore those $Y_i \cup U$ which are empty;
(b) if $codim(Z,X) \le 2$, then $\operatorname{Cl} X \to \operatorname{Cl} U$ is an isomorphism;
A few words about notations:
The condition (*) requires that $X$ is a noetherian integral separated scheme which is regular in codimension one.
The Weil group $\operatorname{Div} X$ consists of Weil divisors $D= \sum n_i Y_i$ where $Y_i$ are prime divisors, the $n_i$ are integers, and only finitely many $n_i$ are different from zero. If all the $n_i \le 0$, we say that D is effective. We obtain $\operatorname{Cl} X$ if we mod out effective Weil divisors from $\operatorname{Div} X$.
For sake of completeness that prime divisor $Y$ is a closed integral subscheme $Y \subset X$ of codimension one (comp. page 130).
The proof of Proposition 6.5 (b):
The groups $\operatorname{Div} X$ and $\operatorname{Cl} X$ depend only on subsets of codimension $1$, so removing a closed subset $Z$ of codimension $ \le 2$ doesn't change anything.
What does it mean "removing a closed subset $Z$ of codimension $ \le 2$ doesn't change anything"? What Hartshorne means concretely by "anything" here? Does it simply mean that $Y_i \cap (X -Z) \neq \varnothing$ for all prime divisors $Y_i$ and $Z \subset X$ of codimension $\le 2$? Or what does this formulation could else mean?
The idea here is that all of the information we need to figure out what's going on with divisors on a scheme $X$ satisfying (*) is accessible by looking just at the codimension-one points: two divisors $D,D'$ are the same iff there's a rational function $f$ so that $D-D'=div(f)$. But we can determine exactly what $div(f)$ is by looking at $f$'s valuation at each codimension 1 point of $X$. Altering codimension-two behavior doesn't change the stalk at the generic point of $X$ nor can it remove the generic point of any codimension-one irreducible variety, so it doesn't change our method of detecting when two divisors are equal or our data we use to perform this task. This is how I'd interpret "anything" in this text.