After taking an algebraic geometry course last year, I've been reviewing the material this year, and I remembered something that struck me as odd, but which I'd neglected to ask about at the time:

Hartshorne's definition of an open subscheme and open immersion (p.85):

An

open subschemeof a scheme $X$ is a scheme $U$, whose topological space is an open subset of $X$, and whose structure sheaf $\mathcal{O}_U$ is isomorphic to the restriction $\mathcal{O}_X|_U$ of the structure sheaf of $X$. Anopen immersionis a morphism $f:X\to Y$ which induces an isomorphism of $X$ with an open subscheme of $Y$.

However, many other sources (including Wikipedia) go with an open subscheme **being** a scheme $Z$ of the form $(U,\mathcal{O}_X|_U)$ where $U\subseteq X$ is an open subset, and then defining an open immersion to be a morphism $f:Y\to X$ that factors through an isomorphism with an open subscheme, i.e. there is an isomorphism of schemes $g:Y\,\stackrel{\sim}{\to} Z$ such that $f=i\circ g$, where $i:Z\to X$ is the inclusion map.

Why make these (quite subtly) more general definitions? Clearly, there is some issue that is addressed by either making the definition of open subscheme, or that of open immersion, contain this "extra" isomorphism. By no means do I treat things that are isomorphic as being "equal", so I understand that this definition is not really equivalent. But surely defining a subgroup of a group $G$ to be "a group $H$ whose underlying set is a subset of $G$ and whose operation is isomorphic to the restriction of the operation of $G$" would sound a bit off?