So far as I can tell, Hartshorne's Algebraic Geometry doesn't define the composition of morphisms of schemes, or the restriction of a morphism to an open subset. Of course it's easy enough to define these in several ways, but not having a fixed definition kind of complicates writing up rigorous solutions to some of the more foundational exercises. Am I just missing something obvious here?
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I agree. On p. 72 locally ringed spaces and their morphisms are defined. Then on p. 73 we have the definition of an isomorphism of locally ringed spaces as a morphism with a two-sided inverse. In principle we need a composition of morphisms for this definition, but Hartshorne doesn't define it there. Instead, he characterizes isomorphisms $(f,f^\#)$ by the property that $f$ is a homeomorphism and $f^\#$ is an isomorphism of sheaves. By the way, with sheaves we have the same problem: On p. 63 isomorphisms of sheaves are defined to be morphisms of sheaves with a two-sided inverse, but not composition of morphisms of sheaves is defined! Finally, on p. 74 morphisms of schemes are defined as morphisms of the underlying locally ringed spaces.
I was a tutor for a lecture on algebraic geometry which was based on the book by Hartshorne. In one exercise one needed to know the precise definition of the composition of two morphisms. Nobody knew it, except for one student. It wasn't explained in the lecture, and the professor didn't even notice that the definition was missing.
One more reason why this book is not the best introduction to algebraic geometry. There are far better introductions, but they often also don't spell out the definition (except of course for EGA I, see Daniel's answer). In Görtz, Wedhorn, Algebraic geometry, the definition is sketched in a remark after Definition 2.29. In Qing Liu, Algebraic Geometry and Arithmetic Curves the defintition is sketched in a remark after Definition 2.20. I could not find the definition in Bosch, Algebraic geometry and commutative algebra; Eisenbud, Harris, The geometry of schemes; Ueno, Algebraic Geometry I.. In most texts such as Vakil, Foundations of algebraic geometry it is just said "there is an obvious notion of composition of morphisms".
It's looking like the answer is "yes." That said, in case it helps anyone who hits this via Google, the relevant definitions are actually written down in: