Is defining an "$S$-scheme" this way just a stylistic choice?

Question

I have a potentially pretty silly question, involving the definition of an "$S$-scheme".

For a fixed scheme $S$, the definition of the category of $S$-schemes is a category with:

• objects: scheme morphisms $X \rightarrow S$ (for schemes $X$)

• morphisms: for two arrows $X \rightarrow S$ and $Y \rightarrow S$, scheme morphisms $X \rightarrow Y$ making the obvious triangle commute $$\array{ X &&\!\!\!\!\!\!\!\longrightarrow\!\!\!\!\!\!\!&& Y \\ & \searrow && \swarrow \\ && S }$$

My question is: would it be misguided to prefer an equivalent definition to the category above to one whose objects are pairs $(X, X \rightarrow S)$ and morphisms are defined in an obvious way. An equivalent question applies, in the spirit of schemes, when people tend to define $R$-algebras as homomorphisms from a ring $R$ to a ring $A$.

I suppose when you specify a morphism $X \rightarrow S$ the target and source is implicitly defined, but it seems very awkward to me that many authors would leave out the scheme $X$.

I wonder if I just have a misguided view on how I should view a category (perhaps it is too "constructive"?) or that this is strictly a matter of style.

Thank you! I hope that that was not confusing.

Answer 1

I wouldn't go so far as to say your view is misguided, but it's just more "definitionally economical" to think about a morphism itself, instead of the pair of a morphism and its domain (since, as you correctly observed, part of the data of the morphism is its domain).

Analogously, one can ask whether a group $G$ should be

• a function $m:G\times G\to G$ satisfying certain properties

• a pair $(G,m)$ of a set $G$ and a function $m:G\times G\to G$ satisfying certain properties

• a triple $(G,m,e)$ of a set $G$, a function $m:G\times G\to G$, and an element $e\in G$ satisfying certain properties

• a quadruple $(G,m,e,i)$ of a set $G$, a function $m:G\times G\to G$, an element $e\in G$, and a function $i:G\to G$ satisfying certain properties

There are times when we want to use (or emphasize) certain things, and other times we don't. Choosing any of the particular definitions above isn't going to change the fact that, in day-to-day thought, a group is just... you know, a group. Similarly, an $S$-scheme is an $S$-scheme.

By the way, the category of $S$-schemes is just an instance of a slice category. Wikipedia uses your approach in its definition, whereas nLab uses the "economical" approach in its definition.

Answer 2

I think the answer has two parts, one formal and one informal. You already noticed the formal aspect: It's not strictly necessary to say what $X$ is, because that information is contained in the morphism $X\to S$. The informal side of the matter is that, despite the official definition, people often say and write things like "Let $X$ be an $S$-scheme" or "Let $X$ be a scheme over $S$." So the emphasis on $X$ (rather than the morphism) that you're looking for is introduced by an informal abuse of language.