Scheme morphism properties that aren't stable under taking triangles?

Question

Let $\mathcal{P}$ be the collection of properties of morphisms of schemes that satisfy the following conditions:

• Stability under arbitrary pullbacks
• Stability under composition

There's a nice list of those on the stacks website.

I'm trying to identify the nice properties I can use freely and intuitively. In particular, are there any properties in that list that don't satisfy the following rule:

Triangle rule: If two of the the collection $\{f,g,f \circ g\}$ satisfy $P$ and all have nonempty domain and codomain then so does the third.

Answer 1

I think there are very few properties $\mathcal P$ that satisfy the following statement (implied by your triangle rule):

Let $X$, $Y$, $Z$ be nonempty schemes, and suppose the diagram $$\begin{array}{ccccc} X & & \xrightarrow{\ \ \ \ \ \ f\ \ \ \ \ \ } & & Y \\ & \searrow & & \swarrow & \\ & & Z & & g \end{array}$$ commutes. If $g \circ f$ and $f$ satisfy $\mathcal P$, then so does $g$.

For example, if $k$ is an algebraically closed field, and $X = Z = \operatorname{Spec} k$, we can take $Y$ any $k$-variety and $f$ a $k$-point (which always exists), implying that any $k$-variety $Y$ has property $\mathcal P$. You can probably come up with similar examples (e.g. $Y$ is a branched cover of $Z$, and $X$ is the locus where it is unramified, etc) that exclude most of the geometrically meaningful properties.

On the other hand, there are some properties that satisfy the other (nontrivial) statement of your rule:

Let $X$, $Y$, $Z$ be nonempty schemes, and suppose the diagram $$\begin{array}{ccccc} X & & \xrightarrow{\ \ \ \ \ \ f\ \ \ \ \ \ } & & Y \\ & \searrow & & \swarrow & \\ & & Z & & g \end{array}$$ commutes. If $g \circ f$ and $g$ satisfy $\mathcal P$, then so does $f$.

Example. Étale morphisms satisfy this property, which is one of the reasons it's possible/useful to talk about the small étale site.

Non-example. Flat morphisms do not satisfy this property. Indeed, if $Z = \operatorname{Spec} k$, then the condition is vacuous. Yet not every morphism of $k$-schemes is flat.

It's probably a good exercise to take some property and construct a counterexample to either of the statements above. For most properties $\mathcal P$, neither of the above hold.