The notion of a projective morphism in algebraic geometry is surprisingly subtle. It is not quite clear what the definition is! For example, the definition in EGA differs from that in Hartshorne. For the purposes of this question, I take the definition in EGA.

Suppose $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ are projective morphisms. I know that $g \circ f$ is projective if $Z$ is quasicompact (see for example Exercise 18.3.B in the August 2012 version of the notes here). Is it true even without $Z$ quasicompact, or is there a counterexample?

(I suspect this is in one of the standard sources, but I haven't stumbled upon it.)

The notion of a projective morphism in algebraic geometry is surprisingly subtle. It is not quite clear what the definition is! For example, the definition in EGA differs from that in Hartshorne. For the purposes of this question, I take the definition in EGA$^1$.

Suppose $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ are projective morphisms. I know that $g \circ f$ is projective if $Z$ is quasicompact (see for example Exercise 18.3.B in the August 2012 version of the notes here). Is it true even without $Z$ quasicompact, or is there a counterexample?

(I suspect this is in one of the standard sources, but I haven't stumbled upon it.)

$^1$ EGA II, 5.5.1-5.5.2: $X$ is called projective over $Y$ if there is a closed $Y$-immersion $X \hookrightarrow \mathbb{P}(\mathcal{E})$ for some quasi-coherent sheaf $\mathcal{E}$ on $Y$ of finite type; equivalently if $X=\mathrm{Proj}(A)$ for some graded quasi-coherent $\mathcal{O}_Y$-algebra $A$ which is generated by $A_1$, which is of finite type.

The notion of a projective morphism in algebraic geometry is surprisingly subtle. It is not quite clear what the definition is! For example, the definition in EGA differs from that in Hartshorne (although not when the target is quasicompact). For the purposes of this question, I take the definition in EGA.

Suppose $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ are projective morphisms. I know that $g \circ f$ is projective if $Z$ is quasicompact (see for example Exercise 18.3.B in the August 2012 version of the notes here). Is it true even without $Z$ quasicompact, or is there a counterexample?

(I suspect this is in one of the standard sources, but I haven't stumbled upon it.)

The notion of a projective morphism in algebraic geometry is surprisingly subtle. It is not quite clear what the definition is! For example, the definition in EGA differs from that in Hartshorne. For the purposes of this question, I take the definition in EGA.

Suppose $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ are projective morphisms. I know that $g \circ f$ is projective if $Z$ is quasicompact (see for example Exercise 18.3.B in the August 2012 version of the notes here). Is it true even without $Z$ quasicompact, or is there a counterexample?

(I suspect this is in one of the standard sources, but I haven't stumbled upon it.)