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## On Diffeomorphisms

Differential Geometry
Tsakanikas Nickos
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### On Diffeomorphisms

Let $f \ \colon M \longrightarrow N$ and $g \ \colon N \longrightarrow P$ be smooth maps between smooth manifolds. Show that the composite $g \circ f$ is smooth. Conclude that the composition of diffeomorphisms is again a diffeomorphism, and show that the converse is not true.

Are there any (reasonable) conditions (imposed on the maps) that guarantee that the converse is also true? The motivation behind this question is the following (for example), coming from Algebraic Geometry: "Let $f \ \colon X \longrightarrow Y$ and $g \ \colon Y \longrightarrow Z$ be morphisms of (noetherian) schemes. If $g \circ f$ is proper and $g$ is separated, then $f$ is also proper."