Restriction and extension of scalars are adjoint

Just a quick note to say it’s almost entirely obvious that if we have a ring map, an -module and a -module , then giving a map is exactly the same as giving a map (where this denotes considered as an -module via the map ). In other words, extension of scalars is a left adjoint to restriction of scalars.

To see this, note that given we can define by , and given we can define by , and these operations are obviously mutually inverse.

Not sure why I felt like blogging about such a triviality. Maybe I think it’s a nice example of something that confused me slightly while solving a problem today and now, with the language of adjoint functors, I should never have to waste time thinking slightly about it again. I guess if you care this is also the basic case of the pullback-pushforward adjunction on quasicoherent sheaves.