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

To see this, note that given b we can define a by m \mapsto b(m\otimes 1), and given a we can define b by m \otimes x \mapsto x.a(m), 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.