Projective, injective, and flat modules

Definitions

An R-module \(D\) is:

1. Projective if \(Hom(D, -)\) is an exact functor.
2. Injective if \(Hom(-,D)\) is an exact functor.
3. Flat if \(D \otimes -\) is an exact functor.

Characterization

We know that \(Hom(D,-)\) and \(Hom(-,D)\) are left-exact and that \(D\otimes -\) is right-exact; so for them to be exact, we only need:

• A module \(D\) is projective when every \(f : B \longrightarrow C\) surjective induces \((f\circ\_) :Hom(D,B) \longrightarrow Hom(D,C)\) surjective.

  https://raw.githubusercontent.com/M42/m42.github.io/images/projective.jpeg

• A module \(D\) is injective when \(f : A \longrightarrow B\) surjective induces \((\_\circ f) : Hom(B,D) \longrightarrow Hom(A,D)\) surjective.

  https://raw.githubusercontent.com/M42/m42.github.io/images/injective.jpeg

• A module \(D\) is flat when \(f : A \longrightarrow B\) injective induces \(f' : D\otimes A \longrightarrow D \otimes B\) injective.

Site proudly generated by Hakyll; written with Emacs and org-mode.