short exact sequence

Definition

A short exact sequence is sequence of morphisms between objects of the form

such that the image of one morphism equals the kernel of the next.

Special properties

This means that the morphism is injective and is surjective.

Behavior with tensoring in -mod

Let

be a short exact sequence of R-modules. Then using the tensor product,

is exact.

Proof

First, we show that is surjective. Given , since the original sequence is exact, is surjective, so there exists such that . Thus, . is generated by basic tensor, and is linear, so it is surjective.

It remains to show that . It is clear that , so only . Let . Since , by the the universal property of the quotient, then induces a map

So we can use this to construct a map

which satisfies , which would mean that is injective (and this is sufficint for the condition we need).

Let and . Fix such that . Then define

This map is bilinear since so the chosen representatives will “pass through the addition”, to be bilinear.

The map is also well defined since if satisfies then

Since , then

This gives