Find a concept
Start typing to search the mathematical index.
A sequence is exact at a term precisely when the incoming image equals the outgoing kernel.
Exactness via kernels and images: A sequence of -modules and homomorphisms
is exact at if and only if .
This rewrites exactness purely in terms of kernels and images, and is used constantly for short exact sequences.
An exact sequence of modules is a sequence of modules and module homomorphisms
such that for every one has , where the kernel and image are taken in the sense of kernels and images. A convenient checklist formulation is given in exactness via kernels/images.
Exact sequences package algebraic information: injectivity, surjectivity, quotients, and splitting phenomena are all phrased as exactness conditions.
Let be a module homomorphism. The kernel of is
It is a submodule, as recorded in kernels are submodules.
Kernels measure injectivity: is injective iff . They also define the notion of exactness (see exact sequences, where kernels match images).
Let be a module homomorphism. The image of is
It is a submodule of ; see kernel/image are submodules for the standard closure argument.
Images measure surjectivity: is surjective iff . Together with kernels, images define exactness via the condition for consecutive maps (see exactness via kernels and images
A short exact sequence is an exact sequence (see exact sequence of modules)
such that is injective and is surjective, and . In elementary terms, is an injective map identifying with a submodule of , and is a surjective map with .
Short exact sequences classify extensions: they encode how is built from a submodule isomorphic to and a quotient isomorphic to .