Exact sequence

An exact sequence is a sequence of maps between algebraic objects in which the image of each map equals the kernel of the next.

Kernel (algebra): The kernel of a homomorphism is the set or subobject mapped to the identity element. Exactness requires this subobject to equal the image arriving from the preceding map.
Short exact sequence: A short exact sequence is an exact sequence with three objects and zero objects at both ends. Its compact form isolates an inclusion, a surjection, and the quotient they describe.
Long exact sequence: A long exact sequence is an exact sequence with many consecutive objects, often arising from a homological construction. It organizes linked kernel and quotient information produced by a short exact sequence.
Left exact functor: A left exact functor preserves finite limits and the initial part of short exact sequences. It preserves kernels but need not preserve surjections or full exactness.
Image (mathematics): The image of a map is the set or subobject of values attained by its inputs. It is compared with the next map's kernel at every position in the sequence.
Split exact sequence: A split exact sequence is a short exact sequence whose surjection has a section, or whose injection has a retraction. A splitting makes the middle object decompose as a direct sum of the outer objects.
Derived functor: A derived functor extends a functor to measure its failure to preserve exactness. Its values often appear as terms in long exact sequences.
Right exact functor: A right exact functor preserves finite colimits and the terminal part of short exact sequences. It preserves cokernels but need not preserve injections or full exactness.
Group homomorphism: A group homomorphism is a function between groups that preserves their operations. Group homomorphisms provide a familiar setting for kernels, images, and exactness.
Quotient group: A quotient group is formed by identifying elements of a group that differ by a normal subgroup. In a short exact sequence of groups, the final nonzero map often identifies this quotient.

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