cokernelbut

The term "cokernelbut" appears to be a misspelling or a non-standard term within mathematics, likely intended to refer to the "cokernel" in abstract algebra or functional analysis. The cokernel is a fundamental concept in category theory and homological algebra, closely related to the kernel.

In the context of a linear map or a homomorphism between two modules, say f: V -> W, the kernel of f, denoted ker(f), is the set of all elements in V that map to the zero element in W. The cokernel, conversely, is defined as the quotient of the codomain W by the image of the map f, denoted coker(f) = W / im(f). It essentially measures the "extent to which" the map f fails to be surjective.

The cokernel can be understood as the dual concept to the kernel. Just as the kernel captures information about the injectivity of a map, the cokernel captures information about its surjectivity. For instance, a linear map f is surjective if and only if its cokernel is the zero module.

The cokernel plays a significant role in constructing resolutions and in the study of short exact sequences. It is also a key component in the definition of various derived functors, such as the Ext functors. The concept generalizes to other categories beyond modules, including topological spaces (where it relates to quotient spaces) and sheaves. If "cokernelbut" was intended to refer to a specific, perhaps colloquial, variation or extension of the cokernel, that usage is not widely established in standard mathematical literature.