Nonexactness

Nonexactness, also written non-exactness, is a property of a sequence or complex that is not exact. In mathematics, a sequence of objects and morphisms is exact at a given object if the image of the preceding map equals the kernel of the following map. Non-exactness occurs when this equality fails at one or more positions, and it is often analyzed using homology or cohomology to measure the failure.

In chain complexes and homological algebra, a sequence of abelian groups (or modules) with boundary maps is exact at a position when im(d_prev) = ker(d_next). Non-exactness at that position is quantified by the corresponding homology group, H_n = ker(d_n)/im(d_{n+1}). A complex is exact in a region precisely when all its homology groups in that region vanish; nonzero homology indicates where the sequence fails to be exact.

In differential geometry and calculus, the de Rham complex provides another setting for exactness: a form is exact if it is the exterior derivative of a lower-degree form. Closed forms satisfy d omega = 0; exact forms are a subset of closed forms. Non-exactness arises when closed forms are not globally exact, which is captured by de Rham cohomology and reflects topological features such as holes or nontrivial cycles in the underlying space.

Consequences and interpretation: non-exactness indicates obstructions to solving equations or to expressing global primitives. It is a formal way to quantify the failure of local data to determine global structure, and it is central to the study of homology, cohomology, and the global properties of spaces and maps. See also exact sequence, homology, cohomology, chain complex, and de Rham cohomology.