A ring in which some finitely generated ideal is not finitely presented; equivalently, there exists a homomorphism between finitely generated free modules whose kernel is not finitely generated. This is the failure of coherence.

Principle

Failure of finite presentation of relations: even though an ideal is generated by finitely many elements, the module of relations among those generators requires infinitely many generators, so relations are not controlled finitely.

Demonstration

A typical concrete situation is the polynomial ring in countably many variables k[x1,x2,...]; the ideal I = (x1,x2,x3,...) is finitely generated only in the sense of countably many generators but not finitely presented as a module over the ring, producing failure of coherence for module kernels tied to I.

Misapplication

Using arguments that assume kernels of maps between finitely generated modules are finitely generated (for instance, treating sheaves as coherent without checking hypotheses) leads to incorrect conclusions; applying homological tools that require coherence gives unreliable results.

Consequence

The category of finitely presented modules need not be exact: kernels of maps may exit the class, obstructing many constructions in algebraic geometry and homological algebra that rely on finite presentability at the level of modules or sheaves.

Reversal

A Coherent Ring, where every finitely generated ideal is finitely presented and kernels between finite free modules are finitely generated.

Boundary

Noncoherence is a statement about finite presentation of finitely generated ideals and kernels; it does not automatically assert absence of other finiteness properties (an example can be noncoherent yet satisfy other local finiteness), and the phenomenon occurs in both commutative and noncommutative settings.

Semantic Tension

‘Noncoherent’ is sometimes conflated with ‘non-Noetherian’, but while all Noetherian rings are coherent, noncoherence pinpoints a different failure: of relations rather than of generation; the tension is between generation and presentation.

Synthesis

A Noncoherent Ring is one where finite generation does not extend to finite presentation: some finitely generated ideals have infinitely many independent relations, so kernels of finite maps are not controlled, breaking exactness properties important in algebraic and geometric constructions.