##  [Failure of Exactness](/failure-exactness-0) 

 Definition

A situation where a sequence of homomorphisms ...→A_{i-1}→A_i→A_{i+1}→... fails to be exact at some position i, meaning the image of the incoming map is not equal to the kernel of the outgoing map; homology groups measure this failure.

 

 

 

 

 

 





## Principle

Principle

Exactness captures the perfect balance of images and kernels; failure of exactness signals obstruction, measured by nonzero homology, and reflects hidden extension, torsion, or relational data not captured by the maps.

 

 

 

 

 





## Demonstration

Demonstration

Consider the sequence Z --×2--> Z --×3--> Z. The image of the first map is 2Z while the kernel of multiplication by 3 is 0, so image≠kernel and the sequence fails to be exact at the middle term.

 

 

 

 

## Misapplication

Misapplication

Using sequences as though exact without verifying — for example applying derived functor vanishing or splitting arguments on nonexact sequences — leads to incorrect homological conclusions.

 

 

 

 

 





## Consequence

Consequence

Nonzero homology groups record obstructions to lifting, splitting, or solvability; they indicate extensions, detect torsion, and obstruct naive computations in derived functors and spectral sequences.

 

 

 

 

## Reversal

Reversal

Exact sequences, where images and kernels match at every position, allow decomposition, lifting, and clean calculations of derived functors; exactness is a hypothesis that enables many structural theorems.

 

 

 

 

 





## Boundary

Boundary

Pertains to chains of homomorphisms in algebra, topology and geometry; failure can be local or global and must be distinguished from lack of surjectivity or injectivity alone since exactness combines both properties.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Between exactness as an ideal condition and practical calculations: short exactness, left- or right-exactness of functors, and the presence of derived functors create a lattice of partial exactness notions.

 

 

 

 

 





## Synthesis

Synthesis

Failure of exactness is the concrete homological signal that images and kernels misalign: homology measures the obstruction, which encodes extension and torsion data and warns against naively transferring properties that require exact sequences.