##  [Noninjective Limit](/noninjective-limit-0) 

 Definition

A pathology of inverse limits (proj lim) in which the inverse limit functor fails to be exact on the right, producing nontrivial derived limits (lim^1, lim^n) and breaking expected exactness properties or identifications.

 

 

 

 

 

 





## Principle

Principle

Inverse limit is left exact but need not be exact; obstructions appear as derived functors lim^i. Noninjective Limit behavior arises in systems with lack of Mittag‑Leffler or other stabilization properties, causing lim to fail to preserve surjections or produce hidden extension data.

 

 

 

 

 





## Demonstration

Demonstration

Consider an inverse system of abelian groups with surjective transition maps whose kernels do not stabilize; then lim^1 may be nonzero and exact sequences of inverse systems need not yield exact sequences after taking limits. This explains phenomena like failure of passage from finite level cohomology to the limit.

 

 

 

 

## Misapplication

Misapplication

Blindly commuting inverse limit with cohomology or assuming that taking limits preserves exactness without checking derived limits leads to mistaken identifications, wrong dimension counts, or loss of extension information.

 

 

 

 

 





## Consequence

Consequence

Recognizing noninjective limit pathology forces computation of derived limits, use of spectral sequences, imposing Mittag‑Leffler conditions, or working in pro‑categories; it clarifies where naive inverse‑limit arguments collapse and where extension classes remain in the limit.

 

 

 

 

## Reversal

Reversal

Under stabilization hypotheses (e.g., Mittag‑Leffler, pro‑zero of higher terms) the derived limits vanish and the inverse limit becomes exact in the relevant range, reversing the pathology into a well‑behaved limit.

 

 

 

 

 





## Boundary

Boundary

This notion pertains to inverse systems and their derived functors; it does not concern direct limits (colimits) which have different exactness properties. It requires homological algebra or category theory to define lim^i and excludes naive pointwise limiting arguments.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Noninjective Limit competes with intuitive 'limits commute with everything' heuristics: the tension is between pointwise/stable behavior where limits behave and pathological systems where derived limits carry essential obstruction data.

 

 

 

 

 





## Synthesis

Synthesis

A Noninjective Limit is the identification of failure of exactness for inverse limits, recorded by nonzero lim^i; it signals where stabilizing conditions are missing and prescribes derived or pro‑categorical remedies to recover the correct global limit behavior.