##  [Nonflatness](/nonflatness-0) 

 Definition

The property of a module or morphism failing to be flat: tensoring with the module does not preserve exact sequences, so base change can produce nonexact results and torsion or Tor groups appear.

 

 

 

 

 

 





## Principle

Principle

Flatness is equivalent to exactness of the tensor functor; nonflatness therefore signals hidden relations or torsion that obstruct the passage of exactness through tensor products and cause base-change anomalies.

 

 

 

 

 





## Demonstration

Demonstration

Consider the short exact sequence 0→Z --×2--> Z → Z/2Z →0 and tensor with Z/2Z; because Z/2Z is not flat over Z, tensoring destroys exactness and Tor_1(Z/2Z,Z/2Z)≠0, demonstrating nonflatness concretely.

 

 

 

 

## Misapplication

Misapplication

Assuming a base change or pullback preserves kernels and cokernels without checking flatness leads to incorrect fiber computations or false dimension statements in algebraic geometry.

 

 

 

 

 





## Consequence

Consequence

Nonflatness causes failure of naive base-change theorems, nonconstancy of fiber dimension, and can produce unwanted torsion in families; it complicates descent, deformation, and cohomology calculations.

 

 

 

 

## Reversal

Reversal

Flatness restores tensor-exactness: flat modules and flat morphisms allow base-change to commute with taking kernels and preserve fiberwise properties, yielding better-behaved families.

 

 

 

 

 





## Boundary

Boundary

A notion for modules and morphisms of schemes or rings; nonflatness does not by itself indicate pathology of all constructions and must be examined relative to the exact sequences or base changes under consideration.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Between flatness, projectivity and freeness: projective or free implies flat, but flat need not be projective; nonflatness should not be conflated with lack of freeness alone.

 

 

 

 

 





## Synthesis

Synthesis

Nonflatness is the systematic failure of tensoring to preserve exactness: it reveals torsion or hidden relations that spoil base-change, fiberwise behavior, and many simplifications one expects in flat families.