Relative dimension

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In mathematics, specifically linear algebra and geometry, relative dimension is the dual notion to codimension.

In linear algebra, given a quotient map

    V
    →
    Q
  

{\displaystyle V\to Q}

, the difference dim V − dim Q is the relative dimension; this equals the dimension of the kernel.

In fiber bundles, the relative dimension of the map is the dimension of the fiber.

More abstractly, the codimension of a map is the dimension of the cokernel, while the relative dimension of a map is the dimension of the kernel.

These are dual in that the inclusion of a subspace

    V
    →
    W
  

{\displaystyle V\to W}

of codimension k dualizes to yield a quotient map

      W
      
        ∗
      
    
    →
    
      V
      
        ∗
      
    
  

{\displaystyle W^{*}\to V^{*}}

of relative dimension k, and conversely.

The additivity of codimension under intersection corresponds to the additivity of relative dimension in a fiber product. Just as codimension is mostly used for injective maps, relative dimension is mostly used for surjective maps.

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