Algebraic Geometry
Since classical algebraic geometry allows to talk about polynomial equations, but not inequalities, the algebraic standard n-simplex is commonly defined as the subset of affine n+1-dimensional space, where all coordinates sum up to 1 (thus leaving out the inequality part). The algebraic description of this set is
- ,
which equals the scheme-theoretic description with
the ring of regular functions on the algebraic n-simplex (for any ring ).
By using the same definitions as for the classical n-simplex, the n-simplices for different dimensions n assemble into one simplicial object, while the rings assemble into one cosimplicial object (in the category of schemes resp. rings, since the face and degeneracy maps are all polynomial).
The algebraic n-simplices are used in higher K-Theory and in the definition of higher Chow groups.
Read more about this topic: Simplex
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