Editing Simplex (section)

== Algebraic geometry ==
Since classical [[algebraic geometry]] allows one to talk about polynomial equations but not inequalities, the ''algebraic standard n-simplex'' is commonly defined as the subset of affine {{math|(''n'' + 1)}}-dimensional space, where all coordinates sum up to 1 (thus leaving out the inequality part). The algebraic description of this set is
<math display=block>\Delta^n := \left\{x \in \mathbb{A}^{n+1} ~\Bigg|~ \sum_{i=1}^{n+1} x_i = 1\right\},</math>
which equals the [[Scheme (mathematics)|scheme]]-theoretic description <math>\Delta_n(R) = \operatorname{Spec}(R[\Delta^n])</math> with
<math display=block>R[\Delta^n] := R[x_1,\ldots,x_{n+1}]\left/\left(1-\sum x_i \right)\right.</math>
the ring of regular functions on the algebraic {{mvar|n}}-simplex (for any [[ring (mathematics)|ring]] <math>R</math>).

By using the same definitions as for the classical {{mvar|n}}-simplex, the {{mvar|n}}-simplices for different dimensions {{mvar|n}} assemble into one [[simplicial object]], while the rings <math>R[\Delta^n]</math> assemble into one cosimplicial object <math>R[\Delta^\bullet]</math> (in the [[category (mathematics)|category]] of schemes resp. rings, since the face and degeneracy maps are all polynomial).

The algebraic {{mvar|n}}-simplices are used in higher [[K-theory|''K''-theory]] and in the definition of higher [[Chow group]]s.