Editing K-theory (section)

=== K<sub>0</sub> of an Artinian algebra over a field ===
One important property of the Grothendieck group of a [[Noetherian scheme]] <math>X</math> is that it is invariant under reduction, hence <math>K(X) = K(X_{\text{red}})</math>.<ref>{{Cite web|url=https://mathoverflow.net/questions/77089/grothendieck-group-for-projective-space-over-the-dual-numbers|title=Grothendieck group for projective space over the dual numbers|website=mathoverflow.net|access-date=2017-04-16}}</ref> Hence the Grothendieck group of any [[Artinian ring|Artinian]] <math>\mathbb{F}</math>-algebra is a direct sum of copies of <math>\Z</math>, one for each connected component of its spectrum. For example,
<math display="block">K_0 \left(\text{Spec}\left(\frac{\mathbb{F}[x]}{(x^9)}\times\mathbb{F}\right)\right) = \mathbb{Z}\oplus\mathbb{Z}</math>