Editing K-theory (section)

=== K<sub>0</sub> of projective space ===
One of the most commonly used computations of the Grothendieck group is with the computation of <math>K(\mathbb{P}^n)</math> for projective space over a field. This is because the intersection numbers of a projective <math>X</math> can be computed by embedding <math>i:X \hookrightarrow \mathbb{P}^n </math> and using the push pull formula <math>i^*([i_*\mathcal{E}]\cdot [i_*\mathcal{F}])</math>. This makes it possible to do concrete calculations with elements in <math>K(X)</math> without having to explicitly know its structure since<ref>{{Cite web|title=kt.k theory and homology - Grothendieck group for projective space over the dual numbers|url=https://mathoverflow.net/questions/77089/grothendieck-group-for-projective-space-over-the-dual-numbers|access-date=2020-10-20|website=MathOverflow}}</ref>
<math display="block">K(\mathbb{P}^n) = \frac{\mathbb{Z}[T]}{(T^{n+1})}</math>
One technique for determining the Grothendieck group of <math>\mathbb{P}^n</math> comes from its stratification as
<math display="block">\mathbb{P}^n = \mathbb{A}^n \coprod \mathbb{A}^{n-1} \coprod \cdots \coprod \mathbb{A}^0</math>
since the Grothendieck group of coherent sheaves on affine spaces are isomorphic to <math>\mathbb{Z}</math>, and the intersection of <math>\mathbb{A}^{n-k_1},\mathbb{A}^{n-k_2}</math> is generically
<math display="block">\mathbb{A}^{n-k_1} \cap \mathbb{A}^{n-k_2} = \mathbb{A}^{n-k_1-k_2}</math>
for <math>k_1 + k_2 \leq n</math>.