Editing K-theory (section)

=== K<sub>0</sub> of a projective bundle ===
Another important formula for the Grothendieck group is the projective bundle formula:<ref>{{Cite journal|last=Manin|first=Yuri I|author-link=Yuri Manin|date=1969-01-01|title=Lectures on the K-functor in algebraic geometry|journal=Russian Mathematical Surveys|language=en|volume=24|issue=5|pages=1–89|doi=10.1070/rm1969v024n05abeh001357|issn=0036-0279|bibcode=1969RuMaS..24....1M}}</ref> given a rank r vector bundle <math>\mathcal{E}</math> over a Noetherian scheme <math>X</math>, the Grothendieck group of the projective bundle <math>\mathbb{P}(\mathcal{E})=\operatorname{Proj}(\operatorname{Sym}^\bullet(\mathcal{E}^\vee))</math> is a free <math>K(X)</math>-module of rank ''r'' with basis <math>1,\xi,\dots,\xi^{n-1}</math>. This formula allows one to compute the Grothendieck group of <math>\mathbb{P}^n_\mathbb{F}</math>. This make it possible to compute the <math>K_0</math> or Hirzebruch surfaces. In addition, this can be used to compute the Grothendieck group <math>K(\mathbb{P}^n)</math> by observing it is a projective bundle over the field <math>\mathbb{F}</math>.