Good people!
I'm asking for some good references here! I need to find the Grothendieck groups for a couple of varieties, and I'm struggling with figuring out how to make this work...
Would anyone of you happen to know of any good references where the author explicitly finds the Grothendieck groups of a couple of not-too-trivial spaces, so I can look at that, figure out the gist of it, and apply the same approach to the varieties I have on hand?
As always, look forward to what you have to offer!
Question: "Good people! I'm asking for some good references here! I need to find the Grothendieck groups for a couple of varieties, and I'm struggling with figuring out how to make this work..."
Answer: If $X$ is any scheme and $E$ is a rank $e+1$ locally trivial sheaf on $X$, it follows there is an isomorphism (the "projective bundle formula")
$$K_0(\mathbb{P}(E^*)) \cong K_0(X)[t]/(t^{e+1}).$$
In particular for projective $n$-space $\mathbb{P}^n_k$ you get
$$K_0(\mathbb{P}^n_k) \cong \mathbb{Z}[t]/(t^{e+1}).$$