References: Grothendieck Groups of Spaces and Varieties

Solution 1:

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

$$K_0(\mathbb{P}^n_k) \cong \mathbb{Z}[t]/(t^{e+1}).$$

Is this module a free $\mathbb{Z} $ module?

Proving an interesting property of absolute values: $\frac{|x+y|}{1+|x+y|}\le\frac{|x|}{1+|y|}+\frac{|y|}{1+|y|}$ [duplicate]

Proof for inequality with absolute values: $\frac{|x+y|}{1+|x+y|} \leq \frac{|x|}{1+|x|} + \frac{|y|}{1+|y|}$

Prove that $A$ is a transitive set iff the following holds:$B \in C$ and $C \in A$ then $B \in A$

If $M \otimes N=0$ then prove that either $M=0$ or $N=0$ [duplicate]

Behaviour of the solutions of a system of differential equations at infinity.

How do I formally show that the Zariski tangent space of the intersection of two closed subschemes is the intersection of the tangent spaces?

Why does Inverse of Linear/Linear Function have same structure as Inverse of Matrix

Calculating complex integral for different values of $n$.