It is a standard fact that for any affine scheme \(X\) and any quasi-coherent sheaf \(F\) on \(X\), the cohomology group \(H^n(X,F)\) vanishes for \(n>0\). In this note, we present two short proofs of this theorem.
This is a note for a study group on class field theory. It covers fundamental topics on central simple algebras and the Brauer group.
The irreducibility of cyclotomic polynomials is a classical yet nontrivial and intriguing phenomenon. In this note, we present a geometric proof of this theorem using the language of schemes.