Uma prova da conjectura do anulamento de Serre via K-teoria algébrica

Abstract

Considerer a commutative local Noetherian regular ring (A, m) and let p and q be two prime ideals of A such that `A(A/p ⊗A A/q) < ∞. In 1958, Jean Pierre Serre defined the intersection multiplicity of A/p e A/q by χ(A/p, A/q) := X∞ i=o (−1)i `A(Tori A(A/p, A/q)), and he conjectured, among other things, that if dim(A/p) + dim(A/q) < dim(A), then χ(A/p, A/q) = 0. Such conjecture became known as Serre vanishing conjecture. This was proven by Paul C. Roberts and, independently, by Henri Gillet and Christophe Soulé in 1985. The main of this dissertation is to illustrate and detail, following the ideas of Gillet e Soulé, how some notions and machinery of algebraic K-theory can be used to prove such conjecture.