Weibel's conjecture explained

In mathematics, Weibel's conjecture gives a criterion for vanishing of negative algebraic K-theory groups. The conjecture was proposed by and proven in full generality by using methods from derived algebraic geometry. Previously partial cases had been proven by,,,, and .

Statement of the conjecture

Weibel's conjecture asserts that for a Noetherian scheme X of finite Krull dimension d, the K-groups vanish in degrees < -d:

K _{i(X)} = 0 for i< - d

and asserts moreover a homotopy invariance property for negative K-groups

K _{i(X)} = K _{i(X x} A ^{r)} for i\le - d andarbitrary r.