Weibel's conjecture

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

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:

$K_i(X)=0\text{ for }i<-d$

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

$K_i(X)=K_i(X\times {𝔸}^r)\text{ for }i\le -d\text{ and arbitrary }r.$

Generalization

Recently, Template:Harvtxt have generalized Weibel's conjecture to arbitrary quasi-compact quasi-separated derived schemes. In this formulation the Krull dimension is replaced by the valuative dimension (that is, maximum of the Krull dimension of all blow-ups). In the case of Noetherian schemes, the Krull dimension is equal to the valuative dimension.

References

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