Weibel's conjecture

In mathematics, Weibel's conjecture gives a criterion for vanishing of negative algebraic K-theory groups. The conjecture was proposed by Weibel (1980) and proven by Kerz, Strunk & Tamme (2018) using methods from derived algebraic geometry.

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 \mathbb {A} ^{r}){\text{ for }}i\leq -d{\text{ and arbitrary }}r.

gollark: Ridiculous. Just make toilet paper out of trees directly.

gollark: And you need entertainment as well, so probably a few hundred terabytes of HDDs so you can store every movie you're ever likely to watch, with redundancy, and you might as well just store every scientific paper and book ever written to help rebuild society.

gollark: I guess you could install that too.

gollark: Also "defensive" lasers for "peaceful purposes only".

gollark: You should also stick entirely independent food production into your bunker, as well as its own nuclear reactor and a thing to condense water from the air.

References

Weibel, Chuck (1980), "K-theory and analytic isomorphisms", Invent. Math., 61 (2): 177–197, doi:10.1007/bf01390120

Kerz, Moritz; Strunk, Florian; Tamme, Georg (2018), "Algebraic K-theory and descent for blow-ups", Invent. Math., 211 (2): 523–577, arXiv:1611.08466, doi:10.1007/s00222-017-0752-2, MR 3748313

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