Weibel's conjecture
From HandWiki
In mathematics, Weibel's conjecture gives a criterion for vanishing of negative algebraic K-theory groups. The conjecture was proposed by Charles Weibel (1980) and proven in full generality by (Kerz Strunk) using methods from derived algebraic geometry. Previously partial cases had been proven by (Morrow 2016), (Kelly 2014), (Cisinski 2013), (Geisser Hesselholt), and (Cortiñas Haesemeyer).
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:
- [math]\displaystyle{ K_i(X) = 0 \text{ for } i\lt -d }[/math]
and asserts moreover a homotopy invariance property for negative K-groups
- [math]\displaystyle{ K_i(X) = K_i(X\times \mathbb A^r) \text{ for } i\le -d \text{ and arbitrary } r. }[/math]
References
- Weibel, Charles (1980), "K-theory and analytic isomorphisms", Inventiones Mathematicae 61 (2): 177–197, doi:10.1007/bf01390120
- Kerz, Moritz; Strunk, Florian; Tamme, Georg (2018), "Algebraic K-theory and descent for blow-ups", Inventiones Mathematicae 211 (2): 523–577, doi:10.1007/s00222-017-0752-2
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