View source for Weibel's conjecture

In mathematics, '''Weibel's conjecture''' gives a criterion for vanishing of negative [[Algebraic K-theory|algebraic K-theory]] groups. The conjecture was proposed by {{harvs|txt|first=Charles|last=Weibel|year=1980}} and proven in full generality by {{harvtxt|Kerz|Strunk|Tamme|2018}} using methods from [[Derived algebraic geometry|derived algebraic geometry]]. Previously partial cases had been proven by 
{{harvtxt|Morrow|2016}}, 
{{harvtxt|Kelly|2014}}, 
{{harvtxt|Cisinski|2013}}, 
{{harvtxt|Geisser|Hesselholt|2010}}, and 
{{harvtxt|Cortiñas|Haesemeyer|Schlichting|Weibel|2008}}.

==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 <&nbsp;&minus;''d'':

: <math> K_i(X) = 0 \text{ for } i<-d </math>

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

: <math> K_i(X) = K_i(X\times \mathbb A^r) \text{ for } i\le -d \text{ and arbitrary } r. </math>

==References==

* {{citation|first=Charles|last=Weibel|title=K-theory and analytic isomorphisms|journal=[[Inventiones Mathematicae]]|volume=61|year=1980|issue=2|pages=177–197|doi=10.1007/bf01390120}}

*{{citation|first1=Moritz|last1=Kerz|first2=Florian|last2=Strunk|
first3=Georg|last3=Tamme|title=Algebraic ''K''-theory and descent for blow-ups|journal=[[Inventiones Mathematicae]]|volume=211| year=2018|issue=2|pages=523–577|mr=3748313|doi=10.1007/s00222-017-0752-2|arxiv=1611.08466}}

[[Category:Algebraic geometry]]
[[Category:K-theory]]


{{Sourceattribution|Weibel's conjecture}}