Weibel's conjecture

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 & Tamme (2018) using methods from derived algebraic geometry. Previously partial cases had been proven by Morrow (2016) harvtxt error: no target: CITEREFMorrow2016 (help), Kelly (2014) harvtxt error: no target: CITEREFKelly2014 (help), Cisinski (2013) harvtxt error: no target: CITEREFCisinski2013 (help), Geisser & Hesselholt (2010) harvtxt error: no target: CITEREFGeisserHesselholt2010 (help), and Cortiñas et al. (2008) harvtxt error: no target: CITEREFCortiñasHaesemeyerSchlichtingWeibel2008 (help).

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 {\displaystyle 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 × A r )  for  i ≤ − d  and arbitrary  r . {\displaystyle K_{i}(X)=K_{i}(X\times \mathbb {A} ^{r}){\text{ for }}i\leq -d{\text{ and arbitrary }}r.}

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, arXiv:1611.08466, doi:10.1007/s00222-017-0752-2, MR 3748313
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