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 in full generality by Kerz, Strunk & Tamme (2018) using methods from derived algebraic geometry. Previously partial cases had been proven by Haesemeyer (2004), Cortiñas et al. (2008), Geisser & Hesselholt (2010), Cisinski (2013), Kelly (2014), and Morrow (2016).

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:

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

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

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

Generalization

Recently, Kelly, Saito & Tamme (2024) 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

• Weibel, Chuck (1980), "K-theory and analytic isomorphisms", Invent. Math., 61 (2): 177–197, Bibcode:1980InMat..61..177W, 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 , Bibcode:2018InMat.211..523K, doi:10.1007/s00222-017-0752-2, MR   3748313

• Cortiñas, Guillermo; Haesemeyer, Christian; Schlichting, Marco; Weibel, Charles (2008). "Cyclic homology, cdh-cohomology and negative K-theory". Annals of Mathematics. 167 (2): 549–573. arXiv: math/0502255 . doi:10.4007/annals.2008.167.549. JSTOR   40345438.

• Cisinski, Denis-Charles (2013). "Descente par éclatements en K-théorie invariante par homotopie". Annals of Mathematics. 177 (2): 425–448. arXiv: 1003.1487 . doi:10.4007/annals.2013.177.2.2. JSTOR   23496531.

• Kelly, Shane (2014). "Vanishing of negative K-theory in positive characteristic". Compositio Mathematica. 150 (8). London Mathematical Society: 1425–1434. doi:10.1112/S0010437X14007472 (inactive 1 July 2025).

• Morrow, Matthew (2016). "Pro cdh-descent for cyclic homology and K-theory". Journal of the Institute of Mathematics of Jussieu. 15 (3). Cambridge University Press: 539–567. doi:10.1017/S1474748014000049 (inactive 1 July 2025).

• Geisser, Thomas; Hesselholt, Lars (2010). "On the vanishing of negative K-groups". Mathematische Annalen. 348 (3). Springer: 707–736. doi:10.1007/s00208-009-0413-1 (inactive 1 July 2025).

• Haesemeyer, Christian (2004). "Descent properties of homotopy K-theory". Duke Mathematical Journal. 125 (3): 589–620. doi:10.1215/S0012-7094-04-12534-5.

• Kelly, Shane; Saito, Shuji; Tamme, Georg (2024). "On pro-cdh descent on derived schemes". arXiv: 2407.04378 [math.KT].