Weibel's conjecture
In mathematics, Weibel's conjecture gives a criterion for vanishing of negative algebraic K-theory groups. The conjecture was proposed by {{harvtxt|Weibel|1980}} and proven in full generality by {{harvtxt|Kerz|Strunk|Tamme|2018}} using methods from derived algebraic geometry. Previously partial cases had been proven by
{{harvtxt|Haesemeyer|2004}},
{{harvtxt|Cortiñas|Haesemeyer|Schlichting|Weibel|2008}},
{{harvtxt|Geisser|Hesselholt|2010}},
{{harvtxt|Cisinski|2013}},
{{harvtxt|Kelly|2014}}, and
{{harvtxt|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:
:
and asserts moreover a homotopy invariance property for negative K-groups
:
Generalization
Recently, {{harvtxt|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
- {{citation|first=Chuck|last=Weibel|title=K-theory and analytic isomorphisms|journal=Invent. Math.|volume=61|year=1980|issue=2|pages=177–197|doi=10.1007/bf01390120|bibcode=1980InMat..61..177W }}
- {{citation|first1=Moritz|last1=Kerz|first2=Florian|last2=Strunk|first3=Georg|last3=Tamme|title=Algebraic K-theory and descent for blow-ups|journal=Invent. Math.|volume=211|year=2018|issue=2|pages=523–577|mr=3748313|doi=10.1007/s00222-017-0752-2|arxiv=1611.08466|bibcode=2018InMat.211..523K }}
- {{Cite journal
| last1 = Cortiñas
| first1 = Guillermo
| last2 = Haesemeyer
| first2 = Christian
| last3 = Schlichting
| first3 = Marco
| last4 = Weibel
| first4 = Charles
| title = Cyclic homology, cdh-cohomology and negative K-theory
| journal = Annals of Mathematics
| volume = 167
| issue = 2
| pages = 549–573
| year = 2008
| doi = 10.4007/annals.2008.167.549
| jstor = 40345438
| arxiv = math/0502255
}}
- {{Cite journal
| last = Cisinski
| first = Denis-Charles
| title = Descente par éclatements en K-théorie invariante par homotopie
| journal = Annals of Mathematics
| volume = 177
| issue = 2
| pages = 425–448
| year = 2013
| doi = 10.4007/annals.2013.177.2.2
| jstor = 23496531
| arxiv = 1003.1487
}}
- {{Cite journal
| last = Kelly
| first = Shane
| title = Vanishing of negative K-theory in positive characteristic
| journal = Compositio Mathematica
| volume = 150
| issue = 8
| pages = 1425–1434
| year = 2014
| publisher = London Mathematical Society
| doi = 10.1112/S0010437X14007472
| doi-broken-date = 5 May 2025
}}
- {{Cite journal
| last = Morrow
| first = Matthew
| title = Pro cdh-descent for cyclic homology and K-theory
| journal = Journal of the Institute of Mathematics of Jussieu
| volume = 15
| issue = 3
| pages = 539–567
| year = 2016
| publisher = Cambridge University Press
| doi = 10.1017/S1474748014000049
| doi-broken-date = 5 May 2025
}}
- {{Cite journal
| last1 = Geisser
| first1 = Thomas
| last2 = Hesselholt
| first2 = Lars
| title = On the vanishing of negative K-groups
| journal = Mathematische Annalen
| volume = 348
| issue = 3
| pages = 707–736
| year = 2010
| publisher = Springer
| doi = 10.1007/s00208-009-0413-1
| doi-broken-date = 5 May 2025
}}
- {{Cite journal
| last1 = Haesemeyer
| first1 = Christian
| title = Descent properties of homotopy K-theory
| journal = Duke Mathematical Journal
| volume = 125
| issue = 3
| pages = 589–620
| year = 2004
| doi = 10.1215/S0012-7094-04-12534-5
| url = https://projecteuclid.org/journals/duke-mathematical-journal/volume-125/issue-3/Descent-Properties-of-Homotopy-K-Theory/10.1215/S0012-7094-04-12534-5.short
}}
- {{Cite arXiv
|eprint=2407.04378
|last1=Kelly
|first1=Shane
|last2=Saito
|first2=Shuji
|last3=Tamme
|first3=Georg
|title=On pro-cdh descent on derived schemes
|date=2024
|class=math.KT
}}
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