mathematics

Weibel's conjecture

Weibel's conjecture is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Weibel's conjecture rather than just read about it. In short: 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.

Key takeaways

Weibel's conjecture belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Weibel's conjecture to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Weibel's conjecture from memory before moving on to harder problems.

Reference excerpt

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:

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.}

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, Charles A. (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).{{cite journal}}: CS1 maint: DOI inactive as of July 2025 (link) 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).{{cite journal}}: CS1 maint: DOI inactive as of July 2025 (link) 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).{{cite journal}}: CS1 maint: DOI inactive as of July 2025 (link) 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].

Worked examples

Example 1 — a first encounter with Weibel's conjecture

Start with the simplest possible case. Write down what Weibel's conjecture claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Weibel's conjecture before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Weibel's conjecture ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Weibel's conjecture

In research: Weibel's conjecture appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Weibel's conjecture in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Weibel's conjecture is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic geometry, Algebraic geometry stubs, K-theory, so understanding it makes those chapters shorter.
In everyday life: Look for Weibel's conjecture outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Weibel's conjecture in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Weibel's conjecture means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Weibel's conjecture out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Weibel's conjecture in simple terms?

Why does Weibel's conjecture matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Weibel's conjecture?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Weibel's conjecture.