mathematics

V-topology

V-topology 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 V-topology rather than just read about it. In short: In mathematics, especially in algebraic geometry, the v-topology (also known as the universally subtrusive topology) is a Grothendieck topology whose covers are characterized by lifting maps from valuation rings. This topology was introduced by Rydh (2010) and studied further by Bhatt & Scholze (2017), who introduced the name v-topology, where v stands for valuation.

Key takeaways

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

Reference excerpt

In mathematics, especially in algebraic geometry, the v-topology (also known as the universally subtrusive topology) is a Grothendieck topology whose covers are characterized by lifting maps from valuation rings. This topology was introduced by Rydh (2010) and studied further by Bhatt & Scholze (2017), who introduced the name v-topology, where v stands for valuation.

Definition A universally subtrusive map is a map f: X → Y of quasi-compact, quasi-separated schemes such that for any map v: Spec (V) → Y, where V is a valuation ring, there is an extension (of valuation rings) V ⊂ W {\displaystyle V\subset W} and a map Spec W → X lifting v.

Examples Examples of v-covers include faithfully flat maps, proper surjective maps. In particular, any Zariski covering is a v-covering. Moreover, universal homeomorphisms, such as X r e d → X {\displaystyle X_{red}\to X} , the normalisation of the cusp, and the Frobenius in positive characteristic are v-coverings. In fact, the perfection X p e r f → X {\displaystyle X_{perf}\to X} of a scheme is a v-covering.

Voevodsky's h topology See h-topology, relation to the v-topology

Arc topology Bhatt & Mathew (2018) have introduced the arc-topology, which is similar in its definition, except that only valuation rings of rank ≤ 1 are considered in the definition. A variant of this topology, with an analogous relationship that the h-topology has with the cdh topology, called the cdarc-topology was later introduced by Elmanto, Hoyois, Iwasa and Kelly (2020). Bhatt & Scholze (2019, §8) show that the Amitsur complex of an arc covering of perfect rings is an exact complex.

See also List of topologies on the category of schemes

References

Bhatt, Bhargav; Mathew, Akhil (2018), The arc-topology, arXiv:1807.04725v2 Bhatt, Bhargav; Scholze, Peter (2017), "Projectivity of the Witt vector affine Grassmannian", Inventiones Mathematicae, 209 (2): 329–423, arXiv:1507.06490, Bibcode:2017InMat.209..329B, doi:10.1007/s00222-016-0710-4, MR 3674218, S2CID 119123398 Bhatt, Bhargav; Scholze, Peter (2022). "Prisms and prismatic cohomology". Annals of Mathematics. 196 (3): 1135–1275. doi:10.4007/annals.2022.196.3.5. MR 4502597. Zbl 1552.14012. Rydh, David (2010), "Submersions and effective descent of étale morphisms", Bull. Soc. Math. France, 138 (2): 181–230, arXiv:0710.2488, doi:10.24033/bsmf.2588, MR 2679038, S2CID 17484591 Voevodsky, Vladimir (1996), "Homology of schemes", Selecta Mathematica, New Series, 2 (1): 111–153, doi:10.1007/BF01587941, MR 1403354, S2CID 9620683

Worked examples

Example 1 — a first encounter with V-topology

Start with the simplest possible case. Write down what V-topology 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 V-topology 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 V-topology 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 V-topology

In research: V-topology 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 V-topology 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: V-topology is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic geometry, so understanding it makes those chapters shorter.
In everyday life: Look for V-topology 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 V-topology in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what V-topology 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 V-topology out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is V-topology in simple terms?

Why does V-topology 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 V-topology?

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 V-topology.