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Ind-scheme

Ind-scheme 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 Ind-scheme rather than just read about it. In short: In algebraic geometry, an ind-scheme is a set-valued functor that can be written (represented) as a direct limit (i.e., inductive limit) of closed embedding of schemes. Examples C P ∞ = lim → ⁡ C P N {\displaystyle \mathbb {C} P^{\infty }=\varinjlim \mathbb {C} P^{N}} is an ind-scheme.

Key takeaways

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

Reference excerpt

In algebraic geometry, an ind-scheme is a set-valued functor that can be written (represented) as a direct limit (i.e., inductive limit) of closed embedding of schemes.

Examples

C P ∞ = lim → ⁡ C P N {\displaystyle \mathbb {C} P^{\infty }=\varinjlim \mathbb {C} P^{N}} is an ind-scheme. Perhaps the most famous example of an ind-scheme is an infinite grassmannian (which is a quotient of the loop group of an algebraic group G.)

See also formal scheme

References A. Beilinson, Vladimir Drinfel'd, Quantization of Hitchin’s integrable system and Hecke eigensheaves on Hitchin system, preliminary version [1] Archived 2015-01-05 at the Wayback Machine V.Drinfeld, Infinite-dimensional vector bundles in algebraic geometry, notes of the talk at the `Unity of Mathematics' conference. Expanded version http://ncatlab.org/nlab/show/ind-scheme

Worked examples

Example 1 — a first encounter with Ind-scheme

Start with the simplest possible case. Write down what Ind-scheme 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 Ind-scheme 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 Ind-scheme 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 Ind-scheme

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

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Ind-scheme means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Ind-scheme out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Ind-scheme in simple terms?

In algebraic geometry, an ind-scheme is a set-valued functor that can be written (represented) as a direct limit (i.e., inductive limit) of closed embedding of schemes. Examples C P ∞ = lim → ⁡ C P N {\displaystyle \mathbb {C} P^{\infty }=\varinjlim \mathbb {C} P^{N}} is an ind-scheme.

Why does Ind-scheme 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 Ind-scheme?

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 Ind-scheme.

Tags

  • Algebraic geometry
  • Algebraic geometry stubs

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