mathematics

Ramanujam–Samuel theorem

Ramanujam–Samuel theorem 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 Ramanujam–Samuel theorem rather than just read about it. In short: In algebraic geometry, the Ramanujam–Samuel theorem gives conditions for a divisor of a local ring to be principal. It was introduced independently by Samuel (1962) in answer to a question of Grothendieck and by C.

Key takeaways

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

Reference excerpt

In algebraic geometry, the Ramanujam–Samuel theorem gives conditions for a divisor of a local ring to be principal. It was introduced independently by Samuel (1962) in answer to a question of Grothendieck and by C. P. Ramanujam in an appendix to a paper by Seshadri (1963), and was generalized by Grothendieck (1967, Theorem 21.14.1).

Statement Grothendieck's version of the Ramanujam–Samuel theorem (Grothendieck & Dieudonné 1967, theorem 21.14.1) is as follows. Suppose that A is a local Noetherian ring with maximal ideal m, whose completion is integral and integrally closed, and ρ is a local homomorphism from A to a local Noetherian ring B of larger dimension such that B is formally smooth over A and the residue field of B is finite over that of A. Then a cycle of codimension 1 in Spec(B) that is principal at the point mB is principal.

References Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS. 32: 5–361. doi:10.1007/bf02732123. MR 0238860. Samuel, Pierre (1962), "Sur une conjecture de Grothendieck", Les Comptes rendus de l'Académie des sciences, 255: 3101–3103, MR 0154887 Seshadri, C. S. (1963), "Quotient space by an abelian variety", Mathematische Annalen, 152: 185–194, doi:10.1007/BF01470879, ISSN 0025-5831, MR 0164973

Worked examples

Example 1 — a first encounter with Ramanujam–Samuel theorem

Start with the simplest possible case. Write down what Ramanujam–Samuel theorem 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 Ramanujam–Samuel theorem 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 Ramanujam–Samuel theorem 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 Ramanujam–Samuel theorem

In research: Ramanujam–Samuel theorem 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 Ramanujam–Samuel theorem 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: Ramanujam–Samuel theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Theorems in algebraic geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Ramanujam–Samuel theorem 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 Ramanujam–Samuel theorem in 20 minutes

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

Frequently asked questions

What is Ramanujam–Samuel theorem in simple terms?

Why does Ramanujam–Samuel theorem 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 Ramanujam–Samuel theorem?

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 Ramanujam–Samuel theorem.