mathematics

Nine lemma

Nine lemma 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 Nine lemma rather than just read about it. In short: In mathematics, the nine lemma (or 3×3 lemma) is a statement about commutative diagrams and exact sequences valid in the category of groups and any abelian category. Consider the commutative diagram to the right.

Key takeaways

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

Reference excerpt

In mathematics, the nine lemma (or 3×3 lemma) is a statement about commutative diagrams and exact sequences valid in the category of groups and any abelian category. Consider the commutative diagram to the right. We have 6 statements:

If all columns as well as the two bottom rows are exact, then the top row is exact. If all columns as well as the two top rows are exact, then the bottom row is exact. If all columns as well as the top and bottom rows are exact, and A 2 → C 2 {\displaystyle A_{2}\to C_{2}} is the zero morphism, then the middle row is exact. By symmetry, exchanging the words "row" and "column" gives 3 more true statements. The nine lemma can be proved by direct diagram chasing, or by applying the snake lemma (to the two bottom rows in the first case, and to the two top rows in the second case).

Variants The sharp nine lemma is slightly stronger. Define a sequence 0 → A → B → C → 0 {\displaystyle 0\to A\to B\to C\to 0} "left exact" iff 0 → A → B → C {\displaystyle 0\to A\to B\to C} is exact. Then:

If all columns as well as the two bottom rows are left exact, then the top row is left exact. If all columns as well as the two bottom rows are left exact, and the first column and the middle row are short exact, then the top row is exact. In Mathematics Made Difficult, Linderholm offers a satirical view of the nine lemma:

Draw a noughts-and-crosses board... Do not fill it in with noughts and crosses... Instead, use curved arrows... Wave your hands about in complicated patterns over this board. Make some noughts, but not in the squares; put them at both ends of the horizontal and vertical lines. Make faces. You have now proved: (a) the Nine Lemma (b) the Sixteen Lemma (c) the Twenty-five Lemma...

in which only (a) is a widely recognized theorem in homological algebra.

References

Illustrations

Worked examples

Example 1 — a first encounter with Nine lemma

Start with the simplest possible case. Write down what Nine lemma 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 Nine lemma 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 Nine lemma 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 Nine lemma

In research: Nine lemma 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 Nine lemma 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: Nine lemma is common in secondary-school and first-year university syllabi. It links to neighbouring topics Homological algebra, Lemmas in category theory, so understanding it makes those chapters shorter.
In everyday life: Look for Nine lemma 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.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Nine lemma in 20 minutes

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

Frequently asked questions

What is Nine lemma in simple terms?

Why does Nine lemma 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 Nine lemma?

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 Nine lemma.