Lazard's universal ring

Key takeaways

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

Reference excerpt

In mathematics, Lazard's universal ring is a ring introduced by Michel Lazard in Lazard (1955) over which the universal commutative one-dimensional formal group law is defined. There is a universal commutative one-dimensional formal group law over a universal commutative ring defined as follows. We let

F ( x , y ) {\displaystyle F(x,y)}

be

x + y + ∑ i , j c i , j x i y j {\displaystyle x+y+\sum _{i,j}c_{i,j}x^{i}y^{j}}

for indeterminates c i , j {\displaystyle c_{i,j}} , and we define the universal ring R to be the commutative ring generated by the elements c i , j {\displaystyle c_{i,j}} , with the relations that are forced by the associativity and commutativity laws for formal group laws. More or less by definition, the ring R has the following universal property:

For every commutative ring S, one-dimensional formal group laws over S correspond to ring homomorphisms from R to S. The commutative ring R constructed above is known as Lazard's universal ring. At first sight it seems to be incredibly complicated: the relations between its generators are very messy. However Lazard proved that it has a very simple structure: it is just a polynomial ring (over the integers) on generators of degree 1, 2, 3, ..., where c i , j {\displaystyle c_{i,j}} has degree ( i + j − 1 ) {\displaystyle (i+j-1)} . Daniel Quillen (1969) proved that the coefficient ring of complex cobordism is naturally isomorphic as a graded ring to Lazard's universal ring. Hence, topologists commonly regrade the Lazard ring so that c i , j {\displaystyle c_{i,j}} has degree 2 ( i + j − 1 ) {\displaystyle 2(i+j-1)} , because the coefficient ring of complex cobordism is evenly graded.

References Adams, J. Frank (1974), Stable homotopy and generalised homology, University of Chicago Press, ISBN 978-0-226-00524-9 Lazard, Michel (1955), "Sur les groupes de Lie formels à un paramètre", Bulletin de la Société Mathématique de France, 83: 251–274, doi:10.24033/bsmf.1462, MR 0073925 Lazard, Michel (1975), Commutative formal groups, Lecture Notes in Mathematics, vol. 443, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0070554, ISBN 978-3-540-07145-7, MR 0393050 Quillen, Daniel (1969), "On the formal group laws of unoriented and complex cobordism theory", Bulletin of the American Mathematical Society, 75 (6): 1293–1298, doi:10.1090/S0002-9904-1969-12401-8, MR 0253350

Worked examples

Example 1 — a first encounter with Lazard's universal ring

Start with the simplest possible case. Write down what Lazard's universal ring 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 Lazard's universal ring 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 Lazard's universal ring 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 Lazard's universal ring

In research: Lazard's universal ring 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 Lazard's universal ring 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: Lazard's universal ring is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic groups, Algebraic number theory, Algebraic topology, so understanding it makes those chapters shorter.
In everyday life: Look for Lazard's universal ring 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.

How to study Lazard's universal ring in 20 minutes

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

Frequently asked questions

What is Lazard's universal ring in simple terms?

In mathematics, Lazard's universal ring is a ring introduced by Michel Lazard in Lazard (1955) over which the universal commutative one-dimensional formal group law is defined. There is a universal commutative one-dimensional formal group law over a universal commutative ring defined as follows.

Why does Lazard's universal ring 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 Lazard's universal ring?

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 Lazard's universal ring.