mathematics

Landweber exact functor theorem

Landweber exact functor 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 Landweber exact functor theorem rather than just read about it. In short: In mathematics, the Landweber exact functor theorem, named after Peter Landweber, is a theorem in algebraic topology. It is known that a complex orientation of a homology theory leads to a formal group law.

Key takeaways

Landweber exact functor 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 Landweber exact functor theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Landweber exact functor theorem from memory before moving on to harder problems.

Reference excerpt

In mathematics, the Landweber exact functor theorem, named after Peter Landweber, is a theorem in algebraic topology. It is known that a complex orientation of a homology theory leads to a formal group law. The Landweber exact functor theorem (or LEFT for short) can be seen as a method to reverse this process: it constructs a homology theory out of a formal group law.

Statement The coefficient ring of complex cobordism is M U ∗ ( ∗ ) = M U ∗ ≅ Z [ x 1 , x 2 , … ] {\displaystyle MU_{*}(*)=MU_{*}\cong \mathbb {Z} [x_{1},x_{2},\dots ]} , where the degree of x i {\displaystyle x_{i}} is 2 i {\displaystyle 2i} . This is isomorphic to the graded Lazard ring

L ∗ {\displaystyle {\mathcal {}}L_{*}} . This means that giving a formal group law F (of degree − 2 {\displaystyle -2} ) over a graded ring R ∗ {\displaystyle R_{*}} is equivalent to giving a graded ring morphism L ∗ → R ∗ {\displaystyle L_{*}\to R_{*}} . Multiplication by an integer n > 0 {\displaystyle n>0} is defined inductively as a power series, by

[ n + 1 ] F x = F ( x , [ n ] F x ) {\displaystyle [n+1]^{F}x=F(x,[n]^{F}x)} and [ 1 ] F x = x . {\displaystyle [1]^{F}x=x.}

Let now F be a formal group law over a ring

R ∗ {\displaystyle {\mathcal {}}R_{*}} . Define for a topological space X

E ∗ ( X ) = M U ∗ ( X ) ⊗ M U ∗ R ∗ {\displaystyle E_{*}(X)=MU_{*}(X)\otimes _{MU_{*}}R_{*}}

Here R ∗ {\displaystyle R_{*}} gets its M U ∗ {\displaystyle MU_{*}} -algebra structure via F. The question is: is E a homology theory? It is obviously a homotopy invariant functor, which fulfills excision. The problem is that tensoring in general does not preserve exact sequences. One could demand that R ∗ {\displaystyle R_{*}} be flat over M U ∗ {\displaystyle MU_{*}} , but that would be too strong in practice. Peter Landweber found another criterion:

Theorem (Landweber exact functor theorem) For every prime p, there are elements v 1 , v 2 , ⋯ ∈ M U ∗ {\displaystyle v_{1},v_{2},\dots \in MU_{*}} such that we have the following: Suppose that M ∗ {\displaystyle M_{*}} is a graded M U ∗ {\displaystyle MU_{*}} -module and the sequence ( p , v 1 , v 2 , … , v n ) {\displaystyle (p,v_{1},v_{2},\dots ,v_{n})} is regular for M {\displaystyle M} , for every p and n. Then

E ∗ ( X ) = M U ∗ ( X ) ⊗ M U ∗ M ∗ {\displaystyle E_{*}(X)=MU_{*}(X)\otimes _{MU_{*}}M_{*}}

is a homology theory on CW-complexes. In particular, every formal group law F over a ring R {\displaystyle R} yields a module over

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Landweber exact functor theorem

Start with the simplest possible case. Write down what Landweber exact functor 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 Landweber exact functor 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 Landweber exact functor 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 Landweber exact functor theorem

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

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

Frequently asked questions

What is Landweber exact functor theorem in simple terms?

Why does Landweber exact functor 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 Landweber exact functor 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 Landweber exact functor theorem.