Stein factorization

Key takeaways

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

Reference excerpt

In algebraic geometry, the Stein factorization, introduced by Karl Stein (1956) for the case of complex spaces, states that a proper morphism of schemes can be factorized as a composition of a finite mapping and a proper morphism with connected fibers. Roughly speaking, Stein factorization contracts the connected components of the fibers of a mapping to points.

Statement One version for schemes states the following: (EGA, III.4.3.1)

Let X be a scheme, S a locally noetherian scheme and f : X → S {\displaystyle f:X\to S} a proper morphism. Then one can write

f = g ∘ f ′ {\displaystyle f=g\circ f'}

where g : S ′ → S {\displaystyle g\colon S'\to S} is a finite morphism and f ′ : X → S ′ {\displaystyle f'\colon X\to S'} is a proper morphism so that f ∗ ′ O X = O S ′ . {\displaystyle f'_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{S'}.}

The existence of this decomposition itself is not difficult. See below. But, by Zariski's connectedness theorem, the last part in the above says that the fiber f ′ − 1 ( s ) {\displaystyle f'^{-1}(s)} is connected for any s ∈ S ′ {\displaystyle s\in S'} . It follows: Corollary: For any s ∈ S {\displaystyle s\in S} , the set of connected components of the fiber f − 1 ( s ) {\displaystyle f^{-1}(s)} is in bijection with the set of points in the fiber g − 1 ( s ) {\displaystyle g^{-1}(s)} .

Proof Set:

S ′ = Spec S ⁡ f ∗ O X {\displaystyle S'=\operatorname {Spec} _{S}f_{*}{\mathcal {O}}_{X}}

where SpecS is the relative Spec. The construction gives the natural map g : S ′ → S {\displaystyle g\colon S'\to S} , which is finite since O X {\displaystyle {\mathcal {O}}_{X}} is coherent and f is proper. The morphism f factors through g and one gets f ′ : X → S ′ {\displaystyle f'\colon X\to S'} , which is proper. By construction, f ∗ ′ O X = O S ′ {\displaystyle f'_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{S'}} . One then uses the theorem on formal functions to show that the last equality implies f ′ {\displaystyle f'} has connected fibers. (This part is sometimes referred to as Zariski's connectedness theorem.)

Worked examples

Example 1 — a first encounter with Stein factorization

Start with the simplest possible case. Write down what Stein factorization 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 Stein factorization 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 Stein factorization 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 Stein factorization

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

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

Frequently asked questions

What is Stein factorization in simple terms?

In algebraic geometry, the Stein factorization, introduced by Karl Stein (1956) for the case of complex spaces, states that a proper morphism of schemes can be factorized as a composition of a finite mapping and a proper morphism with connected fibers. Roughly speaking, Stein factorization contract…

Why does Stein factorization 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 Stein factorization?

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 Stein factorization.