mathematics

Schlessinger's theorem

Schlessinger's 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 Schlessinger's theorem rather than just read about it. In short: In algebra, Schlessinger's theorem is a theorem in deformation theory introduced by Schlessinger (1968) that gives conditions for a functor of artinian local rings to be pro-representable, refining an earlier theorem of Grothendieck. Definitions Λ is a complete Noetherian local ring with residue field k, and C is the category of local Artinian Λ-algebras (meaning in particular that as modules over Λ they are finitel…

Key takeaways

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

Reference excerpt

In algebra, Schlessinger's theorem is a theorem in deformation theory introduced by Schlessinger (1968) that gives conditions for a functor of artinian local rings to be pro-representable, refining an earlier theorem of Grothendieck.

Definitions Λ is a complete Noetherian local ring with residue field k, and C is the category of local Artinian Λ-algebras (meaning in particular that as modules over Λ they are finitely generated and Artinian) with residue field k. A small extension in C is a morphism Y→Z in C that is surjective with kernel a 1-dimensional vector space over k. A functor is called representable if it is of the form hX where hX(Y)=hom(X,Y) for some X, and is called pro-representable if it is of the form Y→lim hom(Xi,Y) for a filtered direct limit over i in some filtered ordered set. A morphism of functors F→G from C to sets is called smooth if whenever Y→Z is an epimorphism of C, the map from F(Y) to F(Z)×G(Z)G(Y) is surjective. This definition is closely related to the notion of a formally smooth morphism of schemes. If in addition the map between the tangent spaces of F and G is an isomorphism, then F is called a hull of G.

Grothendieck's theorem Grothendieck (1960, proposition 3.1) showed that a functor from the category C of Artinian algebras to sets is pro-representable if and only if it preserves all finite limits. This condition is equivalent to asking that the functor preserves pullbacks and the final object. In fact Grothendieck's theorem applies not only to the category C of Artinian algebras, but to any category with finite limits whose objects are Artinian. By taking the projective limit of the pro-representable functor in the larger category of linearly topologized local rings, one obtains a complete linearly topologized local ring representing the functor.

Schlessinger's representation theorem One difficulty in applying Grothendieck's theorem is that it can be hard to check that a functor preserves all pullbacks. Schlessinger showed that it is sufficient to check that the functor preserves pullbacks of a special form, which is often easier to check. Schlessinger's theorem also gives conditions under which the functor has a hull, even if it is not representable. Schessinger's theorem gives conditions for a set-valued functor F on C to be representable by a complete local Λ-algebra R with maximal ideal m such that R/mn is in C for all n. Schlessinger's theorem states that a functor from C to sets with F(k) a 1-element set is representable by a complete Noetherian local algebra if it has the following properties, and has a hull if it has the first three properties:

H1: The map F(Y×XZ)→F(Y)×F(X)F(Z) is surjective whenever Z→X is a small extension in C and Y→X is some morphism in C. H2: The map in H1 is a bijection whenever Z→X is the small extension k[x]/(x2)→k. H3: The tangent space of F is a finite-dimensional vector space over k. H4: The map in H1 is a bijection whenever Y=Z is a small extension of X and the maps from Y and Z to X are the same.

See also Formal moduli Artin's criterion

References Grothendieck (1960), Technique de descente et théorèmes d'existence en géométrie algébrique, II. Le théorème d'existence en théorie formelle des modules, Séminaire Bourbaki, vol. 12 Schlessinger, Michael (1968), "Functors of Artin rings", Transactions of the American Mathematical Society, 130: 208–222, doi:10.2307/1994967, ISSN 0002-9947, JSTOR 1994967, MR 0217093

Worked examples

Example 1 — a first encounter with Schlessinger's theorem

Start with the simplest possible case. Write down what Schlessinger's 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 Schlessinger's 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 Schlessinger's 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 Schlessinger's theorem

In research: Schlessinger's 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 Schlessinger's 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: Schlessinger's 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 Schlessinger's 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 Schlessinger's theorem in 20 minutes

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

Frequently asked questions

What is Schlessinger's theorem in simple terms?

Why does Schlessinger's 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 Schlessinger's 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 Schlessinger's theorem.