Regular embedding

Key takeaways

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

Reference excerpt

In algebraic geometry, a closed immersion i : X ↪ Y {\displaystyle i:X\hookrightarrow Y} of schemes is a regular embedding of codimension r if each point x in X has an open affine neighborhood U in Y such that the ideal of X ∩ U {\displaystyle X\cap U} is generated by a regular sequence of length r. A regular embedding of codimension one is precisely an effective Cartier divisor.

Examples and usage For example, if X and Y are smooth over a scheme S and if i is an S-morphism, then i is a regular embedding. In particular, every section of a smooth morphism is a regular embedding. If Spec ⁡ B {\displaystyle \operatorname {Spec} B} is regularly embedded into a regular scheme, then B is a complete intersection ring. The notion is used, for instance, in an essential way in Fulton's approach to intersection theory. The important fact is that when i is a regular embedding, if I is the ideal sheaf of X in Y, then the normal sheaf, the dual of I / I 2 {\displaystyle I/I^{2}} , is locally free (thus a vector bundle) and the natural map Sym ⁡ ( I / I 2 ) → ⊕ 0 ∞ I n / I n + 1 {\displaystyle \operatorname {Sym} (I/I^{2})\to \oplus _{0}^{\infty }I^{n}/I^{n+1}} is an isomorphism: the normal cone Spec ⁡ ( ⊕ 0 ∞ I n / I n + 1 ) {\displaystyle \operatorname {Spec} (\oplus _{0}^{\infty }I^{n}/I^{n+1})} coincides with the normal bundle.

Non-examples One non-example is a scheme which isn't equidimensional. For example, the scheme

X = Spec ( C [ x , y , z ] ( x z , y z ) ) {\displaystyle X={\text{Spec}}\left({\frac {\mathbb {C} [x,y,z]}{(xz,yz)}}\right)}

is the union of A 2 {\displaystyle \mathbb {A} ^{2}} and A 1 {\displaystyle \mathbb {A} ^{1}} . Then, the embedding X ↪ A 3 {\displaystyle X\hookrightarrow \mathbb {A} ^{3}} isn't regular since taking any non-origin point on the z {\displaystyle z} -axis is of dimension 1 {\displaystyle 1} while any non-origin point on the x y {\displaystyle xy} -plane is of dimension 2 {\displaystyle 2} .

Local complete intersection morphisms and virtual tangent bundles A morphism of finite type f : X → Y {\displaystyle f:X\to Y} is called a (local) complete intersection morphism if each point x in X has an open affine neighborhood U so that f |U factors as U → j V → g Y {\displaystyle U{\overset {j}{\to }}V{\overset {g}{\to }}Y} where j is a regular embedding and g is smooth. For example, if f is a morphism between smooth varieties, then f factors as X → X × Y → Y {\displaystyle X\to X\times Y\to Y} where the first map is the graph morphism and so is a complete intersection morphism. Notice that this definition is compatible with the one in EGA IV for the special case of flat morphisms. Let f : X → Y {\displaystyle f:X\to Y} be a local-complete-intersection morphism that admits a global factorization: it is a composition X ↪ i P → p Y {\displaystyle X{\overset {i}{\hookrightarrow }}P{\overset {p}{\to }}Y} where i {\displaystyle i} is a regular embedding and p {\displaystyle p} a smooth morphism. Then the virtual tangent bundle is an element of the Grothendieck group of vector bundles on X given as:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Regular embedding

Start with the simplest possible case. Write down what Regular embedding 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 Regular embedding 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 Regular embedding 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 Regular embedding

In research: Regular embedding 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 Regular embedding 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: Regular embedding is common in secondary-school and first-year university syllabi. It links to neighbouring topics Morphisms of schemes, Theorems in algebraic geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Regular embedding 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 Regular embedding in 20 minutes

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

Frequently asked questions

What is Regular embedding in simple terms?

In algebraic geometry, a closed immersion i : X ↪ Y {\displaystyle i:X\hookrightarrow Y} of schemes is a regular embedding of codimension r if each point x in X has an open affine neighborhood U in Y such that the ideal of X ∩ U {\displaystyle X\cap U} is generated by a regular sequence of length r…

Why does Regular embedding 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 Regular embedding?

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 Regular embedding.