mathematics

Quasi-separated morphism

Quasi-separated morphism 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 Quasi-separated morphism rather than just read about it. In short: In algebraic geometry, a morphism of schemes f from X to Y is called quasi-separated if the diagonal map from X to X ×Y X is quasi-compact (meaning that the inverse image of any quasi-compact open set is quasi-compact). A scheme X is called quasi-separated if the morphism to Spec Z is quasi-separated.

Key takeaways

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

Reference excerpt

In algebraic geometry, a morphism of schemes f from X to Y is called quasi-separated if the diagonal map from X to X ×Y X is quasi-compact (meaning that the inverse image of any quasi-compact open set is quasi-compact). A scheme X is called quasi-separated if the morphism to Spec Z is quasi-separated. Quasi-separated algebraic spaces and algebraic stacks and morphisms between them are defined in a similar way, though some authors include the condition that X is quasi-separated as part of the definition of an algebraic space or algebraic stack X. Quasi-separated morphisms were introduced by Grothendieck & Dieudonné (1964, 1.2.1) as a generalization of separated morphisms, which require the diagonal map to be a closed immersion. All separated morphisms (and all morphisms of Noetherian schemes) are automatically quasi-separated. Quasi-separated morphisms are important for algebraic spaces and algebraic stacks, where many natural morphisms are quasi-separated but not separated. The condition that a morphism is quasi-separated often occurs together with the condition that it is quasi-compact.

Topological description We say a topological space X is quasi-separated if the intersection of two open quasi-compact subsets of X is quasi-compact. We say that a continuous map of topological spaces f from X to Y is quasi-separated if the inverse image along f of every open quasi-separated subset of Y is quasi-separated. Then a scheme (resp., a morphism of schemes) is quasi-separated in the scheme-theoretic sense if and only if it is quasi-separated in the topological sense, see Grothendieck & Dieudonné (1964, 1.2.6, 1.2.7).

Examples If X is a locally Noetherian scheme then any morphism from X to any scheme is quasi-separated, and in particular X is a quasi-separated scheme. Any separated scheme or morphism is quasi-separated. The line with two origins over a field is quasi-separated over the field but not separated. If X is an "infinite dimensional vector space with two origins" over a field K then the morphism from X to spec K is not quasi-separated. More precisely X consists of two copies of Spec K[x1,x2,....] glued together by identifying the nonzero points in each copy. The quotient of an algebraic space by an infinite discrete group acting freely is often not quasi-separated. For example, if K is a field of characteristic 0 then the quotient of the affine line by the group Z of integers is an algebraic space that is not quasi-separated. This algebraic space is also an example of a group object in the category of algebraic spaces that is not a scheme; quasi-separated algebraic spaces that are group objects are always group schemes. There are similar examples given by taking the quotient of the group scheme Gm by an infinite subgroup, or the quotient of the complex numbers by a lattice.

References Grothendieck, Alexandre; Dieudonné, Jean (1964). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Première partie". Publications Mathématiques de l'IHÉS. 20. doi:10.1007/bf02684747. MR 0173675.

Worked examples

Example 1 — a first encounter with Quasi-separated morphism

Start with the simplest possible case. Write down what Quasi-separated morphism 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 Quasi-separated morphism 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 Quasi-separated morphism 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 Quasi-separated morphism

In research: Quasi-separated morphism 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 Quasi-separated morphism 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: Quasi-separated morphism 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 Quasi-separated morphism 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 Quasi-separated morphism in 20 minutes

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

Frequently asked questions

What is Quasi-separated morphism in simple terms?

Why does Quasi-separated morphism 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 Quasi-separated morphism?

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 Quasi-separated morphism.