mathematics

Zariski–Riemann space

Zariski–Riemann space 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 Zariski–Riemann space rather than just read about it. In short: In algebraic geometry, a Zariski–Riemann space or Zariski space of a subring k of a field K is a locally ringed space whose points are valuation rings containing k and contained in K. They generalize the Riemann surface of a complex curve.

Key takeaways

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

Reference excerpt

In algebraic geometry, a Zariski–Riemann space or Zariski space of a subring k of a field K is a locally ringed space whose points are valuation rings containing k and contained in K. They generalize the Riemann surface of a complex curve. Zariski–Riemann spaces were introduced by Zariski (1940, 1944) who (rather confusingly) called them Riemann manifolds or Riemann surfaces. They were named Zariski–Riemann spaces after Oscar Zariski and Bernhard Riemann by Nagata (1962) who used them to show that algebraic varieties can be embedded in complete ones. Local uniformization (proved in characteristic 0 by Zariski) can be interpreted as saying that the Zariski–Riemann space of a variety is nonsingular in some sense, so is a sort of rather weak resolution of singularities. This does not solve the problem of resolution of singularities because in dimensions greater than 1 the Zariski–Riemann space is not locally affine and in particular is not a scheme.

Definition The Zariski–Riemann space of a field K over a base field k is a locally ringed space whose points are the valuation rings containing k and contained in K. Sometimes the valuation ring K itself is excluded, and sometimes the points are restricted to the zero-dimensional valuation rings (those whose residue field has transcendence degree zero over k). If S is the Zariski–Riemann space of a subring k of a field K, it has a topology defined by taking a basis of open sets to be the valuation rings containing a given finite subset of K. The space S is quasi-compact. It is made into a locally ringed space by assigning to any open subset the intersection of the valuation rings of the points of the subset. The local ring at any point is the corresponding valuation ring. The Zariski–Riemann space of a function field can also be constructed as the inverse limit of all complete (or projective) models of the function field.

Examples

The Zariski-Riemann space of a curve The Zariski-Riemann space of the function field K of a curve over k is the same as the nonsingular projective model of it. It has one generic non-closed point corresponding to the trivial valuation with valuation ring K, and its other points are the rank 1 valuation rings in K containing k. Unlike the higher-dimensional cases, the Zariski–Riemann space of a curve is a scheme.

The Zariski-Riemann space of a surface The valuation rings of a surface S over k with function field K can be classified by the dimension (the transcendence degree of the residue field) and the rank (the number of nonzero convex subgroups of the valuation group). Zariski (1939) gave the following classification:

Dimension 2. The only possibility is the trivial valuation with rank 0, valuation group 0 and valuation ring K. Dimension 1, rank 1. These correspond to divisors on some blowup of S, or in other words to divisors and infinitely near points of S. They are all discrete. The center in S can be either a point or a curve. The valuation group is Z. Dimension 0, rank 2. These correspond to germs of algebraic curves through a point on a normal model of S. The valuation group is isomorphic to Z+Z with the lexicographic order. Dimension 0, rank 1, discrete. These correspond to germs of non-algebraic curves (given for example by y= a non-algebraic formal power series in x) through a point of a normal model. The valuation group is Z. Dimension 0, rank 1, non-discrete, value group has incommensurable elements. These correspond to germs of transcendental curves such as y=xπ through a point of a normal model. The value group is isomorphic to an ordered group generated by 2 incommensurable real numbers. Dimension 0, rank 1, non-discrete, value group elements are commensurable. The value group can be isomorphic to any dense subgroup of the rational numbers. These correspond to germs of curves of the form y=Σanxbn where the numbers bn are rational with unbounded denominators.

References Nagata, Masayoshi (1962), "Imbedding of an abstract variety in a complete variety", Journal of Mathematics of Kyoto University, 2: 1–10, doi:10.1215/kjm/1250524969, ISSN 0023-608X, MR 0142549 Zariski, Oscar (1939), "The reduction of the singularities of an algebraic surface", Ann. of Math., 2, 40 (3): 639–689, Bibcode:1939AnMat..40..639Z, doi:10.2307/1968949, JSTOR 1968949 Zariski, Oscar (1940), "Local uniformization on algebraic varieties", Ann. of Math., 2, 41 (4): 852–896, doi:10.2307/1968864, JSTOR 1968864, MR 0002864 Zariski, Oscar (1944), "The compactness of the Riemann manifold of an abstract field of algebraic functions", Bulletin of the American Mathematical Society, 50 (10): 683–691, doi:10.1090/S0002-9904-1944-08206-2, ISSN 0002-9904, MR 0011573 Zariski, Oscar; Samuel, Pierre (1975), Commutative algebra. Vol. II, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90171-8, MR 0389876

Worked examples

Example 1 — a first encounter with Zariski–Riemann space

Start with the simplest possible case. Write down what Zariski–Riemann space 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 Zariski–Riemann space 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 Zariski–Riemann space 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 Zariski–Riemann space

In research: Zariski–Riemann space 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 Zariski–Riemann space 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: Zariski–Riemann space is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic geometry, Bernhard Riemann, so understanding it makes those chapters shorter.
In everyday life: Look for Zariski–Riemann space 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.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Zariski–Riemann space” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Zariski–Riemann space in 20 minutes

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

Frequently asked questions

What is Zariski–Riemann space in simple terms?

Why does Zariski–Riemann space 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 Zariski–Riemann space?

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 Zariski–Riemann space.