mathematics

Noncommutative projective geometry

Noncommutative projective geometry 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 Noncommutative projective geometry rather than just read about it. In short: In mathematics, noncommutative projective geometry is a noncommutative analog of projective geometry in the setting of noncommutative algebraic geometry. Examples The quantum plane, the most basic example, is the quotient ring of the free ring: k ⟨ x , y ⟩ / ( y x − q x y ) {\displaystyle k\langle x,y\rangle /(yx-qxy)} More generally, the quantum polynomial ring is the quotient ring: k ⟨ x 1 , … , x n ⟩ / ( x i x j…

Key takeaways

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

Reference excerpt

In mathematics, noncommutative projective geometry is a noncommutative analog of projective geometry in the setting of noncommutative algebraic geometry.

Examples The quantum plane, the most basic example, is the quotient ring of the free ring:

k ⟨ x , y ⟩ / ( y x − q x y ) {\displaystyle k\langle x,y\rangle /(yx-qxy)}

More generally, the quantum polynomial ring is the quotient ring:

k ⟨ x 1 , … , x n ⟩ / ( x i x j − q i j x j x i ) {\displaystyle k\langle x_{1},\dots ,x_{n}\rangle /(x_{i}x_{j}-q_{ij}x_{j}x_{i})}

Proj construction

By definition, the Proj of a graded ring R is the quotient category of the category of finitely generated graded modules over R by the subcategory of torsion modules. If R is a commutative Noetherian graded ring generated by degree-one elements, then the Proj of R in this sense is equivalent to the category of coherent sheaves on the usual Proj of R. Hence, the construction can be thought of as a generalization of the Proj construction for a commutative graded ring.

See also Elliptic algebra Calabi–Yau algebra Sklyanin algebra

References Ajitabh, Kaushal (1994), Modules over regular algebras and quantum planes (PDF) (Ph.D. thesis) Artin, Michael (1992), "Geometry of quantum planes", Contemporary Mathematics, vol. 124, pp. 1–15, doi:10.1090/conm/124/1144023, ISBN 978-0-8218-5132-6, MR 1144023 Rogalski, D (2014). "An introduction to Noncommutative Projective Geometry". arXiv:1403.3065 [math.RA].

Worked examples

Example 1 — a first encounter with Noncommutative projective geometry

Start with the simplest possible case. Write down what Noncommutative projective geometry 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 Noncommutative projective geometry 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 Noncommutative projective geometry 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 Noncommutative projective geometry

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

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

Frequently asked questions

What is Noncommutative projective geometry in simple terms?

In mathematics, noncommutative projective geometry is a noncommutative analog of projective geometry in the setting of noncommutative algebraic geometry. Examples The quantum plane, the most basic example, is the quotient ring of the free ring: k ⟨ x , y ⟩ / ( y x − q x y ) {\displaystyle k\langle…

Why does Noncommutative projective geometry 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 Noncommutative projective geometry?

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 Noncommutative projective geometry.