Noncommutative projective geometry

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

${\displaystyle k\langle x,y\rangle /(yx-qxy)}$

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

${\displaystyle k\langle x_{1},\dots ,x_{n}\rangle /(x_{i}x_{j}-q_{ij}x_{j}x_{i})}$

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.

• Elliptic algebra
• Calabi–Yau algebra
• Sklyanin algebra

• Ajitabh, Kaushal (1994), Modules over regular algebras and quantum planes (PDF) (Ph.D. thesis)
• Artin, Michael (1992), "Geometry of quantum planes", Contemporary Mathematics, 124: 1–15, MR 1144023
• Rogalski, D (2014). "An introduction to Noncommutative Projective Geometry". arXiv:1403.3065 [math.RA].