Noncommutative projective geometry explained

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 \langle x, y \rangle / (yx - q xy)

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

k \langle x _{1,} ... , x _{n} \rangle / (x _{i} x _{j} - q _{ij} x _{j} x _{i)}

Proj construction

See also: 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

• • • D . Rogalski . An introduction to Noncommutative Projective Geometry . 2014 . 1403.3065. math.RA .