Noncommutative Projective Geometry

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:
::$k \langle x_1, \dots, x_n \rangle / (x_i x_j - q_ x_j x_i)$

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

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*{{cite arXiv , first1=D , last1=Rogalski , title=An introduction to Noncommutative Projective Geometry , year=2014 , eprint=1403.3065, class=math.RA

Fields of geometry