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:
\( k\langle x,y\rangle /(yx-qxy) \)
More generally, the quantum polynomial ring is the quotient ring:
\( k\langle x_{1},\dots ,x_{n}\rangle /(x_{i}x_{j}-q_{{ij}}x_{j}x_{i}) \)
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
References
Ajitabh, Kaushal (1994), Modules over regular algebras and quantum planes (PDF) (Ph.D. thesis)
Artin M.: Geometry of quantum planes, Contemporary Mathematicsv. 124 (1992).
Rogalski, D (2014). "An introduction to Noncommutative Projective Geometry". arXiv:1403.3065 [math.RA].
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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