One of the values of the field is that it also provides new techniques to study objects in commutative algebraic geometry such as Brauer groups.
Global properties such as those arising from homological algebra and K-theory more frequently carry over to the noncommutative setting.
Another work in a similar spirit is Michael Artin’s notes titled “noncommutative rings”,[1] which in part is an attempt to study representation theory from a non-commutative-geometry point of view.
The key insight to both approaches is that irreducible representations, or at least primitive ideals, can be thought of as “non-commutative points”.
As it turned out, starting from, say, primitive spectra, it was not easy to develop a workable sheaf theory.
As a motivating example, consider the one-dimensional Weyl algebra over the complex numbers C. This is the quotient of the free ring C
This ring fits into a one-parameter family given by the relations xy - yx = α.
This deformation is related to the symbol of a differential operator and that A2 is the cotangent bundle of the affine line.
(Studying the Weyl algebra can lead to information about affine space: The Dixmier conjecture about the Weyl algebra is equivalent to the Jacobian conjecture about affine space.)
-rings can be thought of as interpolating between some derived theories of noncommutative and commutative algebraic geometries.