Noncommutative algebraic geometry

Commutative algebraic geometry begins by constructing the spectrum of a ring. The points of the algebraic variety (or more generally, scheme) are the prime ideals of the ring, and the functions on the algebraic variety are the elements of the ring. A noncommutative ring, however, may not have any proper non-zero two-sided prime ideals. For instance, this is true of the Weyl algebra of polynomial differential operators on affine space: The Weyl algebra is a simple ring. Therefore, one can for instance attempt to replace a prime spectrum by a primitive spectrum: there are also the theory of non-commutative localization as well as descent theory. This works to some extent: for instance, Dixmier's enveloping algebras may be thought of as working out non-commutative algebraic geometry for the primitive spectrum of an enveloping algebra of a Lie algebra. 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. One might imagine this difficulty is because of a sort of quantum phenomenon: points in a space can influence points far away (and in fact, it is not appropriate to treat points individually and view a space as a mere collection of the points).

Due to the above, one accepts a paradigm implicit in Pierre Gabriel's thesis and partly justified by the Gabriel–Rosenberg reconstruction theorem (after Pierre Gabriel and Alexander L. Rosenberg) that a commutative scheme can be reconstructed, up to isomorphism of schemes, solely from the abelian category of quasicoherent sheaves on the scheme. Alexander Grothendieck taught that to do geometry one does not need a space, it is enough to have a category of sheaves on that would be space; this idea has been transmitted to noncommutative algebra by Yuri Manin. There are, a bit weaker, reconstruction theorems from the derived categories of (quasi)coherent sheaves motivating the derived noncommutative algebraic geometry (see just below).

Perhaps the most recent approach is through the deformation theory, placing non-commutative algebraic geometry in the realm of derived algebraic geometry.

As a motivating example, consider the one-dimensional Weyl algebra over the complex numbers C. This is the quotient of the free ring C<x, y> by the relation

xy - yx = 1.

This ring represents the polynomial differential operators in a single variable x; y stands in for the differential operator ∂_x. This ring fits into a one-parameter family given by the relations xy - yx = α. When α is not zero, then this relation determines a ring isomorphic to the Weyl algebra. When α is zero, however, the relation is the commutativity relation for x and y, and the resulting quotient ring is the polynomial ring in two variables, C[x, y]. Geometrically, the polynomial ring in two variables represents the two-dimensional affine space A², so the existence of this one-parameter family says that affine space admits non-commutative deformations to the space determined by the Weyl algebra. 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.)

In this line of the approach, the notion of operad, a set or space of operations, becomes prominent: in the introduction to (Francis 2008) harv error: no target: CITEREFFrancis2008 (help), Francis writes:

We begin the study of certain less commutative algebraic geometries. … algebraic geometry over ${\displaystyle {\mathcal {E}}_{n}}$-rings can be thought of as interpolating between some derived theories of noncommutative and commutative algebraic geometries. As n increases, these ${\displaystyle {\mathcal {E}}_{n}}$-algebras converge to the derived algebraic geometry of Toën-Vezzosi and Lurie.