In the spring semester of 2014, I conducted research in the field of noncommutative algebra and algebraic geometry under the supervision of Professor Jason Bell. To describe my research, I will begin by discussing
related questions arising in geometry and then show how these questions provide motivation for the problem I ended up solving. This solution ended up being part of a paper, written with Jason Bell and Colin Ingalls, which has been submitted to the Bulletin of the London Math Society.
An algebraic variety X is the solution set of a system of polynomial equations in n variables. An algebraic variety X is said to be projective
if it a closed subset of a projective space. Given two projective algebraic varieties X and Y over an algebraically closed field k, a fundamental question in algebraic geometry is to determine whether or not they are birationally equivalent (this is a useful notion of being “the
same” for varieties). This birational classification was completely achieved in the case of surfaces and can be described in terms of a short list of invariants. But to answer these types of questions in general, mathematicians have created many “birational invariants” that control the maps between X and Y. A famous result in this direction is the Riemann-Hurwitz theorem, which implies that if X and Y are smooth irreducible projective complex curves and there is a surjective morphism from X to Y
then the genus of X is greater than or equal to the genus of Y.
In algebraic geometry, there is a dictionary between geometric concepts and algebraic concepts and this dictionary allows one to translate the notion of birational isomorphism of varieties to isomorphisms of their function fields and in particular it makes sense to talk about invariants for fields. As an example, the Riemann-Hurwitz theorem can be formulated by saying that if X and Y are smooth irreducible projective complex curves and the function field of Y embeds in the function field of X then the genus of X is greater than or equal to the genus of Y. Taking this field-theoretic point of view, it is often possible to find noncommutative analogues of invariants that extend geometric invariants to the setting of division rings.
This is the starting point for noncommutative algebraic geometry. To further understand noncommutative geometry and deepen the analogy with classical geometry, Artin gives a conjectured classification of “noncommutative surfaces” akin to the birational classification of surfaces. With Artin’s conjecture as motivation, we proved a noncommutative analogue of the Riemann-Hurwitz theorem mentioned above: Let X and Y
be smooth irreducible algebraic curves over the complex numbers, and let
D (X) and D (Y) be the respective quotient division rings of the rings of differential operators of X and Y. If there is a C-algebra embedding of D (X) into D (Y) then the genus of X is less than or equal to the genus of
Y. Here the quotient division ring of the ring of differential operators on
X plays the noncommutative analogue of the function field of a surface.
To prove this, one first shows that ring of differential operators on X can be described in terms of the ordinary function field of X along with some derivation of the function field. The quotient ring D (X) is then obtained by inverting the non-zero elements. If one chooses a trivial derivation, our statement is just the Riemann-Hurwitz theorem, and this justifies calling our result a “noncommutative” Riemann-Hurwitz theorem. Unfortunately, understanding the quotient division ring D (X) of a curve X is not easy as understanding its function field, as it is a noncommutative localization, which, in general, is horribly behaved. To get around this, we reduced to working with models of our curves over finitely generated rings and then showed that one could reduce modulo various prime ideals of our finitely generated rings while still preserving the relevant geometric information. Finally, we showed that modulo these primes one could reduce to the commutative case, albeit in positive characteristic. After reducing to a commutative problem, we used known results from algebraic geometry to obtain our genera inequality.
It is worth noting that this result does not directly prove the conjecture of Artin. Much more robust invariants are needed to accomplish this. The first half of my research term was spent trying to find a noncommutative analogue of the genus for the ring of differential operators. This noncommutative genus, if it exists, should equal the classical genus when the derivation is trivial. We obtained some partial characterizations, but it didn’t behave as well as we would have liked. However, I was able to fix enough of these problems to obtain our main result. I ultimately found a suitable noncommutative invariant by reducing modulo primes and then borrowing commutative invariants, and this was sufficient to get our theorem. Our result can be regarded as the first step in the search for a noncommutative genus, which, one hopes, will eventually lead to a complete list of noncommutative invariants necessary to obtain a proof of Artin’s conjecture.