I first encountered a non-noetherian subalgebra of a finitely generated commutative algebra in the early 1980’s. Let be the commutative polynomial ring in two variables over a field . The subalgebra is not noetherian. It is a pleasant exercise to show that the ideal is not a finitely generated ideal of . As an ideal of it is equal to .
A non-commutative contrast
If is the first Weyl algebra over a field of characteristic zero with “standard” generators and , then is both left and right noetherian and is a finitely generated -algebra. This was proved in a beautiful paper by Chris Robson, Idealizers and Hereditary Noetherian Prime Rings, J. Algebra, 22 (1972) 45-81; MR0299639 (45 #8687). Google Scholar lists 124 citations to this paper. (Math Reviews lists 26 citations, but the Google Scholar list is more complete.) Robson’s paper contains much more than this example but I find the example a striking illustration of how much the commutative and non-commutative worlds differ. Part of the point, in relation to the previous paragraph, is that the first Weyl algebra is often seen as a non-commutative analogue of the polynomial ring on two variables. In fact, there is a flat family of algebras with the special fiber being the polynomial ring and the other fibers isomorphic to the first Weyl algebra.
I don’t need to tell those who already know this example any more. Those who haven’t seen it before deserve a little more explanation.
The first Weyl algebra
The first Weyl algebra, which I will denote by , over a field , is the free algebra modulo the relation . This is a standard example in non-commutative ring theory. If the characteristic of is zero, which I will assume from now on, then is isomorphic to the ring of differential operators on the polynomial ring in one variable: if that polynomial ring is , then acts on as multiplication by and acts on as the derivative . That these two operators on satisfy the relation is the fact, well-known to our calculus students, that .
The first Weyl algebra is considered a non-commutative analogue of the polynomial ring on two variables. It has the following properties:
- is a -basis for ;
- has a filtration such that the associated graded algebra is isomorphic to the polynomial ring on two variables;
- is left and right noetherian;
- is a domain, i.e., every product of non-zero elements is non-zero;
- has projective dimension 2 as a left module over ;
- is a simple ring: its only two-sided ideals are and ;
- has global homological dimension 1.
The first 5 of these properties, with appropriate changes of notation, hold for the polynomial ring on two variables.
A talk by Lance Small in 1982
In an algebra seminar talk at USC on 10/18/1982, Lance Small provided the ring
as an example of a finitely generated (!) prime subalgebra of matrices over a commutative ring (he was interested in the fact that it satisfies a polynomial identity) whose center, which is isomorphic to , is not finitely generated. I show is finitely generated below. It is probably appropriate to mention the Artin-Tate Lemma at this point because it implies that is not a finitely generated module over its center.
The Artin-Tate Lemma. Let be a field and a finitely generated -algebra. If is finitely generated as a module over a central subalgebra , then is finitely generated. Hence is left and right noetherian.
In his talk, Lance also mentioned that the ring
is noetherian and that is a finitely generated algebra.
Is the commutative example more than a curiosity?
I considered the fact that is not noetherian a curiosity that I didn’t need to pay much attention to. Same with Lance Small’s example, the ring . Good to know, but then time to get on with other business. My attitude changed last summer when Charlie Beil gave a talk in Seattle about his paper Non-noetherian Geometry and Toric Superpotential Algebras.
Charlie showed that the algebra that appeared in Lance Small’s talk appears in a very natural way. Natural in the sense that similar constructions produce very beautiful, very useful, rings that have been intensively studied over the past several years for reasons both internal to mathematics (non-commutative resolutions of affine Gorenstein singularities) and arising from quiver gauge theories (in string theory).
There are several points of view on the context in which to view Charlie’s paper; one is the use of superpotential algebras and associated brane tilings to construct non-commutative crepant resolutions of certain affine varieties; another is that the superpotential algebras lead to quiver gauge theories that are relevant to string theory. These other papers produce non-commutative algebras with good properties that are finitely generated modules over their centers and these centers are finitely generated commutative rings that are coordinate rings of certain affine singular 3-folds. However, for these constructions to work the brane tiling must satisfy a certain “consistency” condition. I don’t want to say more about that here—if you want to know more google on consistent brane tiling.
The starting point for Charlie’s work is that brane tilings that fail to be consistent, or are non-cancellative in Charlie’s terminology, still produce gauge theories of physical interest. The simplest example of a non-cancellative brane tiling corresponds to the algebra where is the following quiver:
A standard way to view a quotient, let’s call it , of a path algebra of a quiver is to use the idempotents (trivial paths) at the vertices to realize as a tiled matrix algebra with “tiles” , . When has only two vertices, labelled 1 and 2 say, we have
The “tile” is spanned by the paths that begin at and end at ; I adopt the convention that means first traverse the path , then . The consistency conditions, which I have not stated, imply that all are isomorphic to a single finitely generated commutative algebra.
Now consider Charlie’s example above. Paths that begin and end at vertex 1 are words in and . The relation tells us that, as elements of , commutes with . It follows that is the polynomial ring . Non-trivial paths that begin and end at vertex 2 are of the form where is a path that begins and ends at . Hence
where the second equality comes from writing . A simple calculation shows that there is an isomorphism
given by the map
What is the appropriate geometric picture for and the non-noetherian scheme ?
This is the question Charlie addresses in his paper. There is a rather well-developed theory for non-commutative rings that are finite, i.e., finitely generated, modules over their centers. The ring is almost a finite module over its center so some variation on that theory should apply to in a useful way.
The first Weyl algebra has a non-noetherian subalgebra
Bavula has shown that the first Weyl algebra over a field of characteristic zero contains a non-noetherian subalgebra. Its construction is a little tricky so I’ll just refer you to his paper if you want to learn more about it.