Non-noetherian subalgebras of finitely generated commutative algebras

First example

I first encountered a non-noetherian subalgebra of a finitely generated commutative algebra in the early 1980’s. Let R=k[x,y] be the commutative polynomial ring in two variables over a field k. The subalgebra S:=k+xR is not noetherian. It is a pleasant exercise to show that the ideal {\mathfrak m} := xR is not a finitely generated ideal of S. As an ideal of S it is equal to xS+xyS+xy^2S+\cdots.

A non-commutative contrast

If A is the first Weyl algebra over a field k of characteristic zero with “standard” generators \partial and t, then k+At is both left and right noetherian and is a finitely generated k-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 A, over a field k, is the free algebra k\langle t,\partial\rangle modulo the relation \partial t-t\partial =1. This is a standard example in non-commutative ring theory. If the characteristic of k is zero, which I will assume from now on, then A is isomorphic to the ring of differential operators on the polynomial ring in one variable: if that polynomial ring is k[t], then t \in A acts on k[t] as multiplication by t and \partial \in A acts on k[t] as the derivative d/dt. That these two operators on k[t] satisfy the relation \partial t-t\partial =1 is the fact, well-known to our calculus students, that (tf)'=tf'+f.

The first Weyl algebra is considered a non-commutative analogue of the polynomial ring on two variables. It has the following properties:

  1. \{t^i\partial^j \; | \; i,j \ge 0\} is a k-basis for A;
  2. A has a filtration F^nA= {\rm span}\{t^i\partial^j \; | \; i+j \le n\} such that the associated graded algebra is isomorphic to the polynomial ring on two variables;
  3. A is left and right noetherian;
  4. A is a domain, i.e., every product of non-zero elements is non-zero;
  5. A has projective dimension 2 as a left module over A \otimes A^{op};
  6. A is a simple ring: its only two-sided ideals are A and \{0\};
  7. A 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

B:=\begin{pmatrix} R & Rx \\ R & k+Rx \end{pmatrix}

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 k+Rx, is not finitely generated. I show B is finitely generated below. It is probably appropriate to mention the Artin-Tate Lemma at this point because it implies that B is not a finitely generated module over its center.

The Artin-Tate Lemma. Let k be a field and A a finitely generated k-algebra. If A is finitely generated as a module over a central subalgebra Z, then Z is finitely generated. Hence A is left and right noetherian.

In his talk, Lance also mentioned that the ring

\begin{pmatrix} A & At \\ A & k+At \end{pmatrix}

is noetherian and that k+At is a finitely generated algebra.

Is the commutative example k+xR more than a curiosity?

I considered the fact that k+xR is not noetherian a curiosity that I didn’t need to pay much attention to. Same with Lance Small’s example, the ring B. 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 B 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 B=kQ/(yba-bay) where Q is the following quiver:

A standard way to view a quotient, let’s call it B, of a path algebra of a quiver is to use the idempotents (trivial paths) e_i at the vertices i \in Q_0 to realize B as a tiled matrix algebra with “tiles” e_i B e_j, i,j \in Q_0. When Q has only two vertices, labelled 1 and 2 say, we have

B=\begin{pmatrix} e_1Be_1 & e_1Be_2 \\ e_2 B e_1 & e_2Be_2 \end{pmatrix}.

The “tile” e_i B e_j is spanned by the paths that begin at j and end at i; I adopt the convention that pq means first traverse the path q, then p. The consistency conditions, which I have not stated, imply that all e_iBe_i 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 y and ba. The relation yba-bay=0 tells us that, as elements of B, y commutes with ba. It follows that e_1Be_1 is the polynomial ring k[y,ba]. Non-trivial paths that begin and end at vertex 2 are of the form apb where p is a path that begins and ends at 1. Hence

B=\begin{pmatrix} k[y,ba] & k[y,ba]b \\ ak[y,ba] & k+ak[y,ba]b \end{pmatrix}=\begin{pmatrix} k[x,y] & k[x,y]b \\ ak[x,y] & k+ak[x,y]b \end{pmatrix}

where the second equality comes from writing x =ba. A simple calculation shows that there is an isomorphism

B \cong \begin{pmatrix} R & Rx \\ R & k+Rx \end{pmatrix}

given by the map

\Phi\begin{pmatrix} p & qb \\ ar & \lambda+asb \end{pmatrix}=\begin{pmatrix} p & qx \\ r & \lambda+sx \end{pmatrix}.

What is the appropriate geometric picture for B and the non-noetherian scheme Spec(k+xR)?

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 B is almost a finite module over its center so some variation on that theory should apply to B 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.

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Drinking and homological algebra?

Popular wisdom says one should not drink and derive but some of you might enjoy a little wine with your derived functors so, with that thought in mind, I suggest you try a glass of this from Tor Vineyards in California.

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Wine for Buchweitz

I have found the perfect wine for Ragnar Buchweitz who is giving two talks on Maximal Cohen-Macaulay modules and Orlov’s Theorem. One might say that the wine is complex and after having one bottle one often needs another, and another, and another. Syzygy is produced in Washington State, and it is good. As are Ragnar’s talks.

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Two-sided ideals in a universal localization

Let Q be a quiver with the property that only finitely many arrows emerge from each vertex—this condition is commonly referred to by saying the graph is row-finite but I find that terminology unsatisfactory; row-finite needs too much explanation (the incidence matrix and the convention about the entries in it).

Pinar Colak proved that every two-sided ideal in the Leavitt path algebra L(Q) is generated by its intersection with the path algebra kQ. See Theorem 8 in his paper. He doesn’t phrase it like this because he doesn’t know that the canonical map kQ \to L(Q) is injective; Proposition 4.1 in Xiao Wu Chen’s paper proves this map is injective.

Colak’s result is very precise. He proves that a two-sided ideal in L(Q) is generated by elements of the form e_i + \sum_{m=0}^n \lambda_m c^m where e_i is the idempotent at a vertex i, c is a cycle at i, and the \lambda_i are scalars in the field k.

My question.

As I said in a previous post, L(Q) is a universal localization of kQ so Colak’s result prompted me to ask George Bergman the following question: if f:R \to S is a  universal localization is every two-sided ideal I in S is generated by its preimage in R, i.e., by f(f^{-1}(I))? The answer is a big fat NO. The simplest example George provided is

R= \begin{pmatrix} k & k+kx \\ 0 & k \end{pmatrix} \subset S=\begin{pmatrix} k[x] & k[x] \\ k[x] & k[x] \end{pmatrix}

The ring S is the universal localization of R at the map g:Re_{11} \to Re_{22} which is right multiplication by  e_{12}. The ring S has many ideals that have zero intersection with  R; for example, any ideal of S that is generated by an ideal of  the central copy of k[x]  whose generator has degree  > 1.

To see that S is the universal localization of R at g, first observe that S \otimes g:S \otimes_R Re_{11} \to S\otimes_R Re_{22} is the map Se_{11} \to Se_{22} given by right multiplication by e_{12} and this is an isomorphism with inverse left multiplication by e_{21}. If R_g denotes the universal localization its universal property says that the inclusion R \to S must factor through R_g. Hence there is a homomorphism R_g \to S with image the subalgebra of S generated by R and e_{21}. But that subalgebra is S so R_g \to S is surjective. I need to do a little more to show that S is the universal localization; I think that should be done by using the fact that R is the path algebra of the Kronecker quiver Q which is

\bullet \, {{\longrightarrow}\atop{\longrightarrow}} \, \bullet.

Let’s label the arrows a and b and adjoin a new arrow b^* in the opposite direction and impose the relation bb^*=e_1 and b^*b=e_2 where e_1 and e_2 are the idempotents at the vertices. Let’s write S' for the path algebra modulo these relations. I think it is clear that S' has the appropriate universal property to be the universal localization of kQ =R at b. A simple calculation will show that S' \cong S.

Contrast with inverting elements in a ring.

If \Sigma is a multiplicatively closed set of non-zero-divisors in a commutative ring A, then every ideal in the localization A[\Sigma^{-1}] is generated by its intersection with A. To prove this we use the fact that elements in A[\Sigma^{-1}] are of the form ax^{-1} where a \in A and x \in \Sigma; an ideal I in A[\Sigma^{-1}] that contains ax^{-1} contains a which is an element of I \cap R so ax^{-1} is in the ideal of A[\Sigma^{-1}] that is generated by I \cap R.

This argument extends to one-sided ideals in localizations of non-commutative rings at Ore sets but there is an extra twist to the argument because one must apply the Ore condition to see that an element ax^{-1} in a left ideal I of the localization is equal to an element of the form y^{-1}b in order to see that ax^{-1} is in the left ideal of the localization that is generated by b, etc. I’m omitting the details.

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The space of Penrose tilings as a non-commutative scheme

I’m in Banff attending the meeting Linking representation theory, singularity theory and non-commutative algebraic geometry. The weather is glorious, sunshine from dawn to dusk! I gave a talk today, a slightly different version of the talk I gave in Leiden a couple of months ago. The slides for today’s talk are here. The videos of all the talks are here.

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Leavitt path algebras, Cuntz-Krieger algebras, and C^*-graph algebras

Invariant basis number.

A fundamental result in linear algebra is that all bases for a given vector space have the same cardinality. This gives rise to a well-defined notion of dimension. It is not difficult to extend this result to free modules over a commutative ring: all bases for a given free module over a commutative ring have the same cardinality. The same result holds for free modules over noetherian rings. Rings with this property are said to have invariant basis number, IBN for short.

Not all rings have IBN. Given any positive integers m and n there is a ring R such that the free left R-module having a basis of cardinality m is isomorphic to the free left R-module having a basis of cardinality n; succinctly, R^m \cong R^n.

Rings without invariant basis number.

In 1956 W.G. Leavitt published a short paper entitled Modules without invariant basis number giving examples of this phenomenon. Let F=k \langle x_1,\ldots,x_n\rangle be the free algebra on n variables over a field k. The set of all words in the letters x_1,\ldots,x_n is a k-vector-space basis for F and multiplication is given by concatenation of words. It follows from this that the left ideal {\mathfrak m}=Fx_1+\cdots+Fx_n is a free left F-module with basis x_1,\ldots,x_n. There is an exact sequence 0 \to F^n \to F \to k=F/{\mathfrak m} \to 0 of left F-modules in which the first map is (a_1,\ldots,a_n) \mapsto a_1x_1+\cdots+a_nx_n. I prefer to write this map as right multiplication by the column vector

X:=\begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix}.

To get Leavitt’s example, we introduce new variables y_1,\ldots,y_n and form the  ring F \langle y_1,\ldots,y_n \rangle modulo the relations:

X\cdot(y_1,\ldots,y_n)=\begin{pmatrix} 1 & 0 & \ldots & 0 \\ 0 & 1 & \ldots & 0 \\ \vdots && & \vdots \\ 0 & 0 & \ldots & 1\end{pmatrix} \phantom{xx}  and \phantom{xx} (y_1,\ldots,y_n)X=1.

Call this new ring R. The map R^n \stackrel{\cdot X}{\longrightarrow} R has inverse given by a \mapsto (ay_1,\ldots,ay_n), which shows that R^n \cong R as left R-modules.

Universal localization.

The construction of R from F is an instance of a general procedure that goes by the name of universal localization. I learned about this from Aidan Schofield’s book Representations of rings over skew fields. The idea is this. Given a ring S and a collection of maps \Sigma between finitely generated projective left S-modules, there is a ring S_\Sigma and a homomorphism f :S \to S_\Sigma with the property that S_\Sigma \otimes \sigma is an isomorphism for all \sigma \in \Sigma and f :S \to S_\Sigma is universal with respect to this property: if g:S \to T is a ring homomorphism such that T \otimes \sigma is an isomorphism for all \sigma \in \Sigma, then g=hf for a unique hmomorphism h:S_\Sigma \to T. The ring S_\Sigma is called the universal localization of S at \Sigma.

In Leavitt’s example, R is the universal localization of F at \Sigma=\{\cdot X\}. The notion of universal localization didn’t emerge until long after Leavitt’s paper but, to me, it is the right context in which to understand Leavitt’s example.

The relations in the localization R.

In preparation for what is to follow I will present R in a slightly different way. First, I will write x_i^*=y_i so R is generated by x_1,\ldots,x_n and x_1^*,\ldots, x_n^*. The relations defining R are x_ix_i^*=1, x_ix_j^*=0 if i \ne j, and x_1^*x_1+\cdots +x_n^*x_n=1.

If I had worked with right modules instead of left modules, i.e., if I had written the kernel of the map F \to F/{\mathfrak m} as x_1F+\cdots+x_nF \to F,  the resulting universal localization would have been generated by x_1,\ldots,x_n and x_1^*,\ldots, x_n^* subject to the relations x_i^*x_i=1, x_i^*x_j=0 if i \ne j, and x_1x_1^*+\cdots +x_nx_n^*=1. Let’s call this ring R'.

The Cuntz algebras {\mathcal O}_n

In 1977, Joachim Cuntz published Simple C*-Algebras Generated by Isometries. The abstract reads as follows:

We consider the C*-algebra {\mathcal O}_n generated by n \ge 2 isometries S_1,\ldots, S_n on an infinite-dimensional Hilbert space, with the property that S_1S_1^*+\cdots + S_nS_n^*=1. It turns out that {\mathcal O}_n has the structure of a crossed product of a finite simple C*-algebra {\mathcal F} by a single endomorphism scaling the trace of {\mathcal F} by 1/n. Thus, {\mathcal O}_n is a separable C*-algebra sharing many of the properties of a factor of type III_\lambda with \lambda=1/n. As a consequence we show that {\mathcal O}_n is simple and that its isomorphism type does not depend on the choice of S_1,\ldots,S_n.

An element p in a C^*-algebra is a projection if p^2=p=p^*. An element S in a C^*-algebra is a partial isometry if S=SS^*S. If S is a partial isometry, then SS^* and S^*S are projections. Cuntz is studying the universal C^*-algebra generated by the S_is so that a full set of relations is given by S_1S_1^*+\cdots + S_nS_n^*=1 and the relations S_i^*S_i=1 and the range projections p_i=S_iS_i^* are pairwise orthogonal. The orthogonality condition is equivalent to the condition S_i^*S_j=0 if i \ne j.

There is an algebra homomorphism R' \to {\mathcal O}_n and the image is a dense subalgebra. For many years I did not understand why Cuntz adopted these relations. Now, through the lens of universal localization, the relations seem natural to me.

The introduction to Cuntz’s paper is easy reading and explains why {\mathcal O}_n is of such interest. The {\mathcal O}_ns settled a number of open problems and exhibited a range of new phenomena. Nevertheless they are relatively easy to work with. Among their important properties is their ubiquity: every simple infinite C^*-algebra contains a subalgebra that has {\mathcal O}_n as a quotient.

The algebras {\mathcal O}_n are themselves simple rings in a very strong way: given a non-zero element x \in {\mathcal O}_n, there are elements a,b \in {\mathcal O}_n such that axb=1. Leavitt had already shown that his algebra R is simple.

Another nice property is that there are, cleverly constructed, inclusions {\mathcal O}_2 \supset {\mathcal O}_3 \supset \cdots. This might not be surprising if you already know there are inclusions F_2 \supset F_3 \supset \cdots where F_n denotes the free algebra on n variables.

The Cuntz algebras are pairwise non-isomorphic because K_0({\mathcal O}_n) \cong {\mathbb Z}/(n-1).

The path algebra of a quiver

For certain people it is natural when considering some aspect of a free algebra to ask if a similar behavior is exhibited by the path algebra of an arbitrary quiver. The free algebra on n variables is itself a path algebra, the path algebra of the quiver having a single vertex and a n arrows from that vertex to itself. A basis for the path algebra is given by all finite paths in that directed graph.

The free algebra on n variables has an alternative description as the tensor algebra over the field k of an n-dimensional vector space, k\langle x_1,\ldots,x_n \rangle \cong k \oplus V \oplus V^{\otimes 2} \oplus \cdots if dim(V)=n. Path algebras of quivers are also tensor algebras.

Let Q be a finite directed graph, or quiver. Parallel arrows and loops are allowed. The path algebra of Q over a field k, denoted kQ, is the vector space with basis the set of finite paths in Q, including the trivial or lazy paths at each vertex, and multiplication given by concatenation of paths, or zero if the two paths do not concatenate. One now has a difficult decision to make: if a and b are arrows with a ending where b begins should one denote the path “first traverse a then traverse b by ab or ba? The virtue of ab is that one reads from left to right. The virtue of ba is that if we wish to assign vector spaces to vertices and linear maps to arrows the convention for composition of functions is that ba means first do a, then do b. The most popular convention amongst those working on Leavitt path algebras is to write ab. The most popular convention among those working on representation theory of quivers is ba!

Here we adopt the convention that ab means “first traverse a then traverse b”. If the end of a is not the start of b we define ab=0 in kQ. In this way kQ becomes an associative ring.

I will often use the letters i and j denote vertices and will write e_i for the trivial, or lazy, or “do nothing”, path at i. Doing nothing twice is the same as doing nothing once, so e_ie_i=e_i, i.e., e_i is an idempotent. Furthermore, if a is an arrow from i to j, then e_iae_j=a. It follows that the sum of all the e_is is the identity element in kQ.

As I said above, kQ can be described as a tensor algebra. Let S denote the  linear span of the e_is and write B for the linear span of the arrows. Then B is an S-bimodule and kQ \cong T_S(B)=S \oplus B \oplus (B \otimes_S B) \oplus \cdots, the tensor algebra of B over S.

It is convenient to introduce the notation s(p) for the vertex that is the start of a path p and t(p) for the vertex where p ends.

Leavitt path algebras

To form the Leavitt path algebra L(Q) we first form \overline{Q}, the double of Q: \overline{Q} has the same vertices as Q and for each arrow a  we introduce a new arrow a^*, the ghost of a, in the opposite direction, i.e., a^* starts where a ends and ends where a starts; thus, if a= e_iae_j, then a^*=e_ja^*e_i.

We define L(Q) to be the path algebra k {\overline Q} modulo the relations a^*a=e_{t(a)}, a^*b=0 if a \ne b for all arrows a and b, and \sum_{a \in s^{-1}(i)} aa^*=e_i for each vertex i at which some arrow ends.

Leavitt path algebras have been intensively studied since they were introduced around 2005.

C^*-graph algebras

Cuntz and Krieger had already introduced the C^*-algebra analogue of Leavitt path algebras in 1980. Let A be an n \times n matrix with entries in \{0,1\}. The Cuntz-Krieger algebra {\mathcal O}_A is the universal C^*-algebra generated by partial isometries s_1,\ldots,s_n subject to the relations s_i^*s_i=\sum_{j=1}^n A_{ij} s_js_j^* and (s_is_i^*)(s_js_j^*)=0 for all i \ne j. Of course, A is the incidence matrix of a directed graph Q, and in that case L(Q) is a dense subalgebra of {\mathcal O}_A. If A_{ij}=1 for all i and j, then {\mathcal O}_A \cong {\mathcal O}_n.

In 1998, the paper Cuntz-Krieger algebras of directed graphs by Kumjian, Pask, and Raeburn, extended the definition of Cuntz-Krieger algebras to arbitrary directed graphs. If Q is the directed graph, they write C^*(Q) for the universal C^*-algebra generated by mutually orthogonal projections p_i, one for each vertex $i$, and partial isometries a, one for each arrow, subject to the relations

a^*a = p_{t(a)} \phantom{xx} and p_i=\sum_{a \in s^{-1}(i)} aa^*.

Raeburn’s 2005 CBMS lectures Graph Algebras gives a comprehensive account of the subject up to 2005. The document Graph algebras: bridging the gap between analysis and algebra  is a useful introduction to graph algebras and Leavitt path algebras that is aimed at algebraists.

I have restricted my discussion to finite directed graphs but almost everything I have said extends in a reasonable way to arbitrary directed graphs.

Why am I talking about this?

I’m in Banff attending the meeting Linking representation theory, singularity theory and non-commutative algebraic geometry. I’m eager to hear Xiao-Wu Chen’s talk today on his paper Irreducible representations of Leavitt path algebras. I have been reading it this week. My interest arises from my paper Category equivalences involving graded modules over path algebras of quivers, where I show that a certain quotient category, QGr(kQ), of the category of graded modules over the path algebra of a quiver (directed graph) Q having a finite number of vertices and arrows is equivalent to the category of graded modules over the Leavitt path algebra L(Q^\circ) associated to the quiver Q^\circ that is obtained by successively removing all sinks and sources from Q.

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Noncommutative Invariant Theory in Seattle, May 26-27, 2012

James Zhang has organized a workshop on Noncommutative Invariant Theory that will take place at the University of Washington, May 26-27, 2012. The description says:

Group actions (or Hopf algebra actions) on noncommutative algebras is one of the main themes in noncommutative invariant theory. This topic is closely related to Artin-Schleter regular algebras and noncommutative algebraic geometry, symplectic reflection algebras, Calabi-Yau algebras and noncommutative resolutions, reperesentation theory, and combinatorics. We emphasize recent developments and new directions in this area. The workshop brings together experts working in various aspects of noncommutative invariant theory and encourages interaction between all participants. In addition to regular research talks, leading experts in the area will give some survey talks.

The schedule is here.

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