Exotic dimensions used in ring theory

Do you ever get the feeling that mathematics uses the word dimension a lot? Well, that's for good reason. The concept of dimension is fundamental in mathematics. What is dimension? You can think of dimension as a numerical invariant characterizing the number of parameters required to do a certain thing. For example, for vector spaces, dimension is the cardinality of a basis, and a basis is a minimal set from which you can specify all vectors via linear combinations. The cartesian plane is two-dimensional because you need two coordinates in order to specify any point.

There are other more exotic types of dimension used in ring theory, and this post aims to be a quick introduction to them.

Rank of a free module

Possible values: all cardinals.

As we've already talked about, the dimension of a vector space is the cardinality of a basis of that space. Every vector space has a basis, and all bases of a given vector space have the same cardinality. Therefore, the concept of dimension in this case is well-defined.

If $R$ is a ring, then we could try and define a dimension (called the rank) for an $R$-module $M$, as the cardinality of a minimal generating set. The only problem is, the rank may not be well-defined. What is true is that if a minimal generating set for $M$ is infinite, then all minimal generating sets for $M$ have the same cardinality. But the conclusion is not necessarily true if there exists some minimal generating set for $M$ that is finite.

Actually, there exists a ring $R$ and two natural numbers $m$ and $n$ such that $R^n\cong R^m$. One such example is the endomorphism ring of a vector space of infinite dimension. Whoa, right? However, suppose a ring $R$ has the property that $R^n\cong R^m$ implies $m=n$. Then $R$ is said to have the invariant basis property. All commutative rings have the invariant basis property, so the concept of rank of a free module always makes sense here. Some noncommutatative rings such as finite rings also have the invariant basis property.

Krull dimension of a commutative ring

Possible values: all natural numbers and infinity.

If $R$ is a commutative ring, its Krull dimension is the supremum over the set of integers $n$ such that there exists a chain
\[ P_0\subset \cdots \subset P_n\] of prime ideals of $R$. For example, fields have Krull dimension zero since the only prime ideal of a field is the zero ideal. The integers (and more generally, Dedekind domains) have Krull dimension one because every nonzero prime in these rings is maximal.

If $F$ is a field, the Krull dimension of the polynomial ring $F[x_1,\dots,x_n]$ is $n$. Algebro-geometrically, the ring $F[x_1,\dots,x_n]$ represents $n$-dimensional $F$-space, and in general, Krull dimension accurately captures the concept of dimension of a algebraic variety. In more general cases, Krull dimension can behave bizarrely.

Embedding dimension of a local ring

Possible values: all cardinals

If $R$ is a commutative local ring with maximal ideal $M$, then the embedding dimension of $R$ is defined to be the dimension of $M/M^2$ as an $R/M$-vector space. For example, consider the commutative ring $\Z/4$. It is local with maximal ideal $(2)$ whose square is zero. Over $\Z/2\cong \Z/4/(2)$, the ideal $(2)$ is one-dimensional. Notice on the other hand that the Krull dimension of $\Z/4$ is zero, so this dimension is not the same as the Krull dimension. In fact:

Theorem. The Krull dimension of a commutative local ring is always less than or equal to its embedding dimension.

In fact, the local rings for which the embedding dimension is equal to the Krull dimension are exactly the regular local rings.

The grade

Possible values: natural numbers and infinity

The grade is not exactly a dimension, but I included it since it is so closely related. For any commutative ring, it is defined as the length of the longest regular sequence on $R$ (the grade also makes sense for $R$-modules). It is close to a dimension and the grade of a commutative ring is always less than its Krull dimension. Rings for which these values are equal are called Cohen-Macaulay. As you might expect, regular local rings are Cohen-Macaulay, but the converse is not true.

Projective dimension of a module

Possible values: natural numbers and infinity

The projective dimension of an $R$-module is the infimum over the set of integers $n$ such that there exists a projective resolution
\[0\to P_n\to\cdots\to P_0\to M\to 0\] of $M$. Of course, if no such integer exists, then the projective dimension is infinity, which is consistent with our definition as the infimum of the empty set is infinity. We write ${\rm projdim}_R(M)$ for the projective dimension of $M$.

The projective dimension is a measure of how many projective modules it takes to "specify" $M$. If ${\rm projdim}_R(M) = 0$ then $M$ is actually projective. Here's an example: if k is a positive integer, then ${\rm projdim}_\Z(\Z/k) = 1$. That is because as a $\Z$-module, $\Z/k$ is not projective but has a projective resolution
\[ 0\to\Z\xrightarrow{k}\Z\to \Z/k\to 0.\] Projective dimension gives rise to a really cool concept: global dimension.

Global dimension of a ring

Possible values: natural numbers and infinity

Because submodules of free abelian groups are free abelian, we see that ${\rm projdim}_\Z(\Z/k) = 1$, and so all $\Z$ modules have projective dimension at most one. Motivated by this, we can define the global dimension of a ring to be the supremum over the projective dimensions of all its modules.

Actually, we have to be a bit careful. We should really define the left global dimension of $R$ to be the supremum over the projective dimensions of all the left modules, and similarly for right modules. The left and global dimensions need not agree, though if one is zero, the other is also zero.

Speaking of global dimension zero: a ring is semisimple if every one of its left modules is projective (equivalently, injective). It turns out this implies that all the right modules are projective. The Artin-Wedderburn theorem says that such rings are finite direct products of full matrix rings over division rings.

Global dimension one rings, or hereditary rings include the integers and many other classes of rings as well. In fact, a ring $R$ has right global dimension one if and only if every submodule of every free $R$-module is projective.

The rings of larger global dimension don't have as nice a classification, but with other characteristics, global dimension provides a useful tool to classify and study rings.

Flat dimension and weak dimension

Possible values: natural numbers and infinity

If we repeat the definitions we had for projective dimension and global dimension but replace projective module with flat module, then we get the concepts of flat dimension of a module (length of smallest flat resolution) and weak dimension of a ring (supremum over flat dimensions of modules). Only, in this case, we don't need to distinguish left and right modules because left weak dimension and right weak dimension are always the same for any ring, whether or not it is commutative.

Rings of flat dimension zero are exactly the von Neumann regular rings: those rings $R$ such that for every $x\in R$ there exists a $y\in R$ such that $x = xyx$. They were named after John von Neumann of course. Commutative von Neumann regular rings are exactly those rings that are reduced and have Krull dimension zero. In fact, I wrote about this more than two years ago.

One strategy to classify rings is to mix various dimensions together, and see if you can come up with some nice hidden characterizations!

Injective dimension of a module

Possible values: natural numbers and infinity

There is also the injective dimension of an $R$-module: it is the infimum over lengths of its injective resolutions. It is the dual notion to projective dimension. If you take the supremum over injective dimensions of a ring, you just get global dimension again. That's also really cool. For example, if a ring $R$ has left global dimension five, then there exists a left $R$-module that has a projective resolution of length five, and there exists another left $R$-module that has an injective resolution of length five, and no shorter resolutions of these modules exist.

Variations of homological dimensions

Possible values: depends on the type of dimension

Projective and injective dimensions are examples of homological dimensions: these are those dimensions that are defined by resolutions of modules. There are many variations, many of them based on relative homological algebra. Perhaps the most famous are the "Gorenstein" dimensions, named after Daniel Gorenstein. These concepts became well-studied because for local rings, finite Gorenstein dimension characterizes whether that ring has finite injective dimension over itself.

I am not very familiar with Gorenstein dimensions, but Lars Winther Christensen wrote a Springer LNM called Gorenstein dimensions, which is a good start to learn more about these cool dimensions.

Gelfand-Kirillov dimension

Possible values: $0, 1, [2,\infty]$ (cool, right?)

The Gelfand-Kirillov dimension is much more mysterious than the other dimensions I defined so far, and is more analytic in nature. It is defined for $k$-algebras $A$ where $k$ is a field. We'll need some preliminary definitions to define the Gelfand-Kirillov dimension.

A finite-dimensional subspace $V$ of $A$ that contains $1$ is called a subframe of $A$. If $V$ is a subframe with ordered basis $\{ v_1,\dots,v_n\}$, we define $V^i$ to be the set of monomials of length $i$ in the $v_1,\dots, v_n$. So for example, $V^3$ contains monomials like $v_1v_2v_4$, etc.

Define $F_n^V = k + V + V^2 + \cdots + V^n$, and define $d_V(n) = \dim_k(F_n^V(A))$. Thus, $d_V(n)$ is a natural number and depends of course on the choice of the vector space $V$ contained in $A$. The number $d_V(n)$ as a function of $n$ measures how fast you can get elements of $A$ by taking longer and longer products of vectors in $V$. It is a growth function.

We define the the Gelfand-Kirillov dimension of $k[V]$ by
\[{\rm GKdim}(k[V]) := \limsup \frac{\log(d_V(n))}{\log(n)}.\] The Gelfand-Kirillov dimension of $A$ is defined to be
\[{\rm GKdim}(A) := \sup_V {\rm GKdim}(k[V]) \] where the supremum is taken over all subframes of $A$ (recall, these are finite-dimensional subspaces that contain the algebra identity). That these quantities exist are left as an exercise for the interested reader.

If $A$ is a finite-dimensional $k$-algebra, then for every subframe $V$, the dimension of the sets $F_n^V(A)$ are bounded, and so ${\rm GKdim}(A) = 0$ in this case. Thus, the Gelfand-Kirillov dimension is only interesting for infinite-dimensional algebras. It is also not hard to show that if $A$ is a free noncommutative $k$-algebra on a two-element set, then ${\rm GKdim}(A)= \infty$. That is because if you take $V$ to be spanned by $\{1,X,Y\}$ then the dimension of $F_n^V(A)$ grows exponentially in $n$.

To produce other basic examples, we can use the following:

Theorem. For any $k$-algebra,
\[{\rm GKdim}(A[x_1,\dots,x_d]) = {\rm GKdim}(A) + d.\]

Thus, the polynomial ring in $d$-variables over a field has Gelfand-Kirillov dimension $d$. These given examples of all possible natural numbers. In fact, non-natural number Gelfand-Kirillov dimensions can only occur for non-commutative rings! In fact, a result in

Borho, Walter; Kraft, Hanspeter. Über die Gelfand-Kirillov-Dimension. Math. Ann. 220 (1976), no. 1, 1–24. MR0412240

is that any number in $[2,\infty]$ can appear as a Gelfand-Kirillov dimension. Unfortunately, this paper is in German so I can't read it and possibly explain it on this blog.

Conclusion

This is by no means an exhaustive list of all dimensions, and I probably will talk a lot more about some of these and others in future posts.

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