 This lecture is part of an online course on commutative algebra, and will be a sort of survey of the dimension of a commutative ring r. So the idea is we would like to define the dimension of a commutative ring in such a way that, for example, if we've got a polynomial ring in three variables, this is the set of ring of regular functions on three dimensional space, so this should have dimension three. And we want to define a similar sort of dimension for all rings. So I'll stop by just giving a survey of attempts to define definition in mathematics. It's usually intuitively obvious what the dimension of something ought to be, but historically it's been quite difficult to define it rigorously. So originally, for example, people assumed the dimension of something was kind of obvious and was just the number of real parameters you needed to define a point in it. Well this sort of fell apart when Cantor showed that there's a bijection from r to r squared and more generally from r to r to the n for any n. So you can't really define the dimension as being the number of real numbers you need to define a point because you only need one real number. These bijections were not continuous, but you would still hope that if you restricted yourself to continuous maps, then you could define the dimension to be the number of parameters needed. Piano came along and showed there was a continuous map from r1 on to r2. This is his famous space-filling curve. So even if you insist that your map should be continuous, you can still parameterize r2 by just one real number. So this is again a bit of a problem for the definition of dimension. Incidentally, there's one rather weird use of this. Norbert Wiener at one point suggested you should actually define integration over r2 using Piano's discontinuous map to reduce this to integrals over r1. For fairly obvious reasons, this attempt to define integration in n dimensions never really took off. Well, if you restrict yourself to differentiable maps, then indeed there's no differentiable map from r1 on to r2. So you can sort of use that to define dimension. Incidentally, it's quite hard to give a rigorous definition of dimension such that it's easy to work with. For example, here's a problem. Can you prove that two-dimensional real vector space is not homeomorphic to three-dimensional real vector space? Well, this is pretty easy to do if you've done an algebraic topology course, but if you haven't, it may need to think quite a bit in order to show that these two things of different dimension are actually not homeomorphic. So one of the earliest definitions of dimension that worked is the Lebesgue covering dimension. Here we see the Lebesgue covering dimensions at most n if every open cover has a refinement, meaning you can make the set smaller, so that each point is in at most n plus 1 sets. For example, if we're working in two dimensions and you take a cover, so if I draw a lot of open sets like this, you can see at these open sets that there are plenty of them that intersect in three points. And if I draw another open set such that it intersects some of them in four points, then you can obviously shrink this blue set that only intersects the green set in three points. However, by making these sets smaller, you can't get fewer than three with an intersection in general. So this sort of informally suggests that R2 has Lebesgue covering dimension 2. It turns out that R to the n has Lebesgue covering dimension equal to n, but this is quite hard to prove. You can try and prove it if you don't believe me, it's difficult. Well, OK, that sort of works if you're doing general topology. We would like to apply this to the spectrum of a ring. And for the spectrum of a ring, this just fails completely. For example, the spectrum of most rings, any two open sets intersect. So the Lebesgue covering dimension of one dimensional affine space is infinite dimensional, which is certainly not what we want. Next, we come to the sort of classical definition of dimension. So this is the definition that works best in general topology, and was due to Brouwer, Menge and Uri's son. And the idea of this definition is that the boundary of X should have smaller dimension. Well, this is a sort of informal principle because the problem is the boundary of a set X quite often has the same dimension of X if it's fairly hairy. So we say definition, a topological space has dimension N, which can be minus one, zero, one, two, three and so on. So it has dimension less than or equal to N. If all points have arbitrarily small neighborhoods with boundary of dimension less than or equal to, sorry, with boundary of dimension less than N. You've got to be a bit careful how you specify this, because if you take a neighborhood of a point and say R2, you can get a neighborhood with a very complicated boundary which still has dimension two. So you've got to say that inside any neighborhood you can find a smaller neighborhood whose boundary has dimension less than N. And then we find the empty set has dimension minus one and is the only set with dimension minus one. Classically, this dimension was only used for separable metric spaces. However, it turns out to work quite well for notary topological spaces that turn up in commutative algebra like the spectrum of a ring. And notary topological spaces are about as far as you can get from separable metric spaces. So it turns out to be quite surprising this definition still works. Anyway, this classical definition does turn out for most sets it gives you the definition you would guess intuitively. And just as for the Lebesgue covering dimension, it's actually quite difficult to prove that N-dimensional real vector space has dimension N. I mean, you're sort of looking at all possible open sets, and there are rather a lot of those. So, at about the same time as the Brouwer-Menger-Urisson definition, Krull came up with the following definition of a topological space. The Krull dimension N is the supremum of N for which there is a chain Z0 contained in Z1 contained in up to ZN of N plus 1 distinct irreducible subsets. For instance, if we look at the spectrum of Kxy, so you remember the spectrum of this is more or less the affine plane, so it's got lots of points and it's got some irreducible subsets that are one-dimensional and it's also got some big irreducible subsets that are two-dimensional. So we can take Z0 to be a point and Z1 to be a line through this point and Z2 to be the plane containing this, so that gives you three irreducible subsets forming a chain. And this sort of suggests that the spectrum of the ring of polynomials and two variables is indeed two-dimensional. Well, this shows the dimension is at least two. It's actually a little bit tricky to show the dimensions at most two because you've got to sort of study all the irreducible subsets of this, which can be quite complicated. So this definition works for notary topological spaces. So this works well for notarian spaces. For non-notarian spaces, it's quite dreadful. So the curl dimension of r to the n, for example, is just zero because the only irreducible subsets are points. More generally for Hausdorff topological spaces, they all have dimension zero. At most, they all just have dimension zero. For notarian topological spaces, in fact, turns out to be the same as the Brouwer-Menger-Urissen definition. So really, there's no reason at all to use the curl dimension because the Brouwer-Menger-Urissen definition is the same as the curl dimension in cases when the curl dimension behaves well and the Brouwer-Menger-Urissen definition actually works well for a lot of other spaces that curl definition fails on. However, for historical reasons, in commutative algebra, we just use the curl dimension rather than the B-MU definition in practice, they're more or less equivalent. Incidentally, this is only normally used for n being finite. If you're into transfinite numbers, you can, if you like, allow the curl dimension of a space to be some sort of transfinite ordinal. You could have infinite well-ordered chains of irreducible subspaces. I don't actually know of any use for this, so I won't say much more about it. Another definition is the Hausdorff dimension. This is a rather nice definition because it's real but not an integer in general. The Hausdorff definition of dimension goes like this. You cover the set X with balls of radius epsilon. You ask how many balls are needed as epsilon tends to zero. By studying the rate of increase of this number as epsilon tends to zero, you can define the dimension. There might be some sort of exponent which might increase very roughly as some power of epsilon and you can use that to define the dimension. This is nice because it's used for fractals. Is there any analog of this in algebraic geometry? Well, I've never come across any serious examples of fractional non-integral dimension in algebraic geometry. Here is a rather wild research project for anyone. Can you find any serious examples of fractional dimension in algebraic geometry? Notice that Hausdorff dimension actually requires the space should be a metric space which already rules out most of the spaces in algebraic geometry. Then there's a really neat definition called the deviation of a poset. You remember a poset is a partially ordered set. The deviation of a poset is defined as this is followed. The deviation of a poset p is said to be less than or equal to alpha where alpha is an ordinal if, for any descending chain, a naught greater than a1 greater than a2 and so on, of p all but a finite number of the intervals from ai to ai plus 1 have deviation less than alpha. So this doesn't at first seem to have much to do with dimension. The way it works for dimension of rings is suppose R is a notarian ring. Then what we can do is we look at the poset of all ideals. So we don't need to restrict the prime ideals. And then it will turn out the dimension of the notarian ring will turn out to be the deviation of this poset of ideals. We won't actually use the deviation of a poset and commutative algebra. However it's a rather nice definition because this also works for, first of all, non commutative rings. We will only be talking about dimension of commutative rings in this course and we'll be using most of the crawl dimension. But if you want to do it with non commutative rings the crawl dimension gets a bit dicey but the deviation of a poset still works well. It also works nicely for dimension of modules. By the way I should warn you that the dimension of a module in this sense has very little to do with the dimension of a vector space. In fact all vector spaces over fields have dimension zero in this sense. It's a different notion of dimension. You have to be a little bit careful with the definition of the deviation of a poset. First of all there are some posets for example the rational numbers that don't actually have a deviation. This definition just fails. So some rings don't either. Also the deviation of a poset can actually be different from the deviation of the poset with the order of verse. You have to be a little bit careful about that. But this seems to me to be about the neatest and most general definition of definition you could have for commutative rings. Next we come across an old algebraic definition or at least several algebraic definitions. Here the idea is that a set of high dimension should have many functions on it. So the idea is there are in some sense more functions on a two dimensional space than on a one dimensional space. Well obviously to make sense of this dimension you need to have defined what you mean by having many functions. And there are several ways of doing this. The simplest one is as follows. Let's take B to be a finitely generated algebra over a field K. And let's suppose that B is an integral domain. Then we can look at the quotient field K of B. And we can define the dimension of B to be the transcendence degree of the quotient field over the base field K. So remember this is the largest number of algebraically independent elements we can find. For example, if you take K x, y, z as your algebra B then the quotient field is just the field of all rational functions in three variables and the transcendence degree is equal to 3 and this is what you expect the dimension of this to be because this is the coordinate ring of a three dimensional space. And this used to be used as the definition of dimension in algebraic geometry and it sort of works fine as long as you're just working with varieties over a field. However, as soon as you start to do more complicated objects this definition goes wrong so it sort of fails for other rings. So it fails for the ring z. So the spectrum of z has dimension equal to 1 if you're working with Krull dimension. However, it's transcendence degree over, if you take the quotient field it's just the rational numbers which has transcendence degree 0. So it doesn't really work for the integers. Even if you're working for algebras over a field this definition kind of goes wrong. For instance, it doesn't really work very well if B is not an integral domain because then it doesn't have a quotient field. It doesn't work very well if B is a local ring in general. I mean, for instance, if you take the completion of B then you can end up with things that have infinite transcendence degree and so on. So this definition sort of works in simple cases but just goes more and more wrong as soon as you start making things, as soon as you start looking at more general rings. Next, we can look at the Gelfand Kirilov dimension. This works for finitely generated algebras R over a field K and here the algebra can be non-commutative. And what we do is we can define the dimension to be the limb soup of log of the dimension of Rn over log of n. So I'll explain what this means. Rn is the subspace generated by monomials of degree at most n in some set of generators. So the idea is you fix a finite set of generators and look at all monomials in those generators and see how this grows. And you might think this depends on the set of generators but in fact it doesn't. So very informally this says that R of n has dimension roughly n to the d. So it's growing sort of like a d-th power of n, possibly plus error terms. And again d need not be an integer. So this gives some algebraic examples where things can have none integral dimension. As I said I don't know of examples of this turning up in algebraic geometry unfortunately. It'd be rather nice to have varieties of dimension two and a half or whatever. So when R is commutative this turns out to be the same as the dimension that we're going to define using crawl dimension or non-commutative or hill polynomials. I should say that this need not be an integer it can be zero, one or any real number greater than or equal to two. So there are some obstructions to it. If you've done von Neumann algebras von Neumann algebras also have modules of none integral dimension which is sort of possibly related to that I'm not quite sure. Now another way of defining dimension using the number of functions is the Hilbert polynomial. Here what you do is you look at a local ring with a maximal ideal m and you think of a local ring as being functions on a variety near a point. So we're trying to look at how many functions how many elements the local ring has. Well obviously that's just infinite so we've got to find some way of making this finite. So what you do is you look at the dimension of R over m to the k for some integer k. Well we shouldn't really use the dimension we should more generally we should use the length of R over m to the k. So this works if R is the local ring of a variety but in general you need to look at the length and if R is notarian then the length of R over m to the k is a polynomial in k for k large and it's not a polynomial in k for all k for small k it quite often goes wrong and this polynomial might have degree d and we define d to be the dimension of the local ring R. So for example if we take the local ring of all power series in x then you can see that the dimension of k of x over x to the k I'm sorry I'm using k in two different ways here it's a field and here it's an integer and so this dimension is going to grow like k so it's a polynomial of degree one so this we say that the ring of power series in k has dimension one which is what you would expect and this turns out in some sense to be the best definition of dimension for commutative algebra I mean it looks a bit round about but it turns out to be really easy to calculate whereas other dimensions like the Krull dimension although they are equivalent to this they're much harder to calculate directly so if I had to pick one dimension to use in commutative algebra I would probably pick this dimension using Hilbert polynomials because it's the easiest to work with I should say that the dimension is local in other words we should really define the dimension at each point for example if you've got a topological space which looks like a plane together with a line going through a point then this is two dimensional but it's really one dimensional at these points here and it's two dimensional at these points here so the dimension isn't really an inventor of a topological space it's really a function on the topological space that varies from point to point and if we've got an algebraic variety say we can then define the dimension of each point to be the dimension of the corresponding local ring where we can define the dimension of the local ring using say Hilbert polynomials as I mentioned in the previous piece of paper so yes another definition of dimension you could try is what about dimension of the tangent space this seems to be an obvious way to define dimension I mean if you've got a smooth manifold its dimension is just the dimension of a tangent space at each point and we can define the dimension of the tangent space if we've got a local ring R then the tangent space is well it's really the dual of M over M squared as a vector space over the field K which is R over M and for notarian rings R this is finite dimensional and the trouble is it gives the wrong answer and it gives the wrong answer in some very easy cases for example if you just take the variety Y squared equals X cubed so it's just a cusp then the tangent space at the origin has dimension equal to 2 and obviously this is wrong because this is just a one dimensional object and the dimension at this point ought to be 1 and what's going wrong is there is a singularity at this point and it turns out the dimension of the tangent space is the right dimension at non-singular points and it's the wrong dimension at singular points in fact this is one way to define a singular point if the tangent space has the correct dimension then you can say that the local ring is regular or non-singular at that point so the dimension of the tangent space can't be used to define dimension but it does actually give you useful information because it tells you where your variety is badly behaved so this is the number of generators of the maximal ideal M there is actually a variation of this that works we can ask the minimal number of generators of a system of parameters sorry the minimal number of elements not generators so what's a system of parameters? well a system of parameters is something that generates an ideal containing M to the N for some N greater than 1 so if we took N equal 1 then this would just be the number of generators of M but it may be possible to find a set of generators for an ideal containing M to the N that has fewer generators than M and it turns out that this actually gives you a good definition of the dimension of a local ring we don't ask for the number of generators of the maximal ideal but only of some ideal containing a power of the maximal ideal and this then works then there are some more homological definitions which make use of homological algebra and there are several ways of doing this you can define a homological dimension as the dimension beyond which various homology groups vanish well what do I mean by homology group? I mean some sort of group you define using homological algebra for instance you can look at the groups X to the I of M N and ask do these all vanish for I bigger than something or you can look at the groups Tor I M N and ask if these all vanish if I is bigger than something and you can ask for the minimal length of a projective resolution or an injective resolution or a flat resolution of modules so there are several variations of these these all give reasonable definitions of dimension they're not all the same and they're not necessarily the same as the dimension of a ring defined using curl dimension although they quite often are if the ring is sufficiently well behaved and doesn't have singularities so there's quite a lot of homological dimensions that I'm probably not going to discuss very much further okay that's the end of a sort of introductory survey of definitions, possible ways of defining dimension what we're going to do in the next few lectures is actually start proving things about these definitions the main three definitions we're going to be using are first of all the curl dimension secondly the dimension using Hilbert polynomials and thirdly the dimension using systems of parameters and what we'll do is we'll prove that these three apparently totally different definitions all the same for notarian local rings