 So, this lecture on algebraic geometry will cover the concept of dimension. So concept of dimension is intuitively obvious, but surprisingly hard to define precisely. So I'll first give a sort of survey of some of the different attempts people have come up with to define dimension. So first of all, dimension is thought of as the number of parameters needed to define a point. For instance, in two-dimensional real space, you need two real numbers to define a point. Notice this fails because Cantor showed that R2, the plane and the real line have the same number of points and Piano and later Hilbert showed that there are continuous maps from R1 onto R2. So in some sense, points of R2 can be specified by just one parameter, so this naive definition of dimension just fails rather drastically. It does, however, make sense if you can find yourself to smooth manifolds. Next attempt was the Lebesgue covering dimension. This says that a set has dimension at most n if every open cover has a refinement such that each point is in at most n plus 1 sets. So for instance, in two dimensions, if you've got some cover, you expect to find a refinement of it which might look something like this, and you can see there are some points in three open sets, but there are no points in four open sets. Well, the Lebesgue covering dimension works fine for Euclidean space, but again, it doesn't seem to be all that useful in algebraic geometry. So we move on to the classical definition. So this is Brauer, Menger, and Bury's son came up with the following definition of dimension. The idea is the boundary of a topological space should have smaller dimension than the topological space, whatever that means, or the boundary of an object. So a topological space has dimension at most n, where n is minus 1, 0, 1, 2, and so on, if all points have arbitrarily small neighborhoods whose boundary has dimension less than n. So for instance, if you take a point in the plane, you can find a small neighborhood of it whose boundary is one dimensional, has dimension less than two. So this sort of suggests that a plane should have dimension less than at most two, because if you take neighborhoods of points, then you can sometimes find them with boundaries of dimension less than two. Of course, you could choose a stupid neighborhood that had a boundary of dimension two, but then inside that, you would find a smaller neighborhood whose boundary had dimension less than two. So traditionally, this definition is only used for separable metric spaces, and the spaces you get in algebraic geometry are certainly not separable or metrizable in general. However, rather amazingly, this definition does actually work perfectly well for definitions of algebraic sets in algebraic geometry, even though they aren't separable or metric. Next we come to the classical definition, the Krull dimension. This is the supremum of the numbers n for which there is a chain, z0 containing z1 and so on containing zn of irreducible subsets. So here zi is not equal to zi plus one, this is strict inclusion. So for example, if we look at a plane, we can take z0 to be a point, z1 might be some sort of curve, and z2 will be the whole plane. So we can find a chain of length two in the plane, but we can't find chains of length three. Notice these have to be irreducible subsets. If you're allowed reducible subsets, then you could have z0 being a point, z1 being two points, z2 being two points and so on, and you could have a chain of arbitrary length. This only works well for Noetherian spaces. For Hausdorff spaces, all have Krull dimension nought because the only irreducible subsets are points, so you just can't find any interesting chains. Most of the topological spaces you get in algebraic geometry are in fact Noetherian topological spaces, so the Krull dimension is fine. However, there's a kind of a bit of a catch. For Noetherian topological spaces, it's the same as the Brouwer-Menger-Urisson dimension that we had earlier. So the Krull dimension for Noetherian topological spaces, the Krull dimension is the same as the Brouwer-Menger-Urisson dimension, and for none Noetherian topological spaces, the Krull dimension appears to be completely useless. So you may well ask, why do we use the Krull dimension? Why don't we use this dimension, which just works better? Well, we use the Krull dimension for historical reasons. Krull started off dimension commutative ring theory and everybody's just been following him. So we will be using the Krull dimension, although maybe it might have been better to use this dimension, but it doesn't really make any difference. Next, we get various variations of the Hausdorff dimension. So the Hausdorff dimension is got by counting the number of balls used to cover a metric space and then seeing what happens to the number of balls as the radius of the balls tend to zero. This dimension is particularly interesting because it can take fractional values and there are lots and lots of fractals whose dimension is some sort of whose Hausdorff dimension is some non-integer value. However, these Hausdorff dimensions seem to be of no use in algebraic geometry. There are very few metric spaces and sets of fractional dimension just don't seem to turn up. So it's a rather interesting concept of dimension, but not really relevant for this course. Another thing you can do is you can look at transfinite dimensions. So definitions of dimension like the Krull dimension or the Brouwer-Menger-Hausdorff dimension can be extended so they work not just for, so dimension is not just a natural number, but can even be some sort of countable or uncountable ordinal number. Again, that doesn't seem to be a lot of use for this in algebraic geometry. Finite dimensional things account for nearly all the applications. Then there's the deviation of a poset. A poset is a partially ordered set. So we say a poset has deviation at most alpha, where alpha is some ordinal number. It's harmless to think of this as being a non-negative integer for the moment. If for all descending chains, all but a finite number of intervals of the chain have deviation less than alpha. Now, the relation of this to the other dimensions is that for notarian rings, what we can do is look at the poset of ideals. Then the deviation is equal to the Krull dimension if you want to be precise. So the deviation of the poset of ideals of a ring also gives a perfectly good version of dimension. Nice thing about this definition of the deviation of a poset is it works well for modules. So instead of looking at ideals of a ring, you can look at sub-modules of a module and it gives a nice notion of dimension of a module. It also works well for non-commutative rings. So if you want to do dimension theory of non-commutative rings, the deviation of the poset of left ideals seems to work quite nicely. Notice by the... it does have one slight disadvantage. For notarian rings, it works fine. However, there are some posets that don't really have a deviation. For instance, the poset of rational numbers. This means there are some rather weird non-notarian commutative rings that don't have a deviation either. However, as I said, for non-notarian rings, there just doesn't seem to be a good definition of dimension. Next, we move on to some algebraic definitions. The idea is that a set of high dimension has many functions on it. And there are many different ways to define what you mean by many. So there are several different ways of doing this. So the simplest one is suppose B is a variety over a field K. Look at the quotient field of the coordinate ring of B. So this will be some field over K. And we just set the dimension of B to be the transcendence dimension of this quotient field. So we recall the transcendence dimension is the largest number of algebraically independent elements of a field. For example, if B is just two-dimensional affine space, the coordinate ring is K, x, y, polynomials and two variables. Quotient field is the field of rational functions in x and y. And it obviously is transcendence dimension two because here are two algebraically independent elements, x and y, and they generate the field. So this works for algebraic varieties. However, it doesn't work all that well for more general objects in algebraic geometry. First of all, it doesn't really work terribly well for algebraic sets that aren't irreducible because they don't have quotient fields. And you can fudge a sort of definition for them, but it's rather clumsy. Also, more generally, algebraic geometry wants to look at more general objects than algebraic varieties. We want to look at more general schemes, for example. And this definition doesn't work terribly well for schemes. For example, it gives the wrong definition for the dimension of the spectrum of the integers. The transcendence dimension is zero, whereas the spectrum of the integers should have dimension one. So this definition, it actually used to be the standard definition for varieties over fields, but it's been mostly abandoned because it doesn't work so well for more general examples. Next, we have the Hilbert polynomial. This definition works very well, but looks a little bit strange. What it does is it works for local rings A. So we recall a local ring is a ring with a unique maximal ideal M. And what you can do is we can look at the dimension of A over M to the K, which is a polynomial in K for large K. I guess it shouldn't really be the dimension of this. It should really be the length of this. So this polynomial of degree D. And we define the dimension of the local ring to be this degree D. And this looks a bit abstract and artificial, but turns out to be really nice because it's easy to calculate and it gives the right answer. For example, and for an algebraic set, we can define the dimension of each point to be the dimension of the local ring at each point, where the local ring is roughly the space of rational functions that are defined at that point. And the dimension of the algebraic set is, of course, just the maximum of its local dimensions. For example, if you look at the field of polynomials K x1 to xn at the point zero, you're looking at the local ring. Let's take the completion of the local ring and just look at all power series in x1 up to xn. Then if you look at the dimension of A over M to the n, the dimension M is just the ideal x1 up to xn. So the dimension of A over M is 1, the dimension of A over M squared is 1 plus n. The dimension of A over M cubed is 1 plus n plus n plus 1 over 2. And this is growing like a polynomial of degree n in this experiment, K. So the ring of polynomials has dimension n. A similar one is the Gelfand Kirilov dimension for a ring, where this applies to finitely generated algebras over a field and is defined to be the limb soup of the log of the dimension of Rn over log of n, where Rn is the subspace generated by polynomials in some set of generators of length at most n. So informally this says that Rn has dimension that's roughly n to the power of d, where d is the slim soup. When R is commutative, this is the same as the dimension using Hilbert polynomials, but for non-commutative rings, the dimension is 0 or 1, or it can be a real number that's at least 2. So again we get this phenomenon of non-integral dimensions. However, again this isn't much use in algebraic geometry because all rings are commutative and we may as well just use the Hilbert polynomial. Next we can define the dimension of the tangent space of a point of a variety. So this works for non-singular varieties or schemes, but if you've got an algebraic set with a singularity, for instance if there's a double point there, the tangent space at this point is going to look two-dimensional, whereas you only want the dimension to be one-dimensional. So this definition fails for varieties with singularities. However it is actually quite useful because you can use this to define the notion of a singular point. You say that a space is non-singular at a point if the dimension of the tangent space is equal to its dimension. Another somewhat technical definition of dimension of a local ring is the minimal number of elements of a system of parameters. This definition sort of works but is rather unmemorable because no one can ever remember what a system of parameters is. It's something like a set of generators for an ideal containing some power of the maximal ideal. So this definition works but just isn't very intuitive and people don't really use it much. Finally, there are several notions of homological dimension. So you can define homological dimension of things by asking when do various homology groups vanish. So we can have, you could for example look at Schief co-homology groups over a topological space and define the dimension n to be a number such that Schief co-homology groups vanish in degree greater than n. Homological dimension works perfectly well as a definition of dimension. The problem is it takes rather a lot of work to define it because you have to define co-homology groups. So homological dimension works but is rather complicated. So to summarize, for notarian topological spaces or notarian rings, there are several different ways of defining dimension. You can define it to be the curl dimension or use the Hilbert polynomial or use the number of elements of a system of parameters or use the Brouwer-Menger-Urisson dimension and for any reasonable object and algebraic geometry these will all give you the same notion of dimension.