 This lecture is part of an online commutative algebra course and will be a somewhat technical lecture about the dimension of a local ring. So in the previous lecture we had several different definitions of the dimension of a local ring and what we want to do is to show that they're all the same. In this lecture we're going to prove an inequality between two of them. So we're going to show that the dimension of a local ring r is less than or equal to the dimension of a local ring r. This is defined using Hilbert polynomials and this is defined using a system of parameters. Now recall what the system of parameters are in a moment. And there was a third definition which was the dimension defined by Kroll. And we're going to prove the following inequalities for this in later lectures. We're going to show that the Kroll dimension is less than the Hilbert dimension which is less than the parameter dimension which is less than the Kroll dimension which will show that they're all the same. So this lecture we're just talking about this bit and ignoring the Kroll dimension. Before doing this I better clear up some confusion in the last lecture. I get confused about this every time I talk about it. And this is there are two different Hilbert polynomials for a local ring because there are two graded rings you can define. You can define the sum of r over m to the n plus 1 or you can define the sum of m to the n over m to the n plus 1. And these both correspond to polynomials. If you look at the degree of lambda of r over m to the n plus 1 or the degree of lambda of m to the n over m to the n plus 1. These are both polynomials in n. However the degrees are different. The degree of the left hand side is equal to 1 plus the degree of the right hand side. So when you talk about the degree of a Hilbert polynomial for local ring it's always a little bit ambiguous because you should really specify which of these rings you're talking about and I forgot to do that last lecture. So which of these degrees is the dimension of r? Well this degree is the dimension of the local ring r. Okay so now let's show how to prove this inequality. So we just recall what a system of parameters is. So a system of parameters is just a set of generators of an ideal q with m contained q contained in m to the r for some positive integer r. And I gave an example last lecture of an ideal q with a system of parameters that had fewer generators than the ideal m. So what we're going to do is let's look at the graded ring where we take sum of r over q to the n plus 1. So I'm replacing m by q here. And this is equal to r over q plus q over q squared and so on. And now you notice that this is an artinian ring. So it's artinian because q contains a power of m to the r. So this actually has finite length over r. So this is an example where we're going to use the Hilbert polynomial for a graded ring whose degree zero piece is no longer a field but an artinian ring. But anyway we find that the length of r over q to the n plus 1 is a polynomial in n provided n is sufficiently large. And what we want to do is to compare the degree of this polynomial with the number of generators of the ideal q. And for that we take a look at the other graded ring. We take sum of q to the n over q to the n plus 1 which is r over q plus q over q squared and so on. And now we notice that the number of generators of the ideal q is equal to the number of generators of this algebra. Sum of q to the n over q to the n plus 1 over the algebra r over q. And now we know that the degree of the Hilbert polynomial of this by which we mean lambda of q to the n over q to the n plus 1 is less than the number of generators of the algebra sum of q to the n over q to the n plus 1. At least provided the number of generators of this is positive. So the degree of lambda r over q to the n plus 1 is 1 plus this. So it's at most the number of generators of this algebra which is equal to the number of generators of q as an ideal. So this is the cardinality of a system of parameters. And this is the Hilbert polynomial of something. The only problem is this isn't quite the Hilbert polynomial we use to define the dimension of the ring. So we need to compare these two Hilbert polynomials. So you remember we define the dimension of the ring as the degree of the Hilbert polynomial where we take the lambda of r over m to the n plus 1. And we want to compare this with the degree of the polynomial lambda of r over q to the n plus 1. So here we've changed. So this is what we use as the definition of the dimension. And this is the expression we came up with at the end of the previous slide or piece of paper, whatever. So how do we compare these? Well, what you do is we notice that we've got maps from r over m to the nr, maps on to r over q to the n, which maps on to r over m to the n. So we find that lambda of r over m to the nr is greater than or equal to lambda of r over q to the n, which is greater than or equal to lambda of r over m to the n. I guess I should have changed n to n plus 1 there, but it doesn't really matter. So we find that f of nr is greater than or equal to g of n, which is greater than or equal to f of n, where f is the Hilbert polynomial that we get by using the ideal m. And this is the Hilbert polynomial using the ideal q. And what we see from these inequalities here is that f and g have the same degree. So apart from having the same degree, they may be totally different polynomials, but they both grow at about the same rate if you rescale the variable, and that implies the polynomials have the same degree. So this now proves our first inequality because the dimension of r using the Hilbert polynomial is equal to the degree of the length of r over m to the n plus 1, which by this inequality is equal to the degree of the length of r over q to the n plus 1, which we showed is less than or equal to the number of generators of q for any q, so that this dimension here is less than or equal to the cardinality of any system of parameters. Okay, so that's the proof of the inequality for this lecture. The next couple of lectures will be proving the other two inequalities that I mentioned at the beginning.