 This lecture is part of an online commutative algebra course, and we will be giving some examples of the spectrum of a ring. So given a commutative ring R, we defined a topological space called its spectrum in the previous lecture, whose points were just the prime ideals. And the first example, we're going to give answers to the question, why is the spectrum of R called spectrum of R? Of course, there's the traditional answer, it's called the spectrum of R because it is the spectrum of R, but the other reason is given by the following example, let's take R to be the algebra over C, generated by A, where A is the matrix in M, N over say the reals, it could be the complex numbers. Now a matrix as a spectrum, which is just its set of eigenvalues. And the name spectrum for the eigenvalues of a matrix sort of came from, it's related to the fact that in quantum mechanics, the spectrum of an element or an atom is something to do with the eigenvalues of a Hamiltonian operator. So the word spectrum is actually related to the word spectrum in physics in a rather indirect way. Anyway, so this is a commutative ring and there's a finite dimensional algebra over, I guess it should be the complex numbers, finite dimensional algebra over the complex numbers. So here the spectrum A is the set of eigenvalues, which are given as follows. If A has minimum polynomial X minus alpha one to the N one, X minus alpha two to the N two and so on, then the spectrum is just equal to the set alpha one, alpha two and so on. On the other hand, this ring R is just equal to C of X quotient out by the minimal polynomial. And the prime ideals are easy to figure out. In fact, in this case, the prime ideals are all maximal. So the prime ideals are just generated by the elements X minus alpha I. So the spectrum of the ring generated by a matrix in the sense of ring theory is the same as its spectrum in the sense of linear algebra given by the set of its eigenvalues. So that's roughly why a spectrum is called a spectrum. Next, we recall that the spectrum of Z was given by something like this. So here we have a generic ideal zero and it had points two, three, five and so on on it. And now what we're going to do is to kind of decompose this a bit. So we can look at the spectrum of the ring Z where we adjoin the element a half. And the spectrum of this looks almost the same except it no longer has a prime two because we sort of killed it off by adding a half. So here we have the prime three and five and so on. On the other hand, we can also look at the spectrum of the ring Z two where we will explain this later when we discuss localization but we'll just give a quick ad hoc definition. It's a set of all rational numbers M over N with N odd. So what this does is we're sort of adjoining a third and a fifth and so on. So it kills off all these ideals here. All we're left with is the maximal ideal two and of course the generic ideal zero. So what has happened is that we can sort of almost decompose spec of Z into two subsets. They're not quite this joint because they both contain the generic ideal. So one of them contains all odd primes in the spectrum and the other contains just the prime two. So what we would do is we use this one if we wish to focus on the prime two. On the other hand, we use this one if we're getting fed up of all the problems caused by the prime two and we just want to eliminate. So this sort of kills the prime two. So these are sort of two opposite things we can do. We can either focus on two or we can eliminate it and you can see this by looking at what you're doing by looking at the spectrum. We're either removing the point two from the spectrum or removing almost everything else. There's also a map from this ring here. So there's a map from the ring Z to the ring Z over two Z where and this gives a map of the spectrum in the opposite direction. So if we draw the spectrum of Z over two Z, it's just a point and you can think of this point as being embedded in this spectrum here. So we really can write the spectrum of Z as a disjoint union of this set here and this set here if you want. Next let's take a look at the spectrum of the Gaussian integers and there's a map from Z to the Gaussian integers. So this induces a map of the spectra going the opposite direction. So we can draw the spectrum of Z a bit like this. So here's the generic point and then we have all the various primes two, three, five and so on. And similarly the spectrum of Z of I also has a generic point. So this is a principal ideal domain. So its points correspond to prime ideals of Z of I and we can figure out the prime ideals of Z of I by decomposing the prime ideals of Z. So two sort of equals one plus I squared up to units. So we get just one prime ideal here was three decomposes as three, it remains prime here but five decomposes as two plus I times two minus I in the Gaussian integers. So what happens is we kind of get two different primes here two plus I and we'll have to make this branch out. We get a second prime two minus I and similarly we can carry on like this. So we've got a prime seven and we've got a prime 11 and we've got two primes corresponding to three plus two I and three minus two I lying above prime 13. So you can sort of see the decomposition of primes integer primes into primes of Gaussian integers as a homeomorphism of topological spaces from the spectrum of Z of I to the spectrum of Z. And you can see that the fibers of this homeomorphism kind of correspond to the decomposition of primes. If you remember from number theory, primes of the form one mod four break up as a product of two primes. So the fibers two elements and primes of the three mod four just break up as one prime and the prime two ramifies which sort of corresponds to the fact there's really a sort of branch point here. So the spectrum of a ring gives you a sort of geometric way of visualizing decomposition of primes in algebraic number fields. Now let's look at the spectrum of the polynomial ring in two variables over an algebraically closed field. And this starts to get a little bit complicated. First of all, we've got some maximal ideals which look like X minus alpha, X minus Y minus beta. And these are going to correspond to the point X, Y in C squared. And then we have some prime ideals of the form F where F is an irreducible polynomial. And then we've got a generic point which is just the prime ideal zero. And if we draw pictures of these, first of all, we just get points for each point of C squared. So here I've drawn C squared and you have to think of C as being one dimensional. So this point corresponds to the ideal X minus alpha, Y minus beta. So this is alpha, this is beta. So we get a lot of maximal ideals corresponding to points. And of course, as I said before, the topology is a bit funny because the topology isn't the same as C squared. The closed sets are just finite unions of points. But then we get some extra points in the spectrum corresponding to these prime ideals. And these points are going to be one dimensional points. So this is the point corresponding to F. So F might be some polynomial in X and Y. And so we think of this as a point but it's also one dimensional in some meaningless sense. And what I mean by this is the closure of this point contains various other points corresponding to maximal ideals. So the point X minus alpha, Y minus beta lies on this point here. If F of X, so if F of alpha beta equals naught. So we can think of this point here as corresponding to the curve given by the polynomial F equals zero. And finally, of course, we have a sort of generic point corresponding to ideal zero, which in some sense is two dimensional. So really huge point because its closure contains everything else. So the closure is everything. Such a big point that its closure is an entire complex plane. So as usual, it's impossible to draw the spectrum in Euclidean space because it's none house store from the best you can do is this funny sort of picture. Well, instead of a polynomial ring, we can look at a power series ring. So what does the power series ring X, Y look like if we look at its spectrum? Well, here it's got a closed point. And this time it's only got one closed point, which is the ideal generated by X and Y, which sort of corresponds to the point naught, naught in C squared. Then we've got some prime ideals, which are none maximal, again, corresponding to F being an irreducible power series. And of course, we've got a generic point zero. And if we draw a picture of this, we've got a closed point here and we've got prime ideals. And these sort of correspond, you can kind of think of these as being infinitely small curves through this point. So this might be a curve F of X, Y equals zero in some sense, except that since F is a power series, it's not really a curve. It's just a sort of ghost of a curve. And then there's a sort of generic two dimensional point. I guess I should have used colors that more distinct than blue and green, but it's too late now. And you can think of this spectrum as being contained in the spectrum of the ring of polynomials in X and Y. Because if you look at the ring of polynomials, X and Y near zero, well, it's got a point at the origin. And it's also got various one dimensional points in some sense passing through it. And it's also got this generic two dimensional point zero. And if you look very closely near the origin, you can see it sort of looks a bit like that. So the sort of ghosts of algebraic curves look a bit like zeros of power series and so on. So there's a, so the homomorphism from the ring of polynomials, the ring of power series gives a sort of homomorphism from the spectrum of this ring to the spectrum of this ring. There's also an example of the spectrum of the ring of polynomials in the integers, but I already made an entire video about this in the algebraic geometry course. So what I'm going to do is to try and put a link to it to the video I made about this. And if I get everything right, there'll be a sort of little white circle somewhere in the top right-hand corner of this video. And if you click on, you can see a different video I made about the spectrum of this ring here. There's a fair amount of overlap between my commutative algebra lectures and algebraic geometry lectures because commutative algebra and algebraic geometry are almost the same, but never mind. Anyway, so instead of that, I will give an example of the spectrum of a Hecker ring. And Hecker rings come from the theory of modular form. So what I'm going to do is I'm going to explain the entire theory of modular forms, which normally takes an entire lecture course. And I'm going to try and do it in about two minutes. So here are two modular forms. There's the modular form E12, which is 691 over 65520 plus sum of sigma 11n q to the n, which is equal to 691 over 65520 plus q plus 2049 q squared and so on. So sigma 11n is the sum of the 11th powers of the devices of n. And another modular form is delta 12, which is q times product over n greater than 1, one minus q to the n, all to the power of 24, just q minus 24q squared and so on. And these are examples of modular forms of level one and weight 12. And if you want to know what that means, you have to go to a modular forms course. And there's something called a Hecker algebra acting on these and the Hecker algebra is spanned by Hecker operators Tn and the Hecker operators Tn act on this two dimensional space spanned by modular forms as follows. So Tn maps E12 to sigma 11n E12 and it maps delta n to tau of n times delta, where I forgot to say this is the sum of n of tau n q to the n. So these numbers are Ramanujan's famous tau function. So we've got a ring, satis generated by Hecker operators acting on this space and these Hecker operators are given explicitly like this. And what I want to do is to work out what is the spectrum of this Hecker algebra? So the spectrums of Hecker algebras are of great interest in number theory. For instance, Andrew Wiles' proof of Fermat's last theorem involved a very detailed study of Hecker algebras acting on modular forms. So it's a sort of generalization of this example. And what I want to do is to draw a picture of this. Or first of all, we need to work out what the Hecker algebra is without actually mentioning modular forms. And the Hecker algebra can be defined in a more elementary way. It's a subring of z times z of all numbers of the form Sigma 11n tau n. So it should be generated by these. Now, if we draw z times z, here I'm going to draw it in the very naive way where we just draw a point for each element. So here's nought nought. And it certainly contains the identity element, which is one, one. And the question is, does it contain any other elements? Well, it obviously does. I mean, it contains Sigma 2 tau 2, which is somewhere way over there. And so since it contains the point 11, in order to find the remaining elements, we want to know are there any congruences? Sigma 11n is congruent to tau of n mod big n. And so if there's a congruence like that, then it will restrict the pairs mn in the Hecker algebra because they will have to satisfy this congruence. And there is one found by Ramanujan, which says Sigma 11n is congruent to tau of n modulo. This is probably not a number you would guess, 691. For example, if n equals two, Sigma 11n is 2049 and tau of n is minus 24. And you can check 2049 plus 24 is three times 691. So this is a very famous congruence. And I want to draw a picture of this congruence. So the Hecker ring consists of all elements in z times z, where first of all, we have the elements n comma n, and then we have the elements n plus 691, n, it's n plus 691, n and so on. And generally we have all numbers mn with m congruent n mod 691. So the ring sort of looks like lots of numbers like that. I haven't drawn this to scale because this should really be 691 units over there. And now we want to know what is the spectrum of this ring? So here's the ring R, it's just a set of pairs in z times z, with these numbers congruent to 691. And we've got a map from R to z times z, to z times z, which just maps the element mn to m in this z and n in that z. So we have a map from the spectrum of z, union the spectrum of z to the spectrum of R. So we've got two copies of the spectrum of z. So here they are, there's two, three, five, here's we've got two, three, five and so on. And here this number is a prime ideal and it corresponds to taking mn to m mod p. So we're mapping m to z over pz for some p. And similarly, this would correspond to mapping mn to n modulo p where p is two, three or five. And the question is, do these two lines meet somewhere? And you can see they actually do meet at the prime 691. Because if we're mapping this to m modulo 691, it's the same as mapping it to n modulo 691 because of Ramanujan's congruence. So the spectrum of this ring looks like two lines, which are copies of the spectrum of the integers, except they meet at this 1.691. So this is a sort of picture of Ramanujan's congruence. So this gives us a way of visualizing modular forms. If you draw the spectrum of the Hecker algebra, then you find each modular form kind of consists of a line inside the spectrum. And these lines sometimes meet. And all these meeting points correspond to congruences between modular forms, which number theorists get very excited about. Incidentally, this is something called an Eisenstein prime. If you come across that phrase, an Eisenstein prime is roughly where two of these lines meet and one of these lines corresponds to the so-called Eisenstein series. I just finished by mentioning there's a funny relation between Ramanujan's congruence and the leach lattice of all things. So Ramanujan's congruence can be deduced from the following formula. It says 691 times theta lambda is equal to 65520 times e12 minus delta. Now this is the theta function of the leach lattice, which is sum over lambda in the leach lattice of q to the lambda squared over two. And all you need to know is that since this is counting things, all the coefficients of this are non-negative integers. So this 691 here is what is giving the congruence between these coefficients here. So in some sense, the reason why these lines meet can is related to the existence of the leach lattice, which is rather famous because Conway discovered that its automorphism group was a new sporadic simple group. So there's a sporadic simple group hiding in this picture here. So the next lecture, we'll be discussing properties of the topology of the spectrum of the ring. So we've seen the spectrum of the ring has this rather bizarre, non-Hausdorff topology in general. So I want to explain how you deal with non-Hausdorff topologies.