 So this lecture is part of an online commutative algebra course and will be about the spectrum of a ring So in the last couple of lectures, we covered two obvious ways of drawing pictures of a ring either by Drawing a point for each element or point for each basis element. The spectrum of a ring is one where you draw a point for each prime ideal So the first two methods Sort of work quite well in a few simple cases, but for most rings, they just don't work very well. I mean most rings It's not very practical to draw every element of the ring and most rings are not vector spaces over fields So you can't draw a basis for it But the drawing a point for each prime ideal works quite well for all rings and I'll start with some motivation So suppose X is a compact House dwarf space then I'm going to take R to be the ring of continuous functions on X Under point-wise multiplication, of course And the idea is that X is a good picture of the ring R so What we do is we draw a picture of X and think of the ring R as being the ring of continuous functions on that space and that's sort of the idea of of defining the Spectrum of a ring. So the ring R in this case is something called a commutative C-start algebra So the theory of C-start algebras was developed Sometime before growth and deconvented schemes and was probably rather influential on on the definition of a scheme Although it's the early history is a little bit unclear So a C-start algebra basically means it's an algebra over the real numbers and it has a norm Where the norm of a function is just the supremum over all X and space X of the absolute value of f of X So there's a whole theory of algebras with a norm that we're not going to worry about too much because that's many of interest to analysts but anyway, the point is that the Space X can be recovered from the ring R So we can go from X to R by taking the ring of continuous functions We can also go back from R to X. So suppose we're given R We can reconstruct X As follows the points of X correspond to the maximal closed ideals of R And the way this works as follows um point X just corresponds to the ideal of points So the ideal of so the ideal of functions with f of X Equals zero so it's just the functions vanishing at X Um, and it follows easily from the stone via strice theorem My strice seems to be turning up rather often in this course for completely different reasons so the stone via strice theorem describes the um Conditions under which a closed sub algebra is equal to the whole ring and from this it's easier to follow that all closed maximal ideals of R are of this form So we can reconstruct the points of the space X. We can also reconstruct the topology on X We can reconstruct the topology in two ways. We can either say a basis of open sets Is given by the functions df um, so I guess I better use the notation that eyes and butters using u f is informally the points where f is not zero well What do we mean by f being zero at a point of x? Well, this is just the points m with f Not in m. So you remember the points are really prime ideals of the ring are so we can just look at the Points where f is none zero and these form of basis of the open sets um, alternatively we we can take any closed ideal And we can form from it the set of maximal ideals containing i which is a closed set Of the space X you can think of it as being the common zeros of all elements of the ideal i So there are two ways of describing the topology and we're going to use these two ways later on for arbitrary ideals By the way, different closed ideals can give rise to the same closed set of X So we don't get a one-to-one correspondence between closed sets and ideals And the idea is to we can copy this for any commutative ring Are we just take a set X to be the maximal ideals of the ring are and define the topology as above in either of the two ways Well, that gives you something called um The maximal spectrum of r Sometimes denoted by spec m of r And it sort of works, but not very well. There's a problem with it as I will explain So if you're looking at c star algebras, if you've got any two c star algebras Um r and s and you've got a map from r to s Then you get a map from the maximal spectrum of s back to the maximal spectrum of r um Let's call that X And this is this can be defined as follows if m is a maximal ideal Of s then you can just look at f to the minus one m as being a point of r Well, there's a bit of a problem if you try doing this for arbitrary rings the problem Is that f to the minus one of a maximal ideal? might not be a maximal ideal For example, if f is the map from the integers to the rational numbers And you take m to be the maximal ideal zero that f minus one m Is again the ideal zero, which is no longer maximal in z. So If we work with maximal ideals We can't really get a map between spaces whenever we've got a map between rings and it's really convenient that homeomorphism between rings ought to correspond in some way to um continuous maps between the corresponding spectra So, um, this isn't too difficult to fix If we just think about what the problem is. So let's let's analyze what the problem is So if you've got a map from rings r to s You remember a maximal ring of i s corresponds to a map from s onto a field So a maximal ideal Is a field that's more or less the definition of a maximal ideal of a ring. It's one such the quotient is a field And now we obtain a map from r to k But the image of r might not be the whole of k So it's a subring of k So it is an integral domain But need not be a field so If this map here is f And the maximal ideal here was m an r over f minus 1 s Is a sub ring of k So it's an integral domain So we can at least say That this is a prime ideal of r You remember an ideal is prime Corresponds to saying that r over p is an integral domain Equivalently this means that if x y is in p This implies x is in p or y is in p, which is an alternative form of this definition You see that's just saying x y is zero in the quotient and that means either x is zero or y is zero Now fit that the problem around here is that a subring of a field is not a field But a subring of an integral domain is an integral domain So the subring of an integral domain is an integral domain And so if f is a homomorphism of rings from r to s and m is a prime ideal in s That way around So f the minus one of a prime ideal Is prime So this suggests that we shouldn't work with maximal ideals, but we should work with prime ideals So now we define the spectrum of r As follows so the points Just correspond to prime ideals Of the ring r And the topology Is defined in either the two ways above we can either choose a basis of open sets given by u of f Which is the set of prime ideals such that f is not in p You can think of this informally as being the points where f is none zero although that's not really quite true Or we can think for for any ideal z we can look at the set of the closed sets Are of the form z of i Which are the set of prime ideals such that i is contained in In the ideal p. So these are two equivalent ways of defining the topology Um, so Um, it might be an idea just to check that this really does give a topology So check That the z of i form a topology So these form the closed sets of a topology not the open sets of a topology So we've got to check that it's closed under unions and that follows easily Because sorry not unions intersections. We've got to show if it's closed under arbitrary intersections because the intersection of all the ideals z of alpha is just the z corresponding to the Ideal generated by all the z of all the i alpha Which is fairly trivial checking that the union of two Of these sets is still one of these sets is a little bit trickier It's actually to equal to z i j where this is generated By all products i j for i in i And j and j it's the usual product of two ideals And now it's not quite obvious that a prime ideal containing One of these two Is the same as the prime ideals containing this and for this you you need to use the fact that if p is a prime ideal then i j is contained in p if and only if i Is contained in p or j is contained in p. So this is an exercise um, so these these sets really do form the topology of On a space And now we should give a few examples of the spectrum of space Let's start with some trivial examples. So first of all if r is the zero ring with no None zero elements. So one is equal to zero the spectrum of r Is the empty set and you actually need to think about this for a few seconds because You have to try and wonder whether or not zero is a prime ideal and zero is not a prime ideal because part of the definition of integral domain Must it must have One is not equal to zero um That's not something you can argue about it just part of the definition It's not something you can figure out by thinking deeply about it So the next example is let's take r to be a field Well, here there's only one prime ideal So the spectrum of r is just the ideal zero Um, and this is just a point So this sort of shows Corresponds to the fact that fields are the easiest sorts of commutative rings their spectra are particularly easy. They're just points So now let's look at a polynomial ring. Let's take r to be the ring of polynomials over the complex numbers So what are the prime ideals of this? Well, this is a this is a principal ideal domain So all ideals are principal. It's really easy to work out the prime ideals There are two sorts of prime ideals. First of all, there are the maximal ones which perform x minus alpha alpha a complex number And there's none maximal one Well, there's just the ideal zero So the spectrum of r Is just the complex numbers union infinity Well, it's not quite because we should think about what the topology is But we shouldn't really think of this as being a pointed infinity. It's more like um, a sort of generic point I mean when you think of a space as being having a point for each complex number plus an extra point You sort of automatically think well, it's something like the Riemann sphere It's just the complex numbers with a point of infinity. And this is just wrong Um, the topology is quite unlike the Riemann sphere and this doesn't behave like a pointed infinity at all So, um, let's first of all look at the topology on the maximal um Um spectrum so so so the spec m of r Looks like c if you ignore the topology, but the topology is a bit weird So the closed sets Are c And all finite sets Remember if you've got an ideal it's just Generated by some polynomial f which is x minus alpha 1 x minus alpha 2 so on x minus alpha n So the only prime ideals maximal ideals containing it are the primae Are the ideals corresponding to the complex numbers alpha 1 alpha 2 up to alpha n in particular this topology Is not house dwarf Um, all open all none empty open sets are dense. I suppose I should say the topology of the other Oh, yes, I do remember to say that um, if you add the extra point Things get even weirder because the point zero is not closed In this topology in fact its closure Is the whole space spec Of of of r So the maximal spectrum is bad enough It's none house dwarf, but at least the individual points are closed Once you start adding this point as well the topology gets even weirder This makes it rather difficult to draw. I mean you can try and draw the complex numbers um as um You you can draw a picture of them as being the usual Euclidean plain Except you've got to remember that you don't have the usual topology on the Euclidean plain the closed sets are just finite sets But when you try and draw this extra point, um Best you can do is you can draw it as a sort of fuzzy generic point um So this is a sort of fuzzy generic point That is everywhere dense So it's a sort of it may be a single point But it's a really big point because its closure contains the whole space um, and this is the main problem about Drawing rings as a spectra. It's really hard to draw them in Euclidean space You have to remember that apology is much weirder than Euclidean space topology, but You'll get used to it after a while So now let's try and do the spectrum of z um again We have the maximal ideals Well, the maximal ideals are just two three five Seven and so on just corresponding to primes and we should also get an extra prime ideal Zero again z is a z is a principal ideal domain So it's easy to find all these ideals and what you notice is that terminology has got messed up the prime ideals Don't quite correspond to primes. They correspond to primes together with the ideal zero and There's nothing you can do about this. You just have to accept that prime ideals don't quite correspond to primes um, so how do we draw a picture of this? well, the closed sets Again easy to work out because they just correspond to ideals of z which you know, so they're finite subsets of two three five and so on and You notice that nought is not in the set So you can find out subsets of these things without nought and the whole space So we've again got this problem that these sets here these points here are closed points But this point of the space is not a closed set and again, it's rather hard to draw this space what you can do is you think of the prime ideal zero as being a big one dimensional point And yes, I know points can't be one dimensional. That's just too bad I'm going to call this a one dimensional point because it sort of behaves as if it were one dimensional And embedded in this huge one dimensional point of these zero dimensional points corresponds to the prime ideals two three five seven 11 and so on So this is the best we can do trying to draw this space in euclidean space as as I just repeat that There is no good way to draw a picture of this euclidean space because any subset of euclidean space is house dwarf And this space is very far from being house dwarf. So we just have to fake it like this Then we can look at the spectrum Of r of x. Let's take polynomials over the reals Now you might think if the spectrum Of complex polynomials is complex numbers union this generic point Then maybe the same is true of the spectrum of the reels. Maybe we just get the real numbers together with an extra point Well, that's not what happens. We do indeed get a copy of the reels corresponding to the maximal ideals x minus alpha for alpha in the reels And we also get a non-maximal ideal zero And this is as usual not closed But we also get some extra maximal ideals Because there are some irreducible polynomials of degree greater than one So We get all Polynomers with two complex roots. So b squared minus four ac It's less than zero. So this is two roots x plus or minus i y with y not equal zero So you see the point Points of the spectrum correspond to the generic point naught plus Sets consisting of x plus i y x minus i y For complex numbers x and y So if y is zero, this just gives us the real point and if y is none zero we get a pair of complex points So the spectrum is equal to the generic point union the complex numbers folded in half In some sense. We sort of can't we identify each complex number with its complex conjugate So in general for fields a similar thing happens if you if you try and form the spectrum of k of x The points correspond to orbits Of the Galois group of the algebraic closure On the algebraic closure of k Together with the point zero So the spectrum of a none algebraically closed field is a little bit more complicated than you might expect you You also need to know what all algebraic extensions of the fields look like Okay, so next lecture. I'll be giving more examples of the spectrum of Of rings