 This talk is part of an online course on commutative algebra and will be about the co-limits and limits. There's been mostly co-limits and limits of modules. So first of all, I'll start by recalling what a co-limit or limit is in a very general and abstract way. And then I'll give lots of examples, which will hopefully make it clear what it is. So for a co-limit, what we do is we take a category C and the category has some objects and some morphisms between them. So I'll sort of draw a picture of a category here has four objects and a few morphisms between them drawn as arrows. And we take a font to F from the category C to the category of modules over a ring. Or more generally, we could do a function to any category, but we'll be mostly talking about co-limits and limits of modules over a ring. What this means is we choose a module for each object of the category and we choose morphisms between the modules corresponding to morphisms of the category C. So we might have modules M1, M2, M3 and M4 and morphisms between them corresponding to these things here. And then a co-limit is a sort of universal object that all these modules map to. So what this means is there is a module M and there are morphisms from all these M i to M making everything in this diagram commute. And furthermore, M is universal with this property. This means that if we take any other module M prime with maps from all the M i to M prime making everything commute, then there's a unique map from M to M prime. So M is the sort of best possible object we can find with maps from all the M i's to M making it commute. So this element M is called a limit, sorry, a co-limit of this functor from C to modules. We define limits in the same way except that instead of having maps from the M i to M we have maps from M to all the M i's. So let me call this N rather than M. So limit is going to be a map, an object with maps to all the M i's making it commute which is universal in the sense that if we've got any other object with maps to all the M's then there's a unique map going like that. Well, it's rather abstract definition is a bit hard to understand when you first come across it. So what I'll do is I'll just give lots of examples of limits and co-limits and hope that this will make it clearer what's gonna happen. So let's start with a simple example. Let's just take the category C to consist of two points with no morphisms between them. So the only morphisms are going to be the identity maps for these two objects. So what does a limit or a co-limit mean? Well, it means you take two modules M one and M two and we're trying to find a module M with a universal which is universal for maps from M one and M two to it. And this co-limit is going to be the direct sum of M one and M two. And you can see this as a co-limit because if you take any other module M prime with maps from M one and M two to M prime then there's obviously unique map from M one plus M two to M prime making this commute. So M one plus M two is the co-limit of this particularly simple function taking that point to M one and that point to M two. Somewhat confusingly it turns out the limit is also the direct sum of M one and M two. This is in general limits are not the same as co-limits but they happen to be in this particular case. So again, if we take the direct sum of M one and M two this maps to M one and M two by projection and it's also universal for this property because if we take any module N with a map to M one and M two then it's easy to see there's a unique map going like that. So this is also a limit. So direct sums and I guess this is really a direct product but products and modules have to be the same as sums at least finite ones. So that wasn't very difficult. Now let's try taking C to be an infinite number of points. So we've got modules M one, M two, M three and so on. And now there's a difference between the co-limit and the limit because the co-limit is the sum of M one and M two and M three and so on. So it's universal for maps like that. So this is the co-limit whereas the limit is now the product which maps to all of these. And the difference between the sum and the product is the sum is a sub module of the product. It consists of all elements of the product such that all but a finite number of entries are zero. So in this case, the limit is no longer equal to the co-limit. This is actually something funny that happens in Abelian categories. The direct sum happens to be the same as the direct product. So now let's take some categories which have morphisms in them. So here I'm going to take C to be the category with two elements and two arrows between these objects. So now let's take a functor from C to the category. It means we have to choose an object for each object of the category. So we pick two modules and now we've got to pick a morphism for each of these two diagrams. And I'm going to map one of these to the zero morphism and the other one to some random morphism F. Now we want to know what is the co-limit? Well, a co-limit is going to be a module M with a map from M2 to M and a map from M1 to M such that everything commutes. But if you think about it a bit, what this means is that M is equal to M2 modulo the image of M1 under F. So here we see that a co-limit is more or less a quotient. If M1 happens to be a submodular M2, then it's exactly a quotient. In general, it's a quotient of M2 by the image of M1. Well, what happens if we try taking a limit of this? Well, for a limit, we've got M1 mapping to M2 by these two maps here. So what we want is a module N which maps to M1 and it maps to M2 and this diagram commutes so this must be the zero map. So the composition of this and F must be the zero map. In other words, N must map to the kernel of F. And in fact, you can see that N is actually equal to the kernel of the map from F, from M1 to M2. So in this case, we see the limit is the kernel of this morphism. So co-cernels or quotients and kernels are special cases of co-limits and limits. Now let's have a look at some slightly more complicated categories. So now suppose we take the categories C to be the following category. I'm going to take an infinite sequence of objects and there are going to be morphism between like this. And you also have other morphisms. For instance, there's a single morphism from there to there but it should be reasonably obvious what this category is. If you want, you can identify this category with the natural numbers and you say there's a morphism from a natural number to another if and only if the first one is less than or equal to the second. And what do co-limits look like? Well, we have to take a functor from this category to the category of modules and this means we have a sequence of modules with each module having a morphism to the next one. And now suppose each MI is actually a sub-module of MI plus one. Then what's the co-limit? Well, the co-limit means we have to find a module M and there must be maps from all the MIs to M making everything commute. And you can then see that M is just the union of all the MI. That's in the special case that the MIs are sub-modules of the next one. So a co-limit in this case is just a union. What's the limit? Well, the limit isn't terribly exciting because for the limit we have to take a module N which is universal for maps going like that. Well, obviously in that case, everything is just determined by the map from M to M0 because then this map must be the composition and so on. So the limit is just M0 which is a rather trivial sort of operation. So the limit of a diagram like that just isn't interesting. If these maps are not inclusion, then the co-limit can be a little bit more complicated. For example, suppose we take Z mapping to Z mapping to Z and suppose each of these maps is zero. Then what's the co-limit? Well, it must be a module such that there are maps from all these Zs to M and everything must commute. Well, this map here must be zero because it factors through this zero map and some other maps here. So this is zero and similarly this is zero and this is zero and so on. So you can see from this that M must actually be the zero module. So the co-limit of this map where they're not inclusion is actually zero even though all these modules are non-zero. So co-limits can sort of unexpectedly collapse things. We can also take limits and co-limits with maps going the other way. So now suppose C is the following category. So now we're going to take the maps going to the left rather than the right. Well, in this case, the co-limit, so what's the co-limit? Well, the co-limit is going to be some element M with maps going like that. So here's M zero and M one and M two. And obviously this map here is determined by this map because it's just the composition and so on. So everything is determined by the map from M zero to M. So we see the co-limit is just M zero. So it's not very exciting. What about the limit? Well, this is my limit. So it's not very exciting. What about the limit? Well, this is more interesting. It's sometimes called the direct limit of these modules here. And what we need is a map from a module M to all these MIs making everything commute. And the construction of this is as follows. M is given by as follows. You take the product M zero times M one times M two, except you take the sub-module of elements of this, of what elements M zero, M one, M two, and so on, such that M i and M j have, sorry, so that M i has image M j whenever i is greater than j. So we must take an element in M zero and an element in M one and an element in M two and so on. And these must all be compatible. The element in M zero we chose must be the image of the element in M one and so on. So this is also something that's called a projective limit and will be used quite a lot when we discuss completions. What happens in practice is we might take an ideal R, so we might take an ideal i inside a ring R, and we might, for instance, form a series of modules R over i, R over i squared, R over i cubed, and so on. So if you've seen the construction of the periodic numbers, this is what you do to construct the periodic numbers. So next you can ask what happens if you've got some sort of really complicated category. Suppose you've got a category that looks like this and there are lots of maps going like that all over the place. I don't know. Some sort of random collection of modules like this. So what does the co-limit look like? Well, the co-limit can be given by generators and relations sometimes. So what we can do is suppose each of these modules is just a copy of the ring R, just to be precise. And suppose each of these is a copy of the ring R. Then what we can think of is that for each of these modules, we get one generator of the module. So we might take a generator of the module R, say one and call that M1 and call this M2 and call this M3 and so on. And then our module is going to be generated by the elements M1, M2, M3, and so on. Except we have to identify some things here. So for each of these, we might get some sort of relation. So if you've got an element X here, say, mapping to the element B here and the element C here, then we would have the relation that B, which is an element of M2, is equal to C in this module M3. So we've got some relation identifying some multiple of M2 with some multiple of M3. And similarly, we can get relations from these other elements here. So co-limits are sort of very closely related to presenting modules by generators and relations. There are various special cases which have a rather confusing collection of names for them, which I quite often get wrong because I always get a bit confused about this. First of all, there's a direct limit. So this is a special case of a limit for a directed post-set. So what's a post-set? Well, a post-set is just a partially ordered set. And we can think of this as a category with at most one morphism between any two elements A and B. And to turn a post-set into a category is very easy. Suppose we've got a post-set with, say, four elements, and that element is less than that, and that element is less than that, and that element is less than that. Then we can form a category out of it just by putting morphisms between them if one element is less than another. So post-sets can be thought of as special sorts of categories with at most one morphism between objects. And so we can form limits and co-limits over post-sets. And these are sometimes called direct limits and direct co-limits provided the post-set is directed. So what does directed mean? Well, directed means that if you're given two elements A and B of your post-set, we can find C with C is greater than A and C is greater than B. Or maybe C is less than A and C is less than B, depending on which way around your co-sets are. There are two versions of directed post-sets, which are obviously sort of equivalent to changing the direction of the order of the post-set. So what this means is given any elements A and B, we can just find an element C like that, or possibly with the arrows going the other way if you want the other convention. So direct limits and direct co-limits usually mean your category is really a directed post-set, which is a rather easy case. Projective limit sometimes means your category looks like this. There's a sort of generalization of directed post-set to categories called a filtered category. And a filtered category is one like this. It means given A and B as objects, we can find an element C such that A and B both map to C. So we can find C and maps from A and B to C. So that's the first condition. There's a second condition, which says that given element A and B and two maps between them, we can find some element C and a map H from B to C. So the composition H, F, and H, G are the same. So the two maps from A to C become the same. Notice the second condition is trivial for a post-set. So if a post-set is being filtered, it's the same as asking for it to be directed. Of course, there's a dual version of this where you turn all the arrows the other way around, but we'll usually be using this version. So for example, direct sums where you take two elements like that and take limits are not filtered limits or other filtered co-limits. Because given these two objects, we can't find an object they're mapping. They're both mapping two. Similarly, co-cernels are not filtered. So here, our category has two objects and maps like that. It satisfies the first condition because given any two objects, we can find one they both map to, but it doesn't satisfy the second condition because given these two arrows, there's no... We haven't given another object such that these two arrows become equal. So co-cernels and direct sums are not filtered co-limits. Some of the infinite unions are. Now I want to give some examples of filtered and non-filtered co-limits. For example, suppose you take a localization of a ring or module. Let's just do a ring at a multiplicative subset S. Then the localization are S to minus one is a filtered co-limit. Here, it's a filtered co-limit of the object R, where we are joined a little... Sorry, I'm not making the difference between capital S and little S the same. So this is a filtered co-limit of rings where you invert one element for S inside this multiplicative subset. What you're doing is you're taking a limit that looks like this. T to minus one and R U to minus one and so on. And we get a map from this to this whenever S divides T. In other words, T is equal to S times S1 for some element of the multiplicative subset. And this is filtered because the subset is closed. S is closed under multiplication. So for example, we've got R U to the minus one and R T to the minus one. And we can map these both to R U T to minus one. So the fact that this is a filtered co-limit is because the subset is multiplicative. If you recall, we also did mention you could do localization of the multiplicative subsets, but that was a little bit more complicated. It wouldn't be a filtered co-limit. And usually what you do is you just make the subset multiplicative. So another example is Q is a filtered co-limit of copies of Z. So we can take Z maps to Z, maps to Z and so on. And we can have this as multiplication by one, this times two, this times three and so on. And this is a filtered set. And if you think about it a bit, the co-limit of this is just Q with this mapping to Q and this mapping to, we would really have to sort of multiply them into Z by half here and times one over six there and so on. So this would make this... So if we do that, then all these maps commute. And you can see the rational numbers as a filtered co-limit of copies of Z. There's a confusing example. So let me put this as a warning. Let's look at the following two examples. Suppose you've got a map from Z to Z to Z to itself and each of these maps is multiplication by two. And you can also ask, what about this map here? If these are all the same Z, why don't you just take a map from Z to itself and that's multiplication by two? And you might think the co-limit of these two things is the same because all we're doing is taking Z mapping to itself under multiplication by two. But in fact, these are quite different co-limits. So the co-limit of the first one is given by Z, the ring Z that joined a half except we're thinking this as being a module because we can map this to... Here we map the map from here to here is given by multiplying an element of Z by a half and similarly we multiply it by a quarter and here by an eighth and so on. And you can see this diagram is now commutative and it's fairly obvious that Z a half is kind of the limit of these. On the other hand, the co-limit of this diagram is actually zero because the element one is mapped to two by this map here. So the elements one and two must map the same element here and the co-limit is actually just zero. So what you've got to be aware of is that you shouldn't think of this. This is not a subcategory of modules over Z. It is a functor from the category like this to Z and quite often if you've got a functor from something to Z if all these things have different images then it's harmless to think of it as being a subcategory but there are cases when these give different answers. This distinction between functors from something to Z to the... It's not Z, that's modules over Z. So the distinction between a functor to modules over Z and a subcategory of modules over Z normally doesn't matter but every now and then it really confuses people. At the moment Mochizuki announced a proof of the ABC conjecture and this proof of course has a certain amount of controversy and confusion and one of the confusing things about his proof is that Mochizuki appears to be using a non-standard definition of limits and co-limits and according to Mochizuki's definition the co-limit of this appears to be zero because he tries to regard this as a subcategory of the modules over the integers rather than as a functor from this category to modules over the integers so there was a certain amount of argument between people about what exactly was going on in Mochizuki's papers. So normally the difference between this doesn't matter but sometimes it does so you really need to remember that this isn't really a subcategory it's a functor from a category to modules. So the next problem is how do co-limits and limits behave with respect to exactness? So in other words if I've got a sequence of exact sequences A i goes to B i goes to C i goes to naught and I take a limit or co-limit of these you might look at naught goes to limit the A i goes to limit B i goes to the limit C i goes to naught we can ask is this exact? So next lecture we'll be discussing exactness of limits and co-limits and the answer is sometimes this is exact and sometimes it isn't