 This lecture is part of an online algebraic geometry course on schemes, and will be about coherent sheaves. So, what is a coherent sheaf? Well, it's a sort of analog of a finite dimensional vector space. So coherent sheaves are related to sheaves in much the same way that finite dimensional vector spaces are related to vector spaces. Now, they're the sort of small sheaves in some sense. So let's first look at the case of modules over a ring. So we're going to look at coherent modules, and the first question is what is a coherent module? Well, the obvious analog of finite dimensional vector spaces is a finitely generated module. So, why don't we just define a coherent module to be a finitely generated one, and the answer is it works fine for notarian rings, but is not so good for non-notarian rings. So let's see the following problem. If M to N is a homomorphism of finitely generated modules, the problem is the kernel of M to N need not be finitely generated. Well, it is if you're working over a notarian ring because any submodule of M, in that case, is notarian. But we can see in the example where this fails, for non-notarian rings, you just take R to be the ring of polynomials in infinitely many variables, and we take the map from R to R, which just maps X i to 0 for all i, and then the kernel is the ideal finitely generated by X i, X i, X i, X i, X i, and so on, which is not finitely generated. So coherent modules solve the problem of finding a nice category. So the problem, we want a category of modules with kernels, co kernels, etc. Technically speaking, we want an abelian category, which can be defined informally as a category with all the nice properties of the category of abelian groups. Well, you can do this by looking at coherent modules, which are slight strengthening of the condition that RB finitely generated. So coherent means first of all, the module M is finitely generated, and secondly, we have this rather odd condition that if R to the N goes to M is a homomorphism, the kernel is finitely generated. So not all submodules of R to the N for the ring are going to be finitely generated, but quite a lot of them are, and it turns out that these are most of the ones you actually need. In particular, this implies M is finitely presented because there's some homomorphism from R to the N on to M, and its kernel is finitely generated. However, as we'll see a little bit later, it's actually stronger than saying M is a finitely presented module. And the coherent modules form an abelian category, which I'm not going to prove. It's a fairly routine and not very exciting. The ring R is called coherent if R is a coherent R module. So it's like you say a ring R is notarian if it's a notarian R module. For coherent rings, it turns out that coherent modules are the same as finitely presented ones. And for notarian rings, it's pretty obvious that coherent modules are the same as finitely generated ones. So let's just have a couple of examples to show some coherent rings that aren't notarian. I'll just go through this fairly quickly because we're not really going to be using them. We're going to stick to notarian rings most of the time. So if you take the field kx1, x2, x3, in infinitely many generators, this is coherent as a ring, but not notarian. Next, if we take the ring R to be kx1, x2, up to xN and quotient it out by all the variables x1, x, all the pairs x1, x2, x1, x3 and so on, then this is not coherent as a module over itself because if we take the map from R to R, taking 1, 2, x1, the kernel is x2, x3 and so on, which is not finitely generated. So R is a finitely presented module that is not coherent. So this is just an example of the sorts of things you have to worry about if you work with non-notarian rings. Anyway, what we really want to do is discuss coherent sheaves, and I'll give a bit of a history to explain why people were worrying so much about non-notarian objects. So coherent sheaves were first introduced for complex analytic manifolds. So Seher and Cartan studied complex analytic manifolds using sheaves, and sheaves were then later introduced into algebraic geometry by Seher based on his work on analytic manifolds. So a lot of the ideas about sheaves originally came from the analytic case, and the problem is, if you look at the sheaf of holomorphic functions on x, where x is a complex manifold, so we might just take, say, x to be the n-dimensional complex space, for example. This is an example. The problem is, you get a lot of non-notarian rings in this sheaf. So if we look at holomorphic functions on, say, even C1, this is not a notarian ring. For example, let's take the ideal i as all functions vanishing on all but the finite number of integers. And then you can easily check that this is an ideal in the ring of holomorphic functions that isn't finitely generated. So the problem is, if you're working with complex analytic manifolds, most of the rings you encounter are simply not notarian, and you need some substitute for it. And Ochre discovered this fantastic substitute for it, which says that the sheaf of holomorphic functions on a complex manifold x is coherent. So what does coherent mean for sheaves? Well, I guess just before explaining what coherence means for sheaves, let me just point out that the fact that rings of holomorphic functions are not coherent is a bit subtle, because you've actually got three rings here. You've got the ring of polynomials, and polynomials are pretty close to holomorphic functions, and holomorphic functions are pretty similar to the rings of power series. And the ring of polynomials and the ring of power series are notarian, and these are what you use if you're doing algebra, but the ring of holomorphic functions, which is what you're interested in if you're doing analysis, is not notarian. So if you're an analyst, you tend to work with non-notarian objects. So what does coherence mean for sheaves? Well, let's take x o x to be a ringed space, and here I'm not assuming x is a scheme, because when x is a complex analytic manifold, and these are the holomorphic functions, then this usually isn't a scheme. So we might take, for example, x is just n-dimensional complex space, and o x is the sheaf of holomorphic functions, which assigns to each open set the holomorphic functions on that open set. Then the definition of the sheaf F to be coherent is sort of a bit similar to the definition for rings. First of all, F is of finite type. What this means is that x is covered by opens u i. So on each u i, we have o x to the n i maps onto, sorry, this restricted to u i, maps onto F restricted to u i. So it means, locally, F is sort of finitely generated, there's a map from n copies of holomorphic functions onto F. This n i may depend on u i, and may actually be unbounded if you've got an infinite number of u i. So this is a slightly subtle concept. It means F is sort of locally finitely generated in some sense. And secondly, we have the coherence condition, the kernel of o x to the n on u to F on u is of finite type for all open u. And you can see these two conditions are very similar to the conditions for a module to be coherent. I think historically coherence was actually first defined for sheaves, and then later defined for modules. So the definition was really rather round about. So in other words, not all sub sheaves of this are of finite type, but ones you're interested in are. And as for rings, you can talk about the space, the sheaf of rings itself being coherent, if it's coherent over itself as a module over itself. So if o x is coherent, then F is coherent, if and only if it's finitely presented. So we can find the sequence o x to the m goes to o x to the n goes to F goes to zero, which is exact. So this is just like saying F is a sort of quotient of two three sheaves. So it's like saying F is a finite, F is a, it's the analog for sheaves of saying that F is a finitely presented module. Here m and n are finite. And then this definition was extended to the case when m and n may be infinite. So F is called quasi-coherent if it's, if we can find o x to the m goes to o x to the n goes to F goes to zero for m and possibly infinite. Now you notice that for modules, this condition is completely vacuous because every module is the quotient of some possibly infinite free module by some other possibly infinitely generated free module. So quasi-coherence for modules over rings is a vacuous condition, but for sheaves it's non-trivial. And then for affine schemes, this is equivalent to the condition that F is locally of the form m twiddle for some module over a ring, which is the definition we gave of quasi-coherence. So that's why this condition has such a weird name of quasi-coherence. Quasi-coherence was originally defined as a reasonably sensible concept and was then applied to affine schemes where for affine schemes, this definition of quasi-coherence is rather round about. For affine schemes, there happens to be a much simpler definition, but for historical reasons we're saddled with this adjective quasi-coherence because quasi-coherence was originally discovered for analytic manifolds when the definition is a bit weirder. So that's the sort of history of why we ended up with such funny terminology for quasi-coherence. Hart-Shorn defines coherence differently as a quasi-coherent sheaf of finite type, and this really is different from the standard definition of coherence. However, it is equivalent to the usual definition for notarian schemes, which is the only case Hart-Shorn is interested in his book, so it doesn't really matter. For non-notarian schemes, you shouldn't use Hart-Shorn's definition as Hart-Shorn said it behaves really badly and you should use the standard definition of coherence. And similarly, if you're doing complex analytic spaces, you should use the standard definition, not Hart-Shorn's. So coherent sheaves form an abelian category. Again, provided you use the correct definition for non-notarian schemes. So quasi-coherent sheaves are ones that over a scheme are ones that look locally like a manifold and coherent, sorry, that look locally like a module, and coherent sheaves are ones that look locally like a coherent module. And if you're a notarian person who only uses notarian rings and schemes, then coherent sheaves are ones that look locally like finitely generated modules. So we can just have an example. The sheaves O, N over P1 are coherent because P1 is covered by two affine sets, A1 and A1. And over each of these affine sets, the sheaf O, N looks like a copy of the ring of regular functions, kx. And kx is a coherent module over kx because kx is a notarian ring. So locally, this is sheaf as coherent and if a sheaf is coherent locally, then it's coherent. So next we have the following problem. Given a sheaf, given a coherent or quasi-coherent sheaf, is f star of f or f lower star of f coherent or quasi-coherent? And this is one of these questions which gets a little bit technical. So let's take f to be a map from x to y where x and y schemes. And f star is fairly easy to deal with. So f star of a quasi-coherent sheaf is quasi-coherent. This is easy to prove and I'm not going to bother. And f upper star of a coherent sheaf is coherent if x is coherent. And again, I'm not going to bother proving this because it's fairly easy and not terribly interesting. So f upper star is nothing very exciting happens. F lower star is a lot more subtle. First of all, let's look at quasi-coherent sheaves. So if f is quasi-coherent, f lower star of f is usually quasi-coherent. Meaning it would be quasi-coherent unless someone has deliberately come up with a pathological counter example when it isn't, as I'm about to do so. So let's have an example when it isn't just to show you the sort of nasty things you might want to watch out for. What we're going to do is let's take x to be the union of an infinite number of copies of the spectrum of r, where r is a discrete valuation ring. For instance, it might be z localized at two. And I'm just going to take y to be the spectrum of r. So the picture we have is something like this. x looks like this. So here we've got x. It's got infinitely many copies of this. And y just looks like this. And I'm going to take my sheaf f to be just the infinitely many. It's just a copy of r twiddle over each copy of the spectrum of r. So I'm just taking the ring of coordinate functions on each of these blue things. And now we can calculate what f star of f is on all the open sets. So this is just two non-empty open sets. So it's got the open set spectrum of k. You can easily see it's just a product of infinitely many copies of k. So we just take a product over all n in n of k. And on the spectrum of r, it's just the product of infinitely many copies of r. Again, it's a very easy calculation. So what's the problem? Well, the problem is if we take the product of an infinitely many copies of r and tensor over r with k, this is not equal to the product of infinitely many copies of k. Because you can see, for instance, here, the product of infinitely many copies of r, if we tensor with k, all the denominators are bounded. Whereas here, the denominators need not be bounded. So, for instance, in the case we had where we were taking the localization of the integers of 2, k would just be the rationals. And we could just take the number a half, a quarter, one eighth, one sixteenth, and so on. And there's no way to get that by taking a product of two adic integers and multiplying by some rational number because the denominator would be bounded. So there are cases when f star of a, if lower star of a quasi-coherent sheaf is not quasi-coherent. Fortunately, we have a theorem if f from x to y is quasi-compact and quasi-separated, then f star of a quasi-coherent sheaf is quasi-coherent. So, I'm not going to prove this because Hart-Shorn gives a sort of proof of this. Well, he proves it under slightly strong conditions. He assumes that f is separated, but you can get away with quasi-separability. What I'll do instead is I'll just point out why you need these conditions and why the proof goes wrong if you don't have these conditions. So what do quasi-compact and quasi-separated mean? Well, quasi-compact means that if u is open in y, then f for minus one u is covered by a finite number of open affines ui. And quasi-separated, quasi-separated, you've probably forgotten the definition of it, just means the map from x to x times over y, x is quasi-compact, which looks like a very technical condition. It's the main use of quasi-separatedness as it implies that ui intersection uj is covered by a finite number of open affines. And again, we've got this important finiteness here. And now the problem with f star is that f star of f plus g is equal to f star f plus f star g. However, f star of an infinite sum of f i is not equal to the infinite sum of f star of f i in general. You can see a counter example in this example here where if you take an infinite sum of copies of our twiddle on each of these, then f star of that is not equal to the sum of f star of the individual things. So the problem is that f lower star doesn't commute with infinite direct sums. And if you look at Hart-Shorn's proof that quasi-compact and quasi-separateness implies this fact here, you'll see it's implicitly uses the fact that f star preserves finite sums of sheaves. And you only need to work with finite sums of sheaves because of these finiteness conditions here. So this is an example of the unfortunate fact that in sheaf theory, there are large numbers of these unmemorable technical conditions you need to add to everything. Finally, let's look at f lower star of a coherent sheaf and ask, is this coherent? And the answer is a big resounding no. It's quite rare for this to happen. And there are very easy examples. We can just take f to be the map from the affine line to a point. You couldn't get anything simpler than that. And let's just take the sheaf here to be kx twiddle. So the sheaf of coordinate functions. And this is coherent. It's a finitely generated module over a notarian ring. So everything's nice and coherent. And f star of k twiddle x is just the sheaf kx twiddle over the spectrum of k, which is a point which corresponds to the module kx over the field k. And this is not finitely generated as a k module. So even in the simplest cases, f lower star does not take coherent sheaves to coherent sheaves. However, we will see later there's one incredibly important case when it does. So suppose f from x to y is proper or finite. Well, finite is a special case of proper. And you need some technical condition. I think it's y is notarian, or probably locally notarian would do. Then f star of a coherent sheaf is coherent. More generally when we define right derived functors of f, we will find that right derived functors of f also take coherent sheaves to coherent sheaves. And this is essentially a generalization of finiteness of cohomology. So we saw an example of this when we looked at f going from p1 to a point. And we saw that f star of o n is finite dimensional. It's a finite dimensional vector space over k, which corresponds to a coherent sheaf over the spectrum of k. A coherent sheaf over the spectrum of k is just a really fancy way of saying a finite dimensional vector space over k. So that's all about Quasi-coherent sheaves for the moment.