 This lecture is part of an online algebraic geometry course on schemes and will be about coherent sheaves on projective space. So the basic original reference for coherent sheaves on projective space is Searle's very famous paper on algebraic coherent sheaves and quite frankly you should really read that paper rather than wasting your time watching YouTube videos if you want to learn algebraic geometry. So we recall last time that we had a correspondence graded modules M over the polynomial ring with quasi-coherent sheaves sheaves f over pr and if you've got a module M we saw how to take a quasi-coherent sheave M twiddle and if you've got a quasi-coherent sheave f then we saw we could go to gamma star of f which was sum over gamma of fn and these correspondences don't give an equivalence of categories. In particular if you start with a graded module M and then you take the corresponding sheave and then you do take gamma star of M twiddle this need not be isomorphic to M so if M for instance is a finite dimensional vector space then this module here is just zero so you can't recover M from it. However it's still pretty close to being an equivalence of categories. Roughly speaking coherent sheaves can be sort of thought of as roughly the same as graded modules as finitely generated graded modules except you sort of ignore the modules of finite length so this roughly means ones that are finite dimensional over the vector space k so we'll sort of see this a bit later this lecture. So what we want to show is that if we start with a sheave and then take the corresponding graded module and then take a sheave corresponding to it this is isomorphic to f at least if f is coherent. Should remark in the other direction although M isn't isomorphic to gamma star of n we do get a natural map from M to gamma star of M twiddle unless the map is the other way around it's very difficult to remember. This map from M to to this space here is neither injective nor surjective in general and the correlation between M and M twiddle is a little bit subtle. Okay anyway now we want to show that f if we take any coherent sheave turn it into a graded module and then turn it back into a sheave we more or less get the sheave we first thought of. So we recall that P r is covered by open sets um of the form d x i which you can think of as being x i non zero somewhat informally and what we're going to do is we're going to show that f is isomorphic to gamma star of f twiddles on each d x i and this will be enough because if we show these two things are isomorphic and if we find an isomorphism between these on each d x i then that will be enough to show that isomorphic. Of course it's not enough to show that these are isomorphic on each d x i we have to show all these isomorphisms are compatible with each other which i will sort of skip because it's kind of obvious the isomorphisms we write will be compatible. So let's have a look on d x i we need to work up what the restriction of this is and if you unravel these definitions you find um what you have to do is you take global sections of gamma n then you can take global sections of sorry not f of n global sections of f of n plus one and you can get from there's a map from these where you multiply by x i multiply by x i again and go to gamma of f of n plus two and so on and we can take a direct limit of these um so we take the limit overall n of global sections of f n this isn't quite the union because these maps aren't injective if these maps were injective then that then this would be the union um and we want to compare this with f of uh d x i so we're we're restricting or we're restricting all these um so we have maps going like this and this induces a map going like that we want to show this is an isomorphism um and um these maps here are neither injective nor surjective in general it's it's only when you sort of take a limit that they become an isomorphism um so we have to show that this map is injective injective and we prove these by sort of just um working through the definitions and taking a look so so let's show that gamma of so the direct limit of the gamma f n to f d x i is injective so to do this let's pick s in um some gamma of f n and suppose s has image nought in f d x i so picture is you know we've got sort of dx one dx two dx three and so on and we're picking some global section and it's zero on one of these this doesn't apply s is zero because for example the chief f might have support um that that that doesn't meet say dx one and then that there will be global sections of it which are actually zero on dx one um well um if f has image zero in d x i then it says image zero in d x i intersection d x j and this means that um x i to the sum power times s is equal to zero in d x j um so um so s is killed by some power of x i in dx j and there are only a finite number of dx j's so s is killed by some power of of x i on p to the r and um this means that s is nought in the direct limit of of the gamma f n so the map is injective um next we want to show the map from the limit of global sections of f n to um um to f of d x i is onto and so what we do is we pick some s in f of d x i so if you go back to our picture we've got d x i d x j d x k and so on and now we can extend f we can extend f to d x j if we multiply by some power of x i because um we know that s is defined on the intersections of these and d x j is sort of got by localize i mean we we get the functions on this intersection by localizing functions on d x j at x i which means we can we can extend any function on this intersection to the whole thing by multiplying by a high power of x i so we can extend it to dx j we have an extension like this and we can also extend to dx k so we have another extension of it like this the problem is the extensions might not be the same on d x j intersection dx k so what do we do about this well we notice that um if we take these two extensions they're at least equal on this region here and that means that they're equal on this region here provide you multiply them by a suitable power of x of x of x i they become equal if multiplied by a power of x i um actually there's a bit of a subtlety going on here we're sort of implicitly assuming that this intersection is affine um the argument still goes through if this intersection is a finite union of affines i mean it's always affine from projective space but in general um when you generalize this argument you need this intersection to be a finite union of affine spaces and this is in general this sort of condition is true if the space you're talking about is quasi separated and you remember the quasi separated property has this nice phenomenon that it ensures intersections of affine spaces and unions of finite number of affines anyway if we work on projective space we can we can ignore that technical complication um so um if you put all this together you see if you multiply s by sufficiently high power of x i then you can extend it to a global section of the whole space and and that says that it's the image of one of these if you twist by n enough so that more or less shows that um that that's the map from um that if you start with a sheaf then it's isomorphic to gamma star um of f twiddle so um so all quasi-coherent sheaves come from graded modules and this means you can transfer a lot of facts about quasi about graded modules to quasi-coherent sheaves for example every um coherent sheaf is a quotient of a finite sum of line bundles of the form on and this is because every finitely generated module is a quotient of shifts of free modules um actually i guess we're using a few facts about coherent sheaves that we haven't quite proved yet um so we're sort of cheating a bit and um assuming that global sections of coherent sheaves are finite dimensional we will be proving that a little bit later um uh yeah um so um in fact we can get a resolution um if we take a um a sheaf f we can get a resolution by locally free sheaves um f nought goes to f one goes to f two and so on because the kernel of the map from f nought to f will also be a coherent sheaf and we can keep on applying this and Hilbert showed that in fact this resolution ends after a finite number of steps so quasi-coherent sheaves all have finite resolutions by um um vector bundles so all of these are sums of sheaves of the form o of n for various values of n um um next um any coherent sheaf is generated by global sections in other words it's generated by a finite number um and what this means is if you've got a sheaf f and we pick any points p in p n then the then then the stalk of f at p is generated sorry um if you've got a coherent sheaf f sorry should have said f twist by n is generated generated by a finite number of global sections for n sufficiently large so what this means is if you take a point p in projective space and the stalk of f n at p is generated by global sections of f and the proof is sort of similar to the proof that um f is f of uh gamma star of f twiddle so i'm going to omit it um we can also see that coherent sheaves are built out of of special coherent sheaves of the form um um um corresponding to um the ideal um of some irreducible subset so p is the ideal of an irreducible subset and i also want to twist these by n this follows from the fact that a finitely generated graded module has um a filtration nought equals m nought contained in m1 so i'm contained in m k equals m with each mi over mi minus one of the form um k x1 zero to xr modulo some ideal again twisted by n so this is a basic result of commutative algebra for ungraded ideals it's a fairly standard result um follows from the fact that any finitely generated module has has an associated prime if it's non zero which gives you the initial step and so for greater modules the proof is very similar so at first sight this seems to make coherent sheaves easy to understand they just correspond you can build them by taking irreducible subsets or projective space and um um um building them out of coherent sheaves that sort of um have support on that irreducible set and they somehow look a bit like a one-dimensional vector bundle on that set um however the problem with that is first of all um in order to classify coherent sheaves you need to understand every irreducible subset of projective space which would mean understanding every possible variety um which is obviously an incredibly difficult problem the other problem is the way these these modules fit together like this is also rather complicated just because you know all these quotients gives you some information about m but there can be enormous numbers of rather complicated way of fitting the quotients together um so this gives us a sort of picture of the blocks you can build coherent sheaves out of but this still doesn't give us a completely clear picture of what a coherent sheave looks like um so i'm going to finish by um showing a basic result that says that if f is coherent on p to the n then the global sections of f is finite dimensional over k um so this this this works for projective space or more generally for projective varieties but you remember it just fails totally for affine varieties if you've got a coherence sheave on an affine variety it corresponds to a module over the coordinate ring and there's absolutely no reason why that module should be a finite dimensional vector space over k um and the proof of this in Hoche on uses some rather tricky commutative algebra that i can never remember um what we're going to do is give a proof using cohomology and the proof using cohomology is very easy and straightforward and there's only one slight problem with it that we haven't actually covered cohomology yet but you know i don't care i'm going to give the proof using cohomology anyway i mean let's face it if i had covered cohomology it wouldn't make any difference because nobody ever bothers looking at the earlier videos i mean i've been checking youtube and everybody watching this channel always stops by watching the latest video i've made as if it's time sensitive information so whatever we're going to give a cohomological proof even though we haven't covered cohomology so let's take a coherent sheave and by using some of the results we gave earlier plus some of the results we didn't really give earlier you can write f as a quotient of a finite sum of line bundles let's call this b so this is going to be a sum of o n's and there will be some kernel a which is also coherent um so this sort of implicitly uses that f is generated by a finite number of global sections and so on and on to get there's a finite sum there so um what you can do is we know the global sections of o n so you could try saying well we've got a map from the global sections of b to the global sections of f let's get my let's get my f's and my gamma's muddled up and since this is finite dimensional and b maps on to f therefore this is finite dimensional well if you argue like that you're a bozo who hasn't been paying attention because just because this sheave maps on to it doesn't mean the global sections map on to I guess that should be a gamma um so cohomology comes to the rescue because what we have here is we can extend this to an exact sequence h1 of a so all we have to do is to prove that h1 of a coherent sheave is finite dimensional then we're finished well that's easy to do because if you've got an exact sequence of sheaves like this then we have an exact sequence of cohomology groups h1 of a goes to h1 of b goes to h1 of f and we know this is finite dimensional because we can calculate the cohomology of sums of line bundles and it always turns out to be finite dimensional so this is the dimension is finite and you're not going to fall into the trap of thinking this map is on to again and as you might guess there's an obstruction which is a second cohomology group of a so we want to show the second cohomology group of any coherent sheave is finite dimensional and you see we can repeat this second cohomology groups of finite dimensional provided a third cohomology group is finite dimensional and this sounds completely pointless we seem to be an infinite regress but we're saved by the following fact that h r plus one of any sheave equals zero if r plus one is greater than the dimension of whatever variety we're working on which is our dimensional projected space and r plus one is greater than the dimension projected space for all these sheaves vanish so we can prove that the space of global sections of any coherent sheave is finite by descending induction on cohomology groups hi of a coherent sheave and it's rather rather a funny proof because you tend to think of the zero than first dimension low-dimensional cohomology groups as being the easy ones to work with but here we're starting with the high-dimensional cohomology groups and working downwards and ending up with the results about the zero-dimensional cohomology groups anyway i just finished by summarizing the results about cohomology groups that we used just that if you would later do cohomology we can check we've got them all so the results about cohomology groups we used but first of all if we've got an exact sequence of sheaves we get a long exact sequence of cohomology groups and so on we've probably get the idea by now we just go on like this the second fact is that hi of a equals naught if i is greater than the dimension of whatever variety we're working on and the third one is that the dimension of the all-cohomology group of any one of these line bundles is less than infinity and this you have to prove by an explicit calculation which was first done by ser okay so i think that's quite enough about coherent sheaves what we'll move on to next lecture is the picard group which is closely related to divisors and line bundles