 This lecture is part of an online algebraic geometry course on schemes and will be about coherent sheaves on projective space So we have the following problem What we want to do is to construct Examples on projective space so We did some examples for P1 by gluing but the problem with gluing is it's really a bit of a mess I mean we could just about get away with it for P1 because we just had to glue together Two things but on bigger projective spaces. We're not gluing large numbers of things And it's really rather tiresome keeping track of the bookkeeping So this is really too messy in general and we would like to find a cleaner way of doing it so for affine schemes So an affine scheme of the form spectrum of R It's really easy because quasi-coherent sheaves are really just the same as Modules M over R Technically speaking there that there's an equivalence of categories between these but for all practical purposes They're the same. So let's look at a projective scheme So a projective scheme might be the spectrum So not spectrum proge of R where R is a graded ring So R is equal to sum over I greater than 0 of some some ring R I and This suggests that quasi-coherent modules should be related to graded modules M Which is sum over I greater than or equal to 0 sum over so I not equal to 0 of M I and There's turns out there's not an exact equivalence between these that the relation between them is a bit more subtle than in the case of affine schemes so what we want to do is to show how to go backwards and forwards between quasi-coherent modules over a protective scheme and graded modules over the corresponding ring So first of all, let's just recall briefly the construction of the Scheme associated to a graded ring So first of all we have the points Correspond to graded primes of R not containing sum over I greater than 0 of R I Now now we need the open sets and we're just going to take a base of the open sets And the base of the open sets consists of the open sets D F Which is the points Where F is not zero Well, so it's not quite the points where F is not zero more precisely. It's the primes Not containing F, but we always think of it as being the points where F is none zero and For each of these base of open sets we have to specify the ring of functions on this so this is the regular functions on the open set D F and You remember that this was equal to R F minus one Zero, so this means Degree zero elements of R F minus one so We'll just recall an example if we take R to be K X naught up to X n then the closed points just correspond to points of projected space over a field up to multiplication by some non zero lambda and this just corresponds to the ideal I of G with G X naught up to X n Equal zero where G is homogenous The only open sets we usually Bother with in defining it in the open sets DX I So I'm just we're just going to look at Very special polynomials F, which is equal to X I and DX is isomorphic to affine space Consists of all the points X naught X 1 up to 1 so this is the I position up to X n and O of D X I is just the degree zero elements of K X naught up to X n with X I inverted Which is just the ring of polynomials in X naught over X 1 X 1 sorry X naught over X I X 1 over X I There's a one in the I place up to X n over X I Which is just a polynomial ring so So This this open set is just isomorphic is a scheme to affine space Okay, now we've reviewed that let's try and do the same thing from modules So I suppose now M is a graded module. We want to define a sheaf on The the the projective scheme of R and Now if M is the graded ring R, you remember we define the sheaf by putting O of D of F I Equals R F Sorry, that shouldn't be enough. I that should just be enough R of F minus one and we took the degree zero elements So this suggests what we should do for M if we want to define a Sheaf M twiddle what we do is is we define M twiddle of D F To be M F minus one And then we take the degree zero elements of that So it's the most obvious thing we can do and we should now check this as a sheaf and this is similar To the case of Showing that the prunge of R is is a scheme So I'll just admit the check. It's just a kind of repeat of it by the way, I'm going to make a sort of Meta comment here The definition we're giving here is a little bit We're defining it in a slightly different way from what Hart-Shorn does the two methods are equivalent So it doesn't really matter and what Hart-Shorn does is he looks at the points of project of R and looks at the stalk of each point and Defines the sheaf using those stalks here. We're sort of ignoring the stalks and just using a base of open sets and This is somehow part of growth index philosophy that the points of a scheme are In some sense, not terribly important What is really important is the open sets together with which open sets cover the others This is actually the basis for something called a growth index apology because growth index realized you could define sheaves by Ignoring the points completely and just working with the open sets you just need to define something for each open set and Check some property holds whenever open sets cover other open sets and growth index or axiomized this and came up with the idea of a growth and Dictapology which turns out to be really useful for defining things like etal co-homology where you Replace open sets with something a little bit more complicated than open sets and you can still define sheaves on those Anyway, so points as we've defined it and twiddle by ignoring the points and just using our knowledge of the open sets anyway Let's get back to some examples so We're just taking r to be k x naught up to x n so prods of r is just n-dimensional projected space and let's put that an r to use the same notation as option that's just r-dimensional projected space over k And now what we can do is we can let v in Pk to the n B s pk to the r B a sub variety and then it corresponds to a graded ideal I and we can form the module M to be r over i and We can form The sheaf M twiddle and M twiddlers are sort of sheaf with support on the Variety V we started with you can easily check the stalk of M twiddle just becomes zero if you're not in V So you can picture M twiddlers being something like a Maybe a little one-dimensional vector space at each point of V and being zero elsewhere You may think why we look at r over i why don't we form I twiddle well I twiddle is sort of generally useless Just common I twiddle is usually uninteresting for example if V is the set where If we take Just the V to be the set hypersurface where f is equal to zero then the ideal i is just equal to f r Which is isomorphic to r so i is actually isomorphic to r as a module and I twiddle is just isomorphic to the sheaf of Coordinate functions on on the projected space So so that's why we we use our over i and not I I just doesn't give an interesting answer Next there's a basic operation on graded modules We can shift the grading so here if M is some over i of M i we can define a new module M N by M N i is N plus i so we just sort of Increased or possibly decreased the grading by N. It's easy to get muddle up about whether you increase or decrease the grading by N So Let's have a look at what happens What about if we take r as a graded module over r and Shift it by N and then take the associated sheaf Well, first of all our naught twiddle is just our twiddle which is just the sheaf of Regular functions on projective space Over our field K so Let's have a look at what our end twiddle is well on on the open set Um D X I And we need to work out what our end twiddle of D X I is and this is just the degree naught elements of K X naught up to X R With X I minus one except we have to shift The grading By By N However, shifting the grading makes no difference or at least it seems to make no difference And the reason is that X I is an isomorphism From the degree Degree letters degree M piece Of K snort up to X R X I to minus one to the degree M plus one piece So shifting the grading by N doesn't seem to make any difference. We haven't affected this up to isomorphism and in other words are and Twiddle is isomorphic to our nought twiddle over each D X I So first site it looks as if this sheath is going to be the same as this one It's the same as this over each open set but there's there's actually a bit of a subtlety going on because The way these sheaves Where you glue these sheaves together is is subtly different However, we see that our end twiddle is locally Isomorphic to our twiddle in other words it is an Invertible sheaf or a line bundle if you prefer the geometric terminology Now let's check that it's actually different so we can ask is our end twiddle the same as our zero twiddle The answer is no unless n is zero of course So we distinguish by calculating the global sections We cut I mean we can't tell the difference by looking locally at them because they're locally isomorphic So we have to do some sort of global operation like taking the sections So let's look at the global sections of our end twiddle and To do that we have to take a section over DFI Which is a degree? N element of K X naught up to X R with X I to the minus one and we have to glue this to something over DFJ, which will be a degree N element of K X naught up to X R This time we take X J to the minus one and we have to glue these for all all the different various values of I and J Well, if we compare these two they have to be the same Over the polynomial ring with X I and X J inverted so if I is not equal to J then this element Can have no poles In X I or X J So it can't really have X I to the minus ones in it because Here this has no X I to the minus ones and it can't really have X J to the minus ones because this is no X J to the minus ones so The degree and element in each of these must be a polynomial so we see That the global sections of our n quiddle is the same as the degree the homogenous degree N polynomials in K X naught up to X R That's provided R is greater than or equal to one because if R is equal to zero Then we can't find two different X I's there in the argument kind of breaks down So let's look at the dimension of Rn twiddle over P zero P one P two P three and so on so here we're looking at the Sorry the dimension of global sections So if we take the global sections of R minus two Global sections of R minus one global sections of R zero Global sections of R one global sections of R two and so on then Here we just get a lot of zeros everything here is zero Here the dimensions are all one Here the dimensions go one two three four five and here the dimensions go one three six 10 15 and so on so so here we've got Pascal's triangle The dimensions are just given by this and up here We've got a sort of trap for the unwary because these are both One so this is a sort of trap Zero dimensional projective space behaves a little bit differently Um So We see that all R zero R one R two and so on are all different We can we can work out the other ones are different by using some easy properties, which I'm not going to check If you work out the tensor products you find that M twiddle tensor and twiddle is isomorphic to M tensor and Twiddle which wouldn't easily check so M and Twiddle Is M twiddle and you notice in one of these the twiddle goes over the whole rock where is there it doesn't and Mn is isomorphic to M Twiddle tensor with our end twiddle so Um in particular our M Twiddle tensor our N Twiddle is isomorphic to our M plus N Twiddle and from that you can sort of shift Up and down by tension with our M So see that all these are also not isomorphic to each other because if you tense it with our M N large it becomes one of these positive ones which are all different. So all these are actually different line bundles Next we've got the following question Can we recover M? From M twiddle so if we give them the sheaf can we recover the original graded module from it So you might expect this by analogy with affine schemes. So for affine schemes You can recover M from M twiddle as being its space of global sections Well, you can't in general do this for projective schemes For example, if M is equal to K in degree zero and Norton degree Not equal to zero then you can check that M twiddle is just equal to zero because if you if you localize this At any X I you just get zero so Here we've got a non-zero module and if you if you turn it into a sheaf you just get zero So so we can't recover M from M twiddle in general However, we can almost do it So you sort of see that more generally if M is finite dimension as a K vector space then again M twiddle is going to be zero and it turns out that this is more or less the only thing that goes wrong So we can recover it as follows suppose. We've got a sheaf over Let's do over our dimensional space over K Then we define f of n to be f tensor with o of n So o of n is of course just our n twiddle and then we're going to define The global sections of f well the global sections of f don't tell us very much because there might not be enough of them So what we're going to do is define a sort of graded version of the global sections of f Where we just take the sum over all i of all n in Z of the global sections of of f of n So for example if we just do this to the coordinate ring, we just get k x naught up to x r because the Gamma that Gamma naught of o is just k gamma one of o is spanned by x naught up to x r and Gamma two is spanned by All the variables x i x j so you can easily check that if you take this graded Module corresponding to the coordinate functions you just pick up the ring r we started with And it's also straightforward to check That gamma star of f is a graded module over gamma star of o Which is just k x naught up to x r That follows from the fact that you've got a map from o m sense f n to f n Plus n Is it quite easy to check so we get this these maps you can take Let's do finitely generated graded modules over K x naught up to x r on the one hand and on the other hand we can talk about coherent sheaves on P to the r K on the other hand, and we've got these inverse maps between them On the one hand you can go from a coherent sheaf to a graded module by applying this function Gamma star F and On the other hand we can go from graded modules m to coherent sheaves by doing this m twiddle Construction and as I said Gamma star them twiddle a knot Inverses in particular we gave an example where if you Take a finitely generated great module and take a twiddle it and apply Gamma star this need Not be isomorphic to On the other hand if you do things in the other order You do get back to a coherent sheaves so if you if you take a coherent sheaf and apply Gamma star and Then twiddle that this actually is isomorphic to f which will be talking a bit about more later So Next lecture would be talking a bit more about this correspondence between coherent sheaves and graded modules