 Good morning, so I wanted to welcome you all here at the ICTP for this school on commutative on algebraic geometry and algebra and prime characteristics. So I hope you had a nice travel and are now fresh and so on so that you can follow the lectures. I wanted to thank everybody who made this possible, so the organizers, Professor Trunn, Rossi and Wermer and obviously all the lecturers. I maybe can say a few words about ICTP, so I mean you know ICTP is a research center which also is devoted to helping development of science in developing countries and so I can mention a few programs that you have that you might be interested in in the future. So one is the diploma program, so maybe you personally would not be interested but you know your students who have their first degree could come here and stay for a year and have some courses and then this should enable them to go to a PhD program afterwards. I see at least one diploma student in the audience. Then we have the show stage where you can apply for this and then you stay associated to us for five years and you have several visits during this time. Then we have visitors and postdocs positions, you know that I also see one of our previous postdocs at least in the audience. So you should look at our homepage and see whether it is you know maybe apply for some of these if you are interested or encourage your students to apply. So in some sense you know this was already all I wanted to say I mean we are already a bit late. So I wish you a very successful stay and I hope that you will learn a lot here and welcome again. Thank you. If I remember when it is the 4th or the 5th school on competitive algebra on this sky at ICTP and this time it is organized by Professor Rossi from Geneva, Professor Verma from India and Professor Chung from Hanoi and this morning we will have a two section. The first one was given by the two lecturers, Professor Brenner from Osnabrige and Professor Keiichi Vantanabe from Japan and then we will have a coffee break and followed by three presentation on the research region. So now I would like to invite Professor Brenner to give you the first lecture. I was sure in some sense I will make some jokes and this so now. Better. Is it loud enough? So maybe we don't need that upstairs. Can we get rid of that? So you can completely concentrate on the blackboard. So the name of the school is what was it, commutative algebra and algebraic geometry in positive characteristic. In my series I would like to demonstrate that the two things go hand in hand and from the commutative side I will talk about tight closure and what is strongly connected Hilbert Kunz theory and today I will start with Hilbert Kunz theory. With tight closure we go on tomorrow and on the algebraic geometry side we will use mainly vector bundle techniques. So the main idea is to use results from vector bundles on projective varieties to get some. If you prefer that you can look upstairs, I don't mind. So what was my sentence? So the idea is to use, so we look at some problems originating in commutative algebra and positive characteristic, we translate it into the more algebraic geometry side in particular vector bundle theory and we solve some of the problems by using methods from this side. So the positive characteristic is P. That's universally accepted that the prime characteristic is denoted by P and what is also universally accepted is that P to the power E is usually denoted by Q so I write it right now because sooner or later I will use it anyway. And say we have a ring and of positive characteristic so we have a field of characteristic P inside and there then we have the Frobenius homomorphism which sends an element F to each piece of power and the first miracle is that this is a ring homomorphism. And of course you can iterate it and then the Frobenius sends F to the power F, Q. And commutative algebra in positive characteristic deals with questions. What can we do with these objects, what can we define, what kind of singularities occur when we look at the action of the Frobenius? No, that's the main idea. 69 looked so in the situation I have written down there, he looked at the following function maybe I should write here, if we have an ideal inside our ring, no then we can look at the ring generated by the image or something like that and that is denoted by I to the power bracket Q, that's also standard notation. And how these guys behave in particular, how they behave asymptotically as Q is getting large. This is one of the central topics in Hilbert Kuhn's theory. So we have two definitions, so the ring is fixed, we have an ideal and the dimension, let's say, let's do it like that. So I suppose that I is primary to a maximal ideal, M of height E, and the main important, there are two important cases that they have a local ring, then we don't have to think about the maximal ideal because there is just one and I will be primary to that maximal ideal and the height of D is the dimension of the ring or we look at the graded or standard graded ring and then the maximal ideal will be the, then everything will be graded and the maximal ideal will be the irrelevant ideal. Okay, we can look at these powers and the corresponding residue class rings and because of this primary condition they have finite length and of course it would not be good to directly look at the length, we have to normalize it by dividing by Q to the height D and then we consider this as a function in E, you could also consider it as a function in Q. So this mapping, it's a mapping from natural numbers to rational numbers, length is a natural number divided by a power of the prime is a rational numbers. So this is called the Hilbert-Kunz function, you say here something like, oh, this is a mysterious function and a very interesting function and it has some surprising behavior and then if something has a surprising or mysterious behavior, nevertheless you would like to understand some, to understand it and to see some structure in it and maybe we call that phi i of E, something like that. Now if we have this function we define the Hilbert-Kunz multiplicity, just by looking at the limit, so we look at the limit as E goes to infinity of the expression I have just written down and hope that it exists. In fact, Kunz in 76 he gave an example that this limit does not exist and then Monsky 70 years later proved that the limit always exists and Monsky was right. So I mean Kunz lost interest because he found this counter example, but then Monsky showed that limit exists not as a positive real number and the natural question he asked where this is a rational number. And today I will talk about some positive results, in fact in higher dimension it will not be always positive, but today I will talk about positive results. Let me mention another important property of the Hilbert-Kunz multiplicity with that you can see that it has something to do with singularity. So I should say, we denote this then like that, the multiplicity, Hilbert-Kunz multiplicity of the ideal and for a local ring, if we just talk about the local ring and we don't fix an ideal we always mean the maximal ideal and therefore then the Hilbert-Kunz multiplicity of the maximal ideal we denote by the Hilbert-Kunz multiplicity of the ring itself. Because literally taking this does not make sense, so here we mean the maximal ideal. The result I want to mention is due to Watanabe who will also give lectures here and Yoshida from 2000 something, I don't know exactly, and now it basically says you have an unmixed condition, I don't write it down, but basically it says that the Hilbert-Kunz multiplicity is always at least one and it is one exactly if and only if R is a regular local ring. So you look here at unmixed local rings, the theory N, and then you have this characterization. So you see this is a nice characterization of the regular case and so as soon as you have a singularity in your ring you will have the Hilbert-Kunz multiplicity will be strictly larger than one and that gives you a measure for a singularity. Okay, now I'm going to explain how this is related to vector bundles. This is, I think we have nobody from category here so I just write equals, she's. If we back to school for a second in linear algebra we consider equations, maybe let's say one equation in N variables and we are interested here in the homogeneous case and then there are and the FIS are numbers inside the field and there are two very different behaviors either all FIS are zero then the complete KN is the solution set, so that's the trivial case. So suppose further that not all of them is zero and say then let's fix one, say F1 is not zero, not then the solution that we can write T1 as, now you have to subtract and divide by F1 and then you have N minus one, three variables and the solution space is a vector space of dimension N minus one. Now let's go one step further, now let's consider that X is, X can be anything topological space or manifold or scheme or the spectrum of a ring and now we consider the case where FI are functions and say differential functions or continuous function or in the algebraic setting algebraic functions, some space object and then we can do the same thing. Out of that we can construct a bundle above X by just looking at, no we have, well let's denote it, V consists of a base point and no we can plug in the point into our functions and then we get this equation we had before, no I should write it like that, V is P and in the second component we have the N variables and this is the equation and not so by, because we have understood that if at least one of the function does not vanish at the point P then we can use this construction and then we see that the fiber, so let's say X is the base space and we have fixed the point P and in the point P not all vanish and then the fiber of this thing, so this is over X is N minus one dimensional vector space. But of course if all the F1 up to FN vanish at that point then suddenly this, the vector space, the fiber explodes to a N dimensional vector space and that we don't like. So, but easily we can get rid of, we only consider it to be the union of this set inside X and DFI is given by the property that FI does not vanish at the point P, no and then we can consider our guy above U and then so this is a basic model for a vector bundle and you see if you look at what happens over DF1 say we can make the same trick here and then this equation provides a trivialization so this guy, so I call it already a vector bundle and we have on DFY we have trivializations which are written down here and if you look at the situation in DFI intersected with DFJ you also see by this construction that the change of the transformation mapping is a linear mapping so that's part of the definition. Okay, so what does that have to do with this situation now we fix generators of our ideal which is M primary as before and also FI are elements inside the ring but no, if you know that's the idea of the spectrum of the ring in some sense so that is not true literally but in some sense it's the elements of the rings are functions on the spectrum but no here what is here is not you cannot write just the field here it's a bit more complicated no but the general idea that you can consider this equation prevails here so let's call again the spectrum of the ring X and again don't have to write it down again we take this notation and now DFI this has to interpret it no the spectrum is the set of all prime ideals this is the DFI is the set of all prime ideals where FI does not belong to the prime ideal or is not zero in the residue class field. Okay, so we look at these elements define a map from our N to R no this is basically the same thing here so you go FI TI so TI are now elements of the spectrum of the elements here and you look at the kernel no and the kernel in commutative algebra is usually denoted by the sieges for the elements no and of course no so this by definition everything what goes here to zero but no this map is in general not subjective because the ideals are primary the ideal is primary to a maximal ideal but it's not the unit ideal no it would be subjective only if it is the unit ideal not but we have this exact sequence and if you write it down like here this is in general not locally free this is locally free this is in general this module is in general not locally free because not you have degenerate behavior where the dimension of the firewall goes up and that is that we don't like so instead we restrict the sequence of modules we look at the corresponding sheaths on our open set you and then no I mean you can always now write a tilde above everything I will not do it also we have that thing we have O U N the Nth direct sum of the structure sheath and because I is no I mean locally on U I always contains a unit so therefore I can write here O U and this is subjective but subjective now means here as sheath homomorphism so this is a exact sequence of locally free sheaths and I prefer to work with locally free sheaths instead of modules now so for example one advantage you have here if you apply the Frobenius to this situation no Frobenius is not exact no pullback by Frobenius is in general not exact so you get some non-exactness and everything gets complicated but Frobenius on if you apply the Frobenius on an exact sequence of locally free sheaths then the pullback is exact again and you don't have to worry about non-exactness now so this is locally free sheaths on U on this open subset of X and in fact you have on each DFI you have trivializations for that so here of course I have the Nth power this is easy this is easy this is more complicated but it's given as the kernel bundle kernel sheath of this subjective sheath homomorphism okay so I will not give an exact definition so in fact I have wrote some notes and so I have checked whether you can find the notes and so the notes is the name of the course vector bundles and tight closure and I tried it with Google with Google you don't find it then I tried it with duck duck go and with duck duck go you find it at least I found it with duck duck go so much better than Google forget about Google but don't try right now you should concentrate here what I am doing here okay this make lives easier but it's still complicated so now we look at the graded case so graded case means we have a standard graded ring and the ideal is homogeneous and all the generators are homogeneous and of course it's primary to the irrelevant ideal so the degree of FI is DI and then our sequence now I can write down the same sequence which is now graded and therefore so here we have U is DI is D of the irrelevant ideal and this is just the spectrum of R without the ideal so really here you might have the point the idea that oh what does it help if I go to a small subset I will lose information somehow but here you only go to the punctured spectrum no this is called the punctured spectrum so you just remove one point in order to get the locally free situation punctured spectrum and because we are graded we can look at the projective variety and no I mean the cone map is defined anyway only outside this point so what happens on the projective variety reflects anyway what happens on the punctured spectrum not what happens on the spectrum itself and so on this guy we basically the same sequence but now we have to allow shifts not just in order to have the complete information I mean here we have now a graded situation but in the graded situation we have shifts and if we go to the projective variety this has the effect that we have here some M no and here what was the OU before becomes now O Y M minus DI I goes from 1 to N and here we have the effect that we have here again we have the also denoted by just the season cheese but now we first we allow here twist and now so here we don't have a free object anymore but they are still locally free yeah say it again I didn't get it it's a standard graded standard graded a zero is the field yeah standard graded yeah and before I forget it later so R will be normal and the dimension will be at least two not dimension zero and one is trivial more or less that's all we have this sequence on the on the on Y and we will work with that sequence but first we have to relate this with the time and we are over an algebraically closed field and of course that is true in every characteristic but later on we will work in positive characteristics okay so not and what are we interested so here we still have this guy and so this is basically in this situation if we are a finer type of an algebraically closed field this is you can forget about the length so this is just this guy the dimension of this residue class ring and what I just said if we apply the Frobenius pullback so now I mean the absolute Frobenius on Y we can pull back everything and maybe I get rid of the M here for a second I pull back and then I plug in M again it's a little bit different different no and no the nice thing that is because of the exactness of the Frobenius on locally free sheaths the Frobenius pullback of the scissor cheese is just the scissor cheese of the P to the E power of the elements no so we get the this sequence to the Q's power so here we have it's just multiplied by Q and no here we have O Y anyway not and then because but we are also interested I have some color now we plug in now we can have to look also at all the M twist no here gets an M and here we just write the M here and our dimension we want to compute here this is now the core kernel of the Frobenius of this map and because we have exactness we have the following formula that the dimension of R modulo I to the power Q no it's a great situation we look at the M degree component and that we can compute compute by evaluating by globally evaluating these guys and taking the alternating sum no and because of this normal and dimension assumption if we evaluate this we get the dimension of the degree M component of the ring so this is H 0 O Y M minus sum so I don't yeah that's enough H 0 of O Y M minus Q times Q minus Q times the I plus H 0 of the Cisigis on the projective spectrum in the M twist no and so this thing is just the K dimension of R M so this is easy to compute this is easy to compute at least asymptotically no you might have in small so if if this twist is close to 0 you might have some some difficulties depending on the genus but if this number is large enough and asymptotically you only have to look at when this is large enough then then this is also easy to compute so say this easy to compute this easy to compute this is difficult to compute it's the hill but couldn't function but then basically we have to look at this guy now the question is did we win anything so we won several things so if we denote this guy by S no so now we can say okay now we have any locally re-chief on a projective variety can we what can we say about the Frobenius pullback of S in twist M and depending on what we have to compute a Q and M no so this one should consider as a kind of Frobenius Riemann-Roch problem no Riemann-Roch problem is basically yeah yeah say something about how global sections of straight vector bundles behave and now here we have also the Frobenius action to understand here somehow S is basically any locally free sheath but in F F is the Frobenius pullback H0 this is the you evaluate the sheaths and this has a K dimension and this is because we are on a projective variety this is finite dimension and this is the H0 because gamma is also capital H0 that's this thing so we are talking about the global sections of vector bundles and their asymptotic behavior as Q and M varies. So what do we gain by that? No we can work over so one thing I said already we work with locally free sheaths in particular taking Frobenius is exact we can no we can use all the machinery of algebraic geometry like intersection theory, ampleness, homology, Riemann-Roch theorems, serial duality not everything which was developed there and but still this is in general a very difficult question so for the rest of my talk today so I guess everything goes a little bit in the direction of evening today I started I think 18 parts I do not want to skip too many things so this is now our new situation and so now from now on so we still keep the graded assumption so R is two dimensional and normal as before everything as before so Y is a branch of R so this is now a smooth projective curve over an algebraically closed field so compared with other stuff easy object no and to have an example in mind you can think of for example Fermat curves are always good to consider in this situation so we have a plane curve over our field think of such a curve that's our curve and no our vector bundles no somehow they look like that my colleague in Sheffield said always that I am only able to draw one picture it always looks like that and then he did the same picture and made fun out of me but so what makes life easier in the curve case so main point is we only have the zeroes and the first comology and they are related by their duality not starting with if we have dimension three then we would have here a surface and on the surface we have H0 H1 H2 H0 and H2 are related by by their duality but H1 can be anything in between and it's difficult to control it not and then we have the notion of the in higher dimensions we the degree of a vector bundle you need a polarization that we don't need here so what is the degree of a locally free sheath say of rank R so there are two important invariance of a locally free sheath the rank which is trivial the rank here is just n minus one no the rank is additive here rank n here rank one you subtract no this is the dimension of the fiber S in the linear algebra example from the beginning and Frobenius taking pull back the rank doesn't change and the degree no you if R is the rank you look at the determinant bundle and this is now an invertible sheath and invertible sheaths correspond to devices and they have a degree no so basically maybe I don't draw a picture for that thing no so here you have an invertible thing so it looks just like that now here I have drawn something of rank two no the fibers should look like a plane and now if you have here any section a rational section so it must not know it might have holes somehow like that no then you just count the zeros and the number of zeros counted correctly with multiplicities is the degree and this number is always the same same is independent of the of the chosen section no okay that's the degree and the degree is also additive so therefore we can know the degree here is easy to compute and so that gives the degree here and with the notion of the degree and the rank we can denote we can define the Mumford same is stability and these notions are now always defined for all locally free sheaths but of course in in the center in the center we have sheaths which come as sheaths from from an ideal no so S is called semi-stable if for all sub bundles sub sheaths we have the no we now we have the sub bundles now we look at the strange combination we divide the degree by the rank and this has always to be smaller or equal the corresponding expression this is also called the slope of the bundle not so we have this condition for all sub bundles no it's so Mumford developed this notion in order to construct modulite spaces no the goal is to to to smash all vector bundles of given rank and degree into one variety as point in one variety and that is not possible for all bundles but the semi-stable notion occurs naturally when you try to build modulite spaces no so look strange at the first at first sight maybe and in positive characteristic we call S strongly semi-stable not in positive characteristic you immediately have the question if S is semi-stable what about the Frobenius pullback of it is it still semi-stable and the answer is no in general not so strongly semi-stable means all Frobenius pullbacks are semi-stable now what is well let's give one at least write it down one example so I said one of the main examples is the ideal is the irrelevant ideal and say if we are in a hyper in a plane plane curve so two-dimensional ring three variables one equation not then the maximum ideal looks just like that and then the facilities no so then the approach of R is a no like like here approach of R is a plane a plane curve into two and the the CZGs they so by definition it's a locally free sheath on the the approach but this is you can it's the same as you look at the co-tension bundle of projective plane and you restrict to this curve so let's call this curve C now so a very natural object no if you so projective projective space they have tangent bundle co-tension bundle and you look you just restricted to close sub schemes no and very natural objects no and how this behaves the global sections under Frobenius is already the the question for Hilbert Kunz a multiplicity of the ring so even in that case it's it's really very natural object I would say okay so what we need is no you cannot so the easiest if everything would be strongly same is stable then it would be much easier that is not the case but we have filtration so we have a filtration S1 in S2 in no I don't know ST maybe ST is the last one we have a filtration of bundles and the property S is that S i modulo S i minus 1 is always semi stable so this is called the hardener or similar filtration long word I don't write down so hardener or similar filtration now and the slopes this is the slopes they are decreasing no so the slope of S1 modulo 0 is the largest one then S2 modulo S1 and so on and they are all same is stable also same is stable of decreasing slopes so this is the slope okay and still not good enough because we need something with strong no and if you start even if you start with a same is stable thing now if you have a same is stable thing the hardener similar filtration will just be the object itself not but now we apply the Frobenius same is stability is destroyed then we apply the Frobenius again and it's more and more destroyed however there is a very important theorem of Langer in this case which says which is also true in higher dimension it says that there exists an E such that the hardener that the hardener is formulated like that such that the hardener or similar filtration of the pullback is stable and that means all the you look at the hardener or similar filtration of this pullback and then all these quotients are strongly semi stable no and that then means that if you go higher and higher you just have to pull back everything and nothing else is changing no you have a stable behavior so maybe up to E you have a chaotic behavior but after E you just pull back the hardener or similar filtration again and again to get the new hardener or similar filtration okay and I wanted to give here one example so if you look at the Fermat Quartic then this guy in characteristic 3 that's the easiest example this guy is semi stable but if we pull it back with characteristic 3 no then we have to write here always a 3 it's not semi stable no so this gives the example of a semi stable bundle but not strongly semi stable no and the point is you can use you can use the curve equation directly to to get a section in the fourth twist of this thing and now if you count the degree so I know you can count the degree of the scissors here you have to count 2 times 4 minus 3 minus 3 minus 3 which gives minus 1 so this has negative degree but it has a global section coming from here and semi stable bundle of negative degree cannot have global sections so no this is I would guess the easiest example but there are many many examples of that type and yeah that's that's one of the main important properties of semi stable bundles that if they have negative degrees then there are no global sections and we want to understand this guy and if we also have semi stable we can say in which degree this will be 0 and then by by said reality everything will be there up to the she's coming from a canonical she's yes that you have if you go to higher powers so you have so suppose E is 0 then you have already so suppose this is a stable a stable partner similar situation if you go back to the first guy now you would always have this guy as I and the statement is that stable means that this are these quotient so for Venus pullback of that module for Venus pullback of that is again see me stable and that holds them for all of them so you cannot not stable in a sense that you always have to for Venus pullback it but then no further complication and the point is that there exists an E you have to go to a very high E but then it will be stable and then everything is just a repetition of what you have found here okay and with that well I can come to I can I think I can come to the end now so so I should say that this method of of understanding Hilbert Kuhn's theory in terms of vector bundle was developed by Trivedi and myself independently Trivedi is also here so she can she can tell everything what I do wrong here and so now let's go back to our situation we have dimension to normal standard graded we have homogeneous elements of degree D I and then we look at the so theorem Trivedi yeah so how do I formulate it so S are the CISG's and because of of longer we know that some for Venus pullback will be have will have a stable hard on our similar filtration and let's write this down so so is now the six number of course you could take a higher number F E of S has hard on our similar infiltration and then you have these objects and they have slopes and these slopes we denote by slopes mu K and if we have these slopes we define mu K now you have somehow to get rid of the E so you have divided by E again and so this is mu K but think of it just we just have to know the numbers coming from the strong hard on our similar infiltration that's the main idea so minus mu K divided by P to the power E and the degree of the curve so you know and then the statement is that the Hubel Kuhn's multiplicity of E is a degree of the curve divided by two and then we have and the rank the rank of these quotients of these quotients are R K no and then we have the sum R K times mu K times mu K times mu K squared no K comes goes from one to T minus the square of the degree of the generators so here we have a nice formula for the Hilbert Kuhn's multiplicity and of course the what you see immediately if you look at this side everything are rational numbers so from that theorem if you want to say it's a corollary the Hilbert Kuhn's multiplicity of the ideal in this dimension two setting is a rational number so this is a positive answer to the question of Monsky in this case okay thank you very much mu K is the slope of this quotient in the strong-hearted of this quotient you use all the mu K's all you have the strong hard on a similar situation therefore you have these quotients and each of them have a slope and here you sum it well in this version and the square of them multiplied by the rank of this guy now so all the information is in the degree and the rank of these guys