 to be Holger Brenner who will be giving his third talk and it will be on vector bundles and tight closure. Yeah, so good morning everybody. So what I forgot to mention yesterday is the direct relation between tight closure and Hilbert Kunz theory. So I have written down this theorem due to Hoxter and Unicke. So, no, typical situation where we talk about tight closure, a local Neuthrian ring of positive characteristic and with some minor extra conditions like analytically unremifried and formally equidimensional. Don't take too much attention to that thing. And I is a primary ideal, primary to the maximal ideal. Not then we have the equivalence that f belongs to the tight closure even only if the Hilbert Kunz multiplicity of the ideal i equals the Hilbert Kunz multiplicity of the ideal generated together by i and the extra element f. No, so that means that if you throw f into, make the ideal a little bit bigger by throwing in f, that does not change the Hilbert Kunz multiplicity. Even only if f belongs to the tight closure. So that's again a way to look at are the tight closures really tight? Meaning that an element f is inside if it does not change the Hilbert Kunz multiplicity. Okay, and the last statement I had yesterday was this. We are in a two-dimensional standard graded normal situation and we consider the corresponding smooth projective curve. We have a homogeneous ideal, primary to the irrelevant ideal generated by elements f1 up to fn of degree di and we look at the corresponding CISIGI module. CISIGI sheaf, CISIGI bundle, whatever. And the condition here is that this is strong, strongly semi-stable and then the statement is that we have an exact degree bound for tight closure. So the element belongs to the tight closure of the ideal or not depends primary on the degree. So we have suddenly a change of the behavior depending on the degree and the degree bound is this number. So the sum of the degrees involved divided by n minus one. And the statement is above this degree bound, everything is in the tight closure, below the degree bound, it's only inside the tight closure if it belongs to the Frobenius closure. And even that usually here Frobenius that you really have to go to Frobenius powers is an exception. So morally you can think only if f belongs to the ideal. And maybe we look at two cases, so let's look at the parameter case. So we have just two generators, then that was known much long before, even for in all dimensions. That in this statement we have just here the sum of the two degrees divided by one. So here the degree bound is d one plus d two. And so this property is here trivial because the synergies of two parameters, this is just the structure sheath but now I have to in degree, I think like that. So that would be the graded isomorphism you have here. And the invertible sheath is always strongly semi-stable, it's always semi-stable and the pullback is invertible. So the concept of semi-stability was not invented for invertible sheaths, it was invented for vector bundles of higher rank. So you get this immediately, and for example, yesterday, Keiichi, Sama, Hubek, no and this is a standard example of tight closure theory, so z square is in the tight closure of x and y. And you see x has degree one, y has degree one, z square has degree two. And maybe I also say now, yesterday I talked about the the cormology class corresponding to this data, so the cormology class would just be here z square, so in Czech notation, z square over x, y. So in a parameter case, it's always just the element divided by the product of the two parameters. So the Czech class is very easy here. So that's the Czech class in H1 over the curve and here. The degree is then, so you are then in the zero shift, which you usually don't write, of course. Let's stick to this ring and make one generator more, which is also a classical example and this is already quite needs some effort to show that the product of the, so in this ring, the product of the variables is inside the tight closure of the ideal generated by the squares of the variables. Now if you count again, two plus two plus two is six divided by two is three and on the left-hand side, we have an element of degree three. So that's the interesting case. So in, ah, if you get rid of x, it's not true. Then you are below the degree bound and that's not wrong. Now so this was not, the first proof of that was by Anurag thing and with the method of the series, this is now very trivial. So we look at on the corresponding elliptic curve, we look at the CZGs and this guy produces a Comology class and I could write it down how it looks like but that's not important right now. We just know that there is a Comology class. And now the curve equation, we can immediately, and here I should give the degree three shift because of that. Now always the degree of the element, we are under, I want to understand where it belongs to the high closure. This degree is here as the degree shift and the curve equation gives us a global section of this bundle just by sending one to x, y, z. Now you see x, y, z is a ZZG because x times x square plus y times y square plus z times z square is zero by the curve equation and so this is injective and on the other hand, not because we know the degree, we know the rank so here we have something of rank one again, rank two and the degree is zero so therefore it also has to be zero and it cannot be another line bundle so it's also the structure sheaf. So we have this nice sequence and by that sequence we see immediately that this is semi-stable. This is semi-stable and we also see immediately that it's strongly semi-stable because if you pull back the whole situation, the left and the right will be the same. There's also even a theorem that on an elliptic curve semi-stable is already strongly semi-stable so the elliptic curve is easier than curves of higher genus so the phenomenon that a semi-stable bundle is not necessarily strongly semi-stable occurs in starting with genus two. So this is strongly semi-stable, it has the degree, so we can apply this theorem, therefore we get this result without any further computation. I mean you can try to do that by hand to really look at the definition of tight closure and good luck with that. Okay, so that's the case of the strong semi-stable bundle now we wanna have a characterization of the general case and I go back to the situation where we study in general a locally free sheath, C is our smooth projective curve and S is locally free on the curve positive characteristic and we have a homology class, a first homology class, I mean there are no other homology classes. Now and you see already here for example in that example that, well it might be, so we have here a homology class in a CZG bundle but in order to understand that we have to look at certain exact sequences and there is no reason why the other sheaths occurring in these sequences should be also CZG bundles in a natural way. So it's better to work in general for locally free sheaths and the question which we came up which was a translation of the tight closure inclusion question is whether the torso defined by this homology class is an affine scheme or not. No, that's the setting. And the main characterization uses again the strong hardener or similar infiltration. No, so and we have mentioned that such a thing exists so that S1 and S2 in ST minus one ST and that is the, now here E is really one E maybe we should write E0 or something. So that's the strong hardener or sim hunt filtration and the slopes of the quotients are decreasing so we have mu as one is larger than mu. So as one is also called the maximal destabilizing sub sheath, this and it goes on like here we have at the end we have S modulo so the slope of that is just to remember and now I mean the interesting case is when some of these slopes are positive and some are negative. If everything is positive it's trivial if everything is negative it's trivial. Another interesting thing if we have here something negative here something positive. And then we look at say I should have the property that SI, I mean the slope is positive if and only if the degree is positive. So this has a non-negative degree the next. So SI plus one modulo SI has degree negative but we'll look some more. I mean if it doesn't if only this exists that's in some sense also considered. And now we look at it seems to be a long theorem. So now we look at this guy modulo SI. Now so basically we have FE S, E S we have here SI which contains all the non-negative stuff in some sense and here we have the quotient which contains all the stuff of negative degree. So that we denote by FE I. You can also think if you go some Frobenius powers higher you can even assume that the Frobenius pullback is the direct sum of these quotients. No it's a direct sum then it gets not really much easier but it's more or less the same. And we have here our homology class C and no due to the long exact sequence in homology this class has here an image class. Do we give it a name? Maybe we don't give it a name. Now we have all the data which we need to formulate our theorem. So the theorem is now that the torso belonging to this class is a fine. I think I formulated it with not a fine, not a fine scheme. So corresponding not belonging to the tight closure. If and only if the image of the homology class of C inside H1 of this guy is not zero but to be fair one has to say it's not killed, not annihilated by Frobenius. So you should think of this theorem as a degree criterion but taking into consideration the strong-hardener or similar infiltration. No you need the strong-hardener or similar infiltration that's the easiest case. You just S is strongly same as table itself. Then it's just a question of having positive or negative degree. Not but here it depends. So if the class is non-zero on the positive side it belongs to the tight closure. If it here goes to zero or is killed by Frobenius then going some it has higher. You can assume it is zero. Then the class comes from that side and then it will be inside the tight closure and the scheme will not be a fine. Now so that's the general statement which characterizes a fine porous. Now today I wanna talk about the plus closure and its relation to tight closure. So maybe I can motivate it from the geometric side. So here we have our curve. We have our torso, it looks like that, TC. And for that it's I think this interpretation with the extension. So projective bundle of an extension without P S. This is quite helpful. So maybe we draw this. No this is the torso which is each fiber is in a fine space but here we have the projective closure and then we have a projective bundle. No but the torso you regain the torso by looking at the projective bundle minus the hyper plane bundle. Projective hyper plane bundle. And for tight closure we are interested in the comology property of that thing whether this guy is a fine or not. And so don't ask me what this noise is. Not a fine. And the easiest way to detect that this torso is not a fine is if we can find a curve inside the torso dominating the base. And by that I mean a curve inside the torso. So of course there are many curves inside. But most curves if you think of that guy most curve will meet somehow this divisor and then they are not completely inside the torso. Now so we are thinking about do there exist curves in PS prime which do not meet this projective sub bundle which are completely inside the torso. And because it's a projective curve as a sub scheme of a projective. Not everything is projective. Sub scheme of something projective. And if it would not meet this thing then it's a projective curve inside the torso and then the torso cannot be a fine scheme because a fine schemes do not contain projective curves. So you can think of that if we know a co homological property does there exist a geometric reason for that? Now if we have you can also, well I mean you can do it on all level. If you have a projective curve inside there will be homology and there's a closed sub scheme also there will be homology of the torso. Non trivial torso. So you should think of the question is there yeah, is there so in case it is not a fine is there a geometric reason? Now the existence of a curve inside the torso would be a geometric reason for the torso not to be a fine. Now that is the question how I will address it. So the corresponding algebraic notion which came first is just the following. F belongs to the same notation to the plus closure if and only if there exists a finite map from R to some ring. No the important thing is finite such that belongs to the extended ideal. No? And we had some so assume that everything is a finite then in particular you can take the Frobenius. No but there are also other relevant finite extensions where something can happen. And it's known that the plus closure is subset of the tight closure that is not a difficult theorem. So whenever you can after going to a finite extension it's inside the extended ideal you know that it's in the tight closure. And so that was cause of question that was in particular a question because of the relevance for the localization problem where this is just the same. And Boxter calls this the tantalizing question so I never heard about this word before but it sounds good, tantalizing question. So tomorrow I will show that this is not true in general today I will show that it's true in this setting with an additional assumption that we are working over finite field. So of course there are many finite maybe one should restrict to a domain to make it life a little bit easier. You could also say we look at the absolute integral closure. Now that is so you go to a quotient field to a algebraically closed closure of the quotient field and you look at the integral closure. So that is called the absolute plus closure. And then you can just say the plus closure is just contraction from this large ring. No I mean this is highly non-Netherian. Maybe Kevin knows something about it. Now one of the nice theorems is that in positive characteristics this is a big coin Macaulay algebra, not highly non-Netherian. Okay, not only of course it contains the perfection of the ring. Okay, so and again, so because I work in the again are normal standard graded two-dimensional I will work with the graded plus closure. So I look only here at graded extension but with degree shifts are allowed. No, if you have Frobenius you will have definitely a degree shift, not a shift shift. You have to multiply by something, not by P. Okay, no and boom, boom, boom, boom. So basically in this situation we can say so F belongs in the, now again I also stick to the primary case. I belongs, F belongs to the graded plus closure of the ideal. No, it means you have here a graded finite map but if you have a graded finite map then you have a corresponding map of curves. No, so then you have there exists a curve C prime to C such that, let's call it phi, the pullback of the homology class inside the, that's the one. The first homology of the curve on the left of the pullback of the CGT bundle, which I denote by S here is zero. And here maybe at first step S is maybe not normal but you can, you can here go to the normalization or you can go here to the normalization so then you can only have to look at smooth projective curves. So the question is basically can you kill a homology class by going to another curve? So how can you characterize that? As before we translate everything to the curve so we have a, we have a locally free sheath, we have a homology class of that locally free sheath, we have a homology class and the question is can we or when, when can we kill C by curve map, of course a non-constant curve map which is then automatically finite C prime to C. And part of it is what can we kill by Frobenius? What can we kill by other stuff? So the Frobenius itself plays or carries a leading role but the Frobenius alone is not enough. What can be killed by Frobenius would be corresponding to the Frobenius closure. Okay, well that's the question. And the lemma relating it with that so equivalent the following are equivalent to this situation. So C exists finite C prime to C such that the pullback of C is thorough. And second thing is the torso T, C contains a projective curve which is exactly this property I was talking about. And it's quite, quite immediately. Now, I mean the main point is the torso of C is the universal object which kills the homology class. If you pull back the class to the torso itself then it's zero. But of course the torso is not finite. And if you have such a curve maybe you have to go to the normalization. So here we don't say anything about that but it has to be a smooth curve. But if it's trivial on the torso then it's also trivial on that curve and then it will be trivial on the normalization of the curve. And the other way is similar. No, so that's quite easy. And so the goal is for today to prove this or the graded case of, which is stronger the graded case of this under the assumption that the field is finite or that the field is the algebraic closure of a finite field. And this is really an essential property we need here. I would like to leave this because I will use it later. Now, and I would say a direct proof in the setting of commutative algebra is more or less out of reach. I think you have to go this way, this interpretation on the curve. So that I don't need anymore. And so what we wanna show is that property which characterizes by this theorem the non-affinous of the torso will be the same criterion which characterizes this property. And so the two properties being non-affine having a projective curve are not directly related but they are related via this yeah, this semi stability stuff and this numerical criterion. So I think of that as a more or less as a numerical criterion for tight closure. And so our goal is what, how to kill a homology classes. And this question is by the way very different in characteristics zero. In characteristics zero, you can basically kill nothing. You can only kill if you have a normal thing, you have the trace map, you can kill nothing. In positive characteristics, you can kill a lot but how much we can kill, we will see. Okay. And so, so we need some concepts here. So S always a locally free sheath is called et al trivializable if exists. So this is trivialization of a bundle. If there exists a map, so I always mean a finite map of smooth projective curve such that the pullback of S is just, S has a certain rank. This is just OC prime to the power R. Of course that you can only expect if you have degree zero. So that is et al trivializable. And so there is the following theorem which is, so I know it from, so it looks like several people have proved this. I know it from a paper of Lange and Stuller but it's also sometimes a sign to Mumford and also to cuts. So whatever. And it basically says you can trivialize a bundle. Now this map should in that case be et al, et al and finite. So S is et al trivializable if and only if you have a repetition in the Frobenius pullbacks. So if there exists some E such that, no now I mean the Frobenius pullback of S is isomorphic to S. No I mean if you have the bundle S you can apply the Frobenius on it how often you want. And maybe they are isomorphic S abstract locally free sheaths or not. And here we require that some Frobenius pullback is isomorphic to S itself. Then you can trivialize it by an et al map. And it's quite surprisingly explicit this statement. You can really write down matrices and derive from that curve equation. Where you can do that. Okay and now we do I mean here we started understanding the tight closure behavior by first understanding the strong semi-stable case that we will also do now. So we first consider the strong least semi-stable case and whether a homology class can be killed or not. And first we need a positive result. So it's first in so now let's it's not the first time and maybe the main time. So K is the finite field. K is the algebraic closure of a finite field. Now that would be also as you please. That doesn't make a difference. And we have our curve. And so S is strongly semi-stable of degree zero. Another degree zero is the interesting case where most of the phenomena really occur. Now then the statement is then exist E prime larger than E such that the ETH prime Frobenius pullback of S is isomorphic to the ETH Frobenius pullback of S. Now this is look similar but it's not the same. Now here we have some some E on both sides, no? And now the point is a proof is quite easy now. Well, it uses heavy machinery in the background but we just look at the family of all Frobenius pullbacks and by our assumption they are all semi-stable because S is strongly semi-stable and they are all of degree zero. Now I mentioned that this whole idea of semi-stable bundles was invented to be in order to construct modular spaces for bundles. And so here it has a certain end the rank is of course fixed. So we, there exists the modular space of degree zero rank R bundles. And so all these objects, if this is the modular space they are all points in the modular space. Now that's the idea of the modular space. And now everything is defined over our finite field or maybe a algebraic closure of a finite field but S is definitely defined over some finite field. Now you only need this data and by doing this process the pullbacks are defined over the same field and the model, now that's the job of the modular space that it's really a variety and the variety has only finitely many points over a finite field, not a thing of projective space you can say which points are over defined over finite field. Now so we have an infinite family in a finite set of points and therefore we have a repetition, that's it. Okay, so you only need boundedness of the family but this is the important step to construct modular spaces. Okay. So next step, theorem, now S is strongly semi-stable and it has the degree is non-negative. And the C is a homology class. Then the statement is then there exists the prime to C such that the pullback of the class is zero, not so five kills the homology class. No, so how does that go? So first, if the degree of S is positive then we can take the Frobenius even because by pulling back, so S has, S has degree one then the pullback has degree P. And even you have to multiply by the rank I think. So the degree is then by looking at the Frobenius pullbacks the degree is getting arbitrary large. And they stay semi-stable and therefore there will be no homology anymore. No, they are getting ample you can say. No, the Frobenius pullback of S, if S is positive Frobenius pullback is very, very positive and at one point these bundles will become ample and then there is no homology inside anymore by serduality and semi-stability that you need here. Okay, so here you can Frobenius. So now the case where the degree is zero then we are in this situation. And then we first go to the smaller number we go by Frobenius and then after this pulling back the situation to C via the E Frobenius we have that this guy and a further pullback is isomorphic but then we are in this situation. So here on after Frobenius pullback we arrive this situation and of course so always now a finite field. So that is really important or over a finite field. And then by this thing we have here C prime so here Frobenius and here Etale and so as the pullback of the complete thing here is so now we have a homology class C, no, prime is here. Now we have a homology class inside the structure sheath or the half power of the structure sheath but that we now we can consider the components. We consider the components and no, we have to kill the components and then either we can use the parameter theorem of Karen Smith that type closures plus closure for parameter ideals or even easier we can even show that either so we can no we look what does the Frobenius to that thing and the Frobenius kills part of it maybe nothing maybe everything and what is not killed by Frobenius there is a basis where the pullback of the homology class goes to itself and from that we can build so-called Artin Schreyer extensions to kill the class so either kill by Frobenius or Artin Schreyer sequences or Artin Schreyer extensions, sorry. Okay, that's this, that's this. Okay, no and so the last theorem is now maybe I do it with blue for plus so now we change this theorem now we say over a finite field no so this, this and that, this basically is the basis to prove the same theorem for containing a projective curve so here instead of not being not a fine we write contains a projective curve no and projective curve is the same as being killed by a finite extension no, no and then finally so for C we can say now finally the finite field that is TC is not a fine if and only if TC contains so there is a geometric reason for it contains a projective curve or the class is killed by going to a finite extension no and so that's a theorem and going back to our algebraic setting our normal two-dimensional, two-dimensional finite field we have that tight closure, tight closure equals graded plus closure and then it's also plus closure and so this is for the homogeneous ideals deeds has extended this to non-homogeneous ideals but so still in the homogeneous ring graded case dimension two but then we found a way to generalize starting from this also to non-homogeneous ideals and I think that's it for today, thank you Are there any questions? Why is that true? This is zero, I go here and here I have this isomorphism where we have E and E prime but if we first do the E for binius then we have here something and here basically we have E so that is the first step and here say if T is Fe of S then for T we have I think something like that E prime minus E of T is then isomorphic to that guy and for this T we apply Lange Stuller to get the trivialization of T and so all together we get applying for binius and applying the ethyl map coming from Lange Stuller we get the trivialization of the vector bundle that's the point Yeah, yeah sure but still if I have a strongly semi-stable bundle defined over algebraic closure of a finite field then the data which are relevant will belong to a finite field and then I do it and the for binius pullbacks of them they are also defined over this finite field and so all the points which occur here are over this finite field that's the argument of course the modular space is infinite but the FQ points are finite Are there any more questions? Not we'll just thank the speaker and our next stop will begin in three minutes