 the n minus one-dimensional subvector space of KN, unless all the Fs are zero, in that case, the complete space is the solution set. And in a second step, we took a geometric object, say, and now we consider functions, differentiable functions, algebraic functions, on this topological space or manifold or whatever. And then we can form the set consisting of all base point and variables t1 up to tn with the property that the sum fi p in certain times t i is zero. And this gives an object, a geometric object above. X and p is the base point. And the fibers, as soon as one of the fi does not vanish at p, the fiber will be a n minus one-dimensional space. And, but in general, it will happen that the fi's have a common zero and there the fiber goes up. I'm sorry, it looks something like that. Here you have, say, n is two, then it looks like that, then suddenly at a point where the all functions vanish, the dimension will go up. And that makes it more complicated. So usually we restrict, we restrict to the open subset outside the locus where all the functions vanish. And then we have a constant dimension of the fibers and then, say, be restricted to u over u is then a vector bundle. And in the case of commutative algebra, our functions are now elements inside the ring. And we look at the CZG module of these elements which live inside, the best thing is to express it with a short exact sequence. I is the ideal generated by these elements. This is just the kernel and the map is just given by the matrix given by these elements. Now, and again, this is only a module, but if we look at the, so X will now be the spectrum of R and U is the subset where not all of the FI vanish. Now, and if we look at the sheaf version of that, then we get here a vector bundle, so a locally free sheaf. No, this restricted to U and then here we have the structure sheaf of U and time and if here we get the structure sheaf of U again once. No, and yesterday I explained how, and in the graded case, we get the same thing on the corresponding projective variety. No, and yesterday I explained how to get results on Hilbert Kuhn's theory by looking at properties of these vector bundles on the projective variety. In particular, in the situation where R is a two dimensional ring and then the normal and then the variety is a smooth projective curve. Well, that was what we did yesterday. Now today, so this is a homogeneous linear equation. Now today, we do the more or less the same thing but we make it a little bit more complicated. So the left hand side of the equation is the same but now on the right hand side we have another element F. So this is an inhomogeneous linear equation in N variables. No, and you know how the solution behavior is. No, and what we also learn in school, say if this is the solution space here, T is the solution space here. I mean the solution space can be empty but if it's not empty, then we can pick, or let's say, yeah, we have an action of the solution space of the homogeneous equation acting on the solution space of the inhomogeneous equation by just adding. No, in school we say solution to the homogeneous equation plus solution to the inhomogeneous equation gives a new solution to the inhomogeneous equation. And if we fix special solution, we even get an isomorphism, a bijection, better say a bijection, but this is not canonical. No, we cannot, so the solution space is not a vector space here. Maybe we say solution set. Here the solution set is a space, vector space, here not. Now let's look on this level. Now we have S before, we have our functions to K, so think of K as a complex numbers or whatever, but also F is now a function and now we can build the same thing but here now we don't have zero, but we have the value of this extra function at the point P, so that is now, maybe I write it with new variables. Oh, maybe the old variables. No, and now the condition is just F, I, P times P, I has to be equal to F, no, this function at the point P. No, and the image is more or less the same. The only new thing is that the fiber might be empty, no, but still basically the image looks the same. No, so the fibers, yeah, the fibers are still K to the power I, but for example, here you have a zero section. You have a zero section mapping every point to zero in each component, no, so you have here somehow. The zero section, here in general, you do not have a section at all or at least not a continuous section or algebraic section, no, there's no reason for that. So here we have very new global properties which might be interesting. And again, we have this action just by looking at this action fiber wisely. So this bundle acts on this guy. We will soon give a name to that. And again, we don't like so much that the fibers may have higher dimensions for certain points, so usually we will restrict to the same open subset and then we have such an action just by addition. And now here we have a vector bundle, oh, sorry, of course, T, and this guy, we will later give a definition. This is called a torso or a principal homogeneous bundle or there are many, many of fiber, principal fiber bundle, there are many names for that. So I will usually say torso, that's from a French school, I would say. Okay, and now in the algebraic setting, we just, no, we have still our n elements and we have an extra element, r, and this sequence is as before, nothing happens. So what can we do with f? And we will see that, so we are basically then interested in the question, whether f belongs to the tight closure of the ideal, the ideal is still given by the n functions and we are interested, we have now an extra function, an extra element of the ring, does it belong to the tight closure of the ideal generated by the given functions? No, and we will present, in fact, several representations of this problem, of this question where an element belongs to the tight closure of an ideal in terms of this torso, no? So this is a geometric interpretation of the inclusion problem for tight closure in terms of torsos over, yeah, over the, an open subset of the spectrum. That is basically what we are doing today. So, yeah, let's give the scheme theoretic definition of a torso, so where do I have it? I changed a little bit my, just to make clear what we are talking about. So we have a scheme, I call now the base scheme again, X, but it's rather this U, but later on it will be the brush. So let's have a neutral letter X and we have a geometric vector bundle, no, that's the geometric version of a locally free chief vector bundle. So you should really think, no, maybe we draw the base is one dimensional and no, we have, yeah, for each point we have a vector space of six dimension R, which is called the rank and locally it looks like an assigned space over the base and the transition map is a linear, that's important. So in particular, we have an addition here on the vector bundle. And then the definition is a scheme, P over the base X is called a V torso, not if we have an action over the base X to P and the point is locally this action should look like the action of the vector bundle on itself. And the union of UIs in a certain index set and identity, no, a scheme isomorphisms from P restricted to UIs to V restricted to UIs such that the following diagrams commute, no, so, no, I mean from that you just restrict, you go to U, I don't write UI, I just write U, not that this must be true for all I, no, you have this thing and then, no, you have here, you go down to the vector bundle. So that's a U, that's a V and here you have also this thing, no, and downstairs you just have the addition in the vector bundle and here you have phi, oh, phi, and here you have, no, I mean here the identity and phi, no, and that has to commute. No, so here this is the action of the vector bundle on that and yeah, so another point is locally it looks like the action of the vector bundle on itself but only locally, that doesn't mean any global statement so globally it's not a vector bundle, you don't have addition, you don't have zero section on T. Now it's a more complicated object, okay. And yeah, this, so the question is, how do you classify torsos? And the main point is that it's given by the first homology of the vector bundle. So say X is a scheme, maybe you need separated scheme or something and you have our vector bundle and maybe usually I identify vector bundles with locally free sheaves, here I wanna be, because here it's really important that it's a geometric object, I wanna be clear so here we take the sheaf of sections in V which is the corresponding locally free sheaf, no, and then there is a correspondence between, so on one side we have V torsos like that and the first homology, sheaf homology inside the sheaf of sections. Now so for example, over a fine scheme, every torsor is trivial, and trivial means here is the vector bundle, but usually you have homology, no and I denote, so here we have a class and the corresponding torsor we denote like that. And not the easiest, so I will not give the proof but think of you represent your homology class, first homology class as a Czech homology class and there you have transition mappings and out of these you construct T, no? Now and the point maybe here I've said for the vector bundle situation I said somewhere that the transition mappings are linear, here the transition mappings are only a fine linear so you have a linear plot but also you allow shifts, that's the only difference. Okay so we have this representation, another nice interpretation of this object is, so we will get at least two nice interpretations, so. If you look at this guy, well this has many interpretations so here I look at the, now it's always difficult to remember in which direction I have to go, I think like that, no? So you have this identification, X1 is the shift homology, X1 where in the first entry is the structure shift and no but this you can really interpret as extension and as an extension so now let's fix the class here, consider the class here, we look at the corresponding extension and then everything turns around, you have a short exact sequence and in the middle, no the object in the middle together with this mappings constitute the extension and the thing is made in such a way that if you look at the global one here this will be mapped under the connecting homomorphism of shift homology to this class. So one, no if you look at the long exact sequence for the starting point, no you have here this, usually denoted by delta, one is mapped to the class you are representing by a torso and now we look, so S is our vector bundle we start with, S prime is also vector bundle because it's in between two vector bundles and for, so we consider it as maybe it's, I do the, I go back to my vector bundle notation so we have a vector bundle of rank say if V has rank R, this has rank R plus one and we have a vector bundle and a sub vector bundle of rank R, rank R plus one and here we look at the corresponding projective bundles. So this is exactly the process you do when you construct projective space, you start with a vector space and you look at all the lines going through the origin but that you do now point wise. So here you have, here you have your vector bundle and now for each fiber you look at the corresponding projective space. Now, so we get from this inclusion, it's a strict inclusion, we get that inclusion and now we look here, it's a projective bundle inside a projective bundle co-dimension one and now we look here at the complement. So projective bundle, so I mean projective bundle, I draw like a fine space, but here you have your sub bundle and now you look, so point wise you look at say P, no the dimension has gone down by one, so we have here the other dimensional projective space without some linearly embedded R minus one projective space and then you have it every fiber. That's this object and this object, not therefore you can say the fiber, the fibers of this object over every point is a fine space again because not projective space minus a projective linear hypersurface hyperplane is just a fine space. So we know immediately that the fibers here over the base are a fine spaces and this is one interpretation of the torso. No, very, so you can choose. Either you do here, check homology, you construct the torso or you know what the extensions are and from the extensions you go this way and then you get the torso. The main point is you have a homology class is represented by a geometric object torso. Okay, now we've said that we wanna look at this situation. So we wanna understand that as a torso and there are several ways to do it. Maybe the easiest thing, if you look at that, no we have, on one hand we have our CZGs, F1 up to FN and now we just consider the new element F as the next element and this again gives us CZGs. So we have here, not just by, so the last new coefficient will be just zero, that is this map. So FN but then also F and then it goes to O by, just looking at the last component and no we have such a sequence and in fact, no this looks like that. So that, if you go through this construction you get on the spectrum of the ring you get also the torso. No so, as soon as you have a set of ideal generators and the next new element you will get a torso which puts together all this information. So that's one way to do it. Another way maybe, yeah a little bit more direct I would say is to look at so-called forcing algebras. So this is a construction of hoax that's done in a context of solid closure, I will come to that. So we have our ring, we have N elements. I think it sounds stupid if I say we have N elements and another element but I could say we have N plus one elements but the role of the element is very different and F elements in the ring and now we form the following algebra. So we join for each element for each generator variable. We one up to TN and now we mod out just one equation and that looks familiar. Basically we take this equation and now we can discuss do we like a plus or a minus? That is, no we have to decide. Today I write plus. And then if I say something like it's corresponds to the torso it might be true that it's only true up to a sign but that doesn't change much. So plus or minus is not very nice. Algebra very easy. So what does this algebra do? It has a universal property. In this algebra F belongs to the ideal generated by F1 up to FN. But you do not know anything about the coefficients because the coefficients are variables. And so as soon as you have any algebra where in the extended ideal F belongs to the extended ideal it will factor through the forcing algebra. But not uniquely. Here we have a bit, be a bit careful. So that's a universe. That's the forcing algebra. That's an R algebra. And if we look at the corresponding morphism of the spectra this will be the map from T to the base. So here we have in general higher dimensional fibers. So fibers are not constant in that situation. Now you can write down whatever you want to if you wanna have a concrete example. I will give some examples later. Now so this is a so-called forcing algebra. And to get the relation with the torsors is that if we again go U is again the union of the DFIs and if we restrict the spectrum of the forcing algebra to U then, so that is also proposition. This is a model, the torsor given by F in this case. Now at the torsor given by F we have either you do it in that way or we go back to the version on U. So here we have O, U, N goes to O, U not short exact sequence of really free sheaves. And not because it's a short exact sequence we have here. We have here, we can look at the long exact sequence of comology and an element F is an element in R and this goes, this is a global section here. And now we go by global evaluation. No, this gives a comology class by the connecting homomorphism inside, not first comology class inside the open subset of the CZG bundle. And as it is a comology class by this theorem it gives a torsor where the CZG is act on and but this you can also realize directly in that sense. Not so many things come together and it's good to have all these possible interpretations. So to remind you of the definition of tight closure so we have a domain, so I restrict to a domain and we have an ideal so in containing a field of positive characteristic and then we say F belongs to the tight closure of I if and only if there exists. Now most people would write here a C but for the multiplicator in tight closure theory but C is my comology class so I don't write C, I write Z, so Z not zero, I assume the main and Z to the power FQ belongs to the I to the bracket Q. That's the definition and okay so what has that to do with the forcing algebra? So in general forcing algebras and really in that sense are a very good tool to study closure properties of ideals or of modules in general by looking at corresponding properties of this map. So now the question is to the tight closure, how do we see it in this map? For example just to give you a very glimpse when does the element F belongs to the ideal in the ring itself, that's the case if and only if you have a section here and of course then other properties will be other closure operation will have a more complicated interpretation. But here we concentrate on tight closure and here the statement is tight closure is solid closure but now we have to say of course what does that mean? So it means that F belongs, so let's say R is normal, normal excellent domain, I think a local and then we say or then the statement is F belongs to the R and I, I is M primary ideal then we have that this is equivalent to the statement that so let D be the dimension of R that the local homology with support in M of the forcing algebra is not zero. Now this is definitely surprising at first glance. In particular on the right hand side you do not have any positive characteristic. One should say that in characteristic zero if dimension is larger than three or starting with three this does not give a tight closure, tight theory. So it's still one should think of this as a statement in positive characteristic. And so if you look at this, this basically means I mean you have downstairs the local homology of R you know that it's not zero by the ordnate theorem no if D is the dimension and you have a natural map to A and then the question is so here no we have said yesterday tight closure it should be tightly to the ideal. So if you smash F into the ideal and that is what the forcing algebra does it should not change too much. And for example, so this theorem says it doesn't change that this local homology is non zero. No it's still non zero. No so the converse is if F is not inside the tight closure then the homological dimension of the top of the local homological height I should say drops. No it goes down there. And so in particular and so if the dimension is at least two we can we have an isomorphism with sheath homology so the homological degree goes down by one and U is again the punctured spectrum. So the question is this is upstairs so I should write it like that. It's a pre-image of the punctured spectrum where everything is locally free. Now for D being two dimension of the ring being two and yesterday at the end we could only prove something in dimension two and that will be the same here. D being two here we have then a one and then the question is or then it translates I is inside the tight closure if and only if. No of course MA is not the maximal ideal in the forcing algebra anymore but it's an open subset in the spectrum of the forcing algebra and if this subset is not a fine not a fine in the sense not a fine scheme it's not isomorphic to the spectrum of a ring. So the situation of course downstairs if the punctured spectrum in dimension two is not a fine it's quasi-affined not a fine but now we look at this forcing algebra so the DMA is the complement of the exceptional fiber and this might be a fine or not a fine and that characterizes tight closure. Okay now I could go into the direction of results what we can do with that what kind of characterization we can some minutes past nine but not many maybe two or three minutes after nine. Yeah let's get rid of everything. So now for the rest of the talk today R will be normal two dimensional standard graded over say a field of positive algebraically closed field of positive characteristic and I will be primary to the maximal ideal of the standard graded homogeneous maximal ideal of the standard graded ring and we fix ideal homogeneous ideal generators of degree D1 up to Dn and we have another element homogeneous of degree M and this gives and C will be the corresponding smooth projective curve now we saw yesterday that then we have again same notation but different object now we have a scissor G bundle on the curve going to OZ and if we twist it again by M we twist it by M we get here and M we get here and M and then our element F defines a global section here and then by the connecting homomorphism the delta F gives us a first homology class inside the scissor G bundle in the twist M and then it gives you so these data because it's a homology class gives, let's denote this by S gives an S torso over the curve also we have here our nice smooth projective curve we have here our torso and then we want to know is the torso an affine scheme, huh? If it's a vector bundle it's not an affine scheme then we have a section, the zero section and then we have a projective curve inside and the affine scheme cannot contain projective curves in fact, that's a very, but then I will talk tomorrow the question when does this torso contain projective curves? That is tomorrow so today I'm interested when is this a fine or not a fine, a fine scheme and as yesterday, so it's more natural to ask this question a little bit more general so we have not because again we want to use the hardener of similar infiltration or the strong hardener and similar infiltration of that guy and in that the filtering objects will not be scissor Gs anymore so it's better to write from the beginning say we have a locally free sheath on our curve we have a comology class with the torso and we ask is it an affine scheme or not? That's the question and so let's come to the results so this situation and so this has a lot to do with the degree of an affine scheme so this is for the case and then you reduce to that case in general so theorem if S is strongly semi-stable, strongly semi-stable then the torso is a fine if and only if the degree of S is negative and I mean if the class is zero it can never be a fine because then it's the vector bundle and then it has a section and then it's not a fine but it can also happen but this is rather exception that it's not zero in negative degree but some Frobenius pullback I write it down so that would be the version in characteristic zero no, it's affine if and only if the degree is negative and the class is non-zero that's the in characteristic zero and in positive characteristic we have, we need the condition that the class is not annihilated by the Frobenius but this is in negative degree it might happen but it's rather an exception and if you are in a relative family it's for almost all prime numbers this does not occur but that is a subtlety okay so here we have what goes into this so we have results of Hartrong and Geesecker about the ampleness of vector bundles maybe I don't say more to that yeah, let's write down already what does that mean for tight closure yeah, I think I finished then by only doing the strongly semi-stable case so I will do the general case tomorrow we go back to this situation and now suppose that for some reason we know that the CZG module is strongly semi-stable that's an assumption here which is in general not true but as I said already by going to the strong-hardener or symbol infiltration we can reduce it to the strong semi-stable case and then what does the right-hand side says it only says that the degree of the CZGs has to be negative but the degree we can compute by looking at the sequence the degree of that guy here and then we get the result that so the degree of F is again M not then we have, so it's two statements if M so this I call the degree formula for no, you take just the sum of the degrees divided by N-1 which is the rank of the CZG then F belongs to the tight closure of I if M is below this bound F does not belong to the tight closure unless F is itself inside the ideal or I have to say that F belongs to the Rubinius closure so not but this is a rare phenomenon so morally you should think of you should think of like that unless F belongs to the ideal itself but to be exact it should be like that so here you have and maybe one degree tuple which will be relevant in the last lecture is so two elements is the parameter case easy, no then it's just the sum of the degrees divided by one so the next interesting case and that is already complicated is three elements and say all three elements have degree 4 not then we have here 12 divided by 2 so the degree bound is 6 in degree 6 we have interesting behavior above 6 we know it's in the at least if it's strongly semi stable it's in the tight closure below it's not in the tight closure unless it's in the ideal itself but so here just by the by the statement itself it will be inside but if it isn't strongly semi stable then the interesting behavior is in degree 6 so for example count example to localization problem will be of this degree type three generators of degree 4 one element of degree 6 behaving weirdly all the other behaviors are quite regular but in this degree if something can go wrong then it will go wrong in this degree type ok thank you very much