 Okay. Then I want again to welcome to all of you and let me thank my co-organizer, Professor Verma and Chung for the beautiful idea in my opinion to provide to young researchers a training in the foundations of commutative algebra and algebraic geometry in prime characteristics. We can say that we start in the best way. It is a pleasure and honor for me to introduce Professor Craig Eunice of the University of Virginia. Craig Eunice and Melvin Oxter developed first the theory of the tight closure in 1988. They published one of the first papers on the subject in the new series of bulletin of my American mathematical society. And let me recall that in the true introduction, they wrote with the fine, the tight closure of an ideal in characteristic P and use it to give a new much simpler proofs of many theorems in a great strangeness form. This is the beginning of a fundamental part of international mathematics. Then we can start the first lecture by Professor Craig Eunice and the title of the talk is where does the tight closure come from? Please Craig. Thank you very much. This is a can, first of all, I should ask, can everyone hear me all right. It's good. I'm going to share my screen and hope it works. Everyone see that. Okay, I will start so. Everyone can read that. I can't see the chat while I do this, so I want to encourage anyone that has questions to ask them at any time. And if somebody could check the chat for me and relay them thanks Kyle. That would be great. I also want to thank ITCP for hosting this event. I love going there, the times I've been there. Everyone has a chance to participate in another program in person there sometime. The organizing committee has been fantastic. And I want to thank them. Also, it's a huge amount of work to organize something like this. The list of tutors who are going to be giving the lectures is incredible. Many, actually all of them have contributed greatly to this theory in one way or another over the years. And finally, I want to hope the participants have a really good time and have as much fun learning this as I did, helping to start it. So with that, let me let me go ahead and start I don't have a lot prepared I have a few remarks I also wanted to leave time for questions about not so much the actual details of the theory but maybe questions about to expand on where it came from or how we originally did it. So I'm going to give three answers to start with, I guess I should say, in terms of the years, literally, the, the start was a talk I gave in the University of Illinois in 1986 at a conference there I believe in November. And I spoke on something involving the integral closure of ideals which I'll actually talk about in a little bit. So the talk Mel Hoxter came up to me we'd been talking about various things over the years, and he said, basically he said, what's in answer three down there he said, maybe we should think about the crucial idea in the proof and try to conceptualize. So that sounded great, and we arranged that I would visit the University of Michigan and as soon as possible in the next year which was sometime I think in February of 1987. And we spent a very intense week, working this out. This magical week of my own mathematical career, it was, it literally was almost like magic, we had this idea and we started thinking about how to apply it, and suddenly all these results just sort of came for free almost. And we were very excited at that time. And since that time of course it's been developed by a huge number of people in many different directions which will be hearing about. So let me go to my first answer here which you probably have already read where did it come from. Well, through the late. Well through the decade of the 70s basically characteristic P methods started coming into community value. And it already existed in number theory for a long time, for instance, the proof of irreducibility of cyclotomic polynomials and things like that that those are characteristic P proves. But really in community of algebra, the fundamental paper of Pesquina and Spiro which proved some of the homological conjectures by reducing them to characteristic key was a crucial step. So Joel Roberts proof that rings of invariance or Colin McAulay which you'll hear about in these lectures. Keichi Watanabe go toe and others started taking up the idea of F purity which you'll also hear about Paul Roberts use characteristic P methods to also study homological methods. And then Lipman and Reese are very important in terms of their work on integral closures, even though they didn't directly use characteristic P methods. So that's my first answer it came from a long history of characteristic P methods and problems to which you could apply. Another answer was really to some extent although Keichi Watanabe and others were also very important here. And that was the desire to find algebraic proofs of results which were purely algebraic but at that time had only been proved by analytic methods, or in some cases algebra geometric methods. The professor gave a very important series of talks at George Mason University, where Neil Epstein is right now. In 1919, I forgot the year 1980, maybe, where he talked about analytic methods, proving algebraic results, and he threw down the gauntlet to try to find algebraic proofs of these. And finally, the third answer is one I learned from Mel Hoxter also, and I already mentioned it, whenever you have a germ of an idea that proves something, a method, a lima, whatever it is, conceptualize. And what that means is make a definition if you must, but take that nugget of an idea and make a concept of it and start studying that concept. And it's amazing how far that will go sometimes. So, let me start not with tight closure, but with closure operations in general, because tight closure is a closure operation. So here's a question for all the students. My rings are going to be in the theory and it's not always necessary but I, it's much better to stay in that case. I is going to be an ideal inside are I hope you can read this. Here's a question to define a new ideal which contains I by the following property. Say an element is in J this bigger ideal, if and only if for every homomorphism from our to a field. The image of X is always in the image of I in that field. Now of course, it may not look like you're saying much because, you know, fields have no proper ideas so the image is either going to be zero or the whole field. Well, I'll give you a second to think about it. Especially the students. Anybody have a guess they want to give in the chat students only. Anybody brave enough by Bob suggests the radical of I exactly very good. The nil radical. So J is actually the nil radical of I, which is the set of all X and R. Such that there exist in X to the innocent. In fact, let me give a theorem here. Which says the following are equivalent. First of all, the first property is what I just said that for all maps to fields the image of X is in. The second is that X is in the intersection of all prime ideals, which contain I. The third is that X is in all the minimal primes containing I, and they're only finitely many of those because it's a material ring. And the last is the property that usually defines the nil radical. It's not hard to prove one implies two implies three. Three implies four is a in a team McDonald or common technique involving localization and four implies one is more or less trivial. Do it for an exercise if you want. But the thing I want to emphasize here is that this property right here. The fourth property is is an equation. And it's very useful as you all know. But the first one has to do with maps to arbitrary fields if they're very different. And it's by combining both of these that you often find a huge amount of strength in applications. All right, let's do another closure operation. Second question. This one. I don't think the students would necessarily know because they may have never seen it before. So the same setup. I have a ring are a netherian I have an ideal, and I wanted to find another ideal. I wanted to define it by the property that for every homomorphism from our into a one dimensional regular local ring, or also known as a discrete valuation, rank one discrete valuation. The same property I talked about earlier is that the image of X, which is fee of X is in the extension of it to be. By the way, I maybe I should say it just for commonality. The field of course, right here this is just a zero dimensional regular local funny way of putting it. But that's all the no radical is and now I'm just sort of popping the dimension up by one and saying well what if you do the same thing but you only look for maps to discrete valuation. So the integral closure bar of iron, not our, and it's usually denoted with a bar. And I believe this understanding the integral closure was at least for me, a very important part of the development of tight closure. Now let me give you the theorem about integral closures which parallels the theorem. I gave earlier about the new rack. So basically I want to assume right here. Maybe I should mark this in red right here. That I is not in the union of the minimal primes of our so for instance if ours reduced or domain. I just say ours a domain I just be saying I is non zero. And I have an element X and R then. This means the following are equivalent. So first it has the property I just talked about right here. That for every map to a DVR I should have said that's a DVR. This is a DVR. So one dimensional regular local. The images in IV. The second one is a very useful one. If you know more about the integral closure of rings which is what typically you study first. The fastest the integral closure of the restring. RIT. So this is the subring of the polynomial ring, which is generated by the elements little I times T. So just take the integral closure, then inside our tea, then the property one is equivalent to saying that X times T is an S third one is an equational. So this is the equation characterization. It says that something very much like the nil radical. It says there's a monic polynomial satisfied by X, such that the coefficients get into higher and higher powers of by a one is an I, a two is an I squared, etc. Asa been is an IDV. So this obviously by the way implies that X is in the nil radical. And since everything here is in I already. So this certainly says X to the n is. And the fourth is maybe the most important for tight closure. It says that you don't actually have to check one equation. You don't have to check maps to rings, but you can check. There are infinitely many equations in some sense that there exists a C which is not in the minimal primes of R the union, such that C X to the n is I the end and I should have quantified in here so this is for all in sufficient. So all these are equivalent characterizations of the integral closure. I see a chat here. I can see the chat didn't realize I could. Good. Yes, they're in the George Mason. That was that George Mason conference was, I believe, the very first time I met Kichi Watanabe, which played a huge role in all this. It was a real pleasure to meet him. I believe you can correct me if I'm wrong. So I'm just going to repeat. As I said, the set of all X satisfying these equivalent conditions called the integral closure. And it's an ideal containing I and contained in the mill radical. And it's extremely important ideal for many reasons. But there are two directly related to tight closure. And I want to mention one of them and give you a small proof about the other. So, I think I said right at the start that several things had been proved using analytic methods that there was no known algebraic proof for. And this is what Hoxter based many of his lectures on, and one of them was something called the Briance and Skoda theory. So I'm going to give you the Briance and Skoda theory in the statement of it. I hope you can see that. You have are in this case is a power series ring and in variables and an arbitrary ideal. Then the Briance and Skoda theorem says that the integral closure of the nth power of I is contained in I. The end here. Notice these ends actually match up. So this in and this in are the same in that was very critical. This was inspired by a question of Mather actually originally. What do I have on the next slide. Oh, good. Maybe I should mention that. Mather's question was based on the following so are here is a. R is again a power series ring over the complex numbers and F is in R. It was known that F is always integral over its part. This is, to me, one of the fundamental relations about integral closure, which is always fascinating. And Mather asked. Now remember the integral closure is always inside the nil radical of the same idea. There exists some in such that F to the end is in the partial. This is not obvious. Try to prove it directly sometime. You'll find how hard it is. And Mather's question was that. Well, I think his original question is can in be chosen. And the Branson's code of theorem gives this answer and I'll say why it's because. So let me go back up to what it says. It says that the nth power of any the integral closure of the nth power of any ideal is in the idea. And all I really need to do is note here that if you take the integral closure. to the nth power, it's always inside this engine. This is very easy to prove using the equivalent characterizations again. So if you go down here. If you combine this statement. With the theorem of Branson scota. You see that in fact, F to the end is. The Branson scota theorem answered in the affirmative. The question of Mather, and it did so by giving actually. It actually told you what the uniform number was. By the way, while I mentioned it on this. You might ask what it means if in is one. It's a very famous theorem of Saito, which says that if, if F equals zero is an isolated singularity, then in is one if and only if F is so called quasi homogeneous that there's some waiting that can make it homogeneous. So this was the focus of one of the focuses of Hoxters lectures at the George Mason meeting. And then satay gave an algebraic proof soon after that. But I was a colleague Ellipman at that time, and I was working hard on the integral closures of ideals David Reese visited Purdue at that time. And I started thinking about more about powers and integral closures of ideas, and I want to show you an argument. This was a time that was, I wrote up in paper and I think 86 and it was what I was talking on at this Illinois conference, because this shows you a tight closure argument without ever mentioning tight closure. So I want to end with this basically because to some extent this was the beginning for me of this journey. Let X one through XD be a regular sequence. By the way, I guess I should have said there is a very easy tight closure proof of brands discovered, which you'll probably see in the lectures. I'm not going to give it now, of course. So, remember for the students here, you've just started studying some of this this means X one is a non zero divisor and Xi is a non zero divisor on our mod the first I minus one. And the yoga is that regular sequences. I'll use this a little bit behave. As if they are variables as much as they can, at least when they interact with themselves. You can make this precise using the Cohen structure theorem flatness but I don't think I want to do that in this talk right now. Here's a theorem. It may not look like much, but it actually has a lot of consequences. And this was, I should give credit this was done by Eto. In all cases. And by myself, if it contains a field. And I'm basically there's a more general statement but I'm going to prove the simplest case of this right now because I want to show you how characteristic P can come in to help you. So the theorem is very simple. But always an Ethereum ring. As you have a regular sequence. And I is the integral closure of the ideal. If you want a specific example you can think of the case where F is a hyper surface with an isolated singularity. The partials would be a regular sequence in this case. And we just saw that F is always in the integral closure for instance so that would be an example of such a phenomenon. Then, if you look at this intersect I squared. Even I squared bar. Maybe I should put bar here. It's just the X's times the ideal. So what does this mean. So that's the statement. If the summation of our X side is in the integral closure. Then, the R is can be chosen and the integral closure. Now in fact, because all the scissor G's on a regular sequence are are generated by the casual scissor G's. You can see that in fact all choices of our I will have to be an I bar, but literally it doesn't say that start. I just want to stress something in my own thinking about tight closure that one of the things in community of algebra that I've always been very curious about is, when you have some relation like this. I mean, it could be a summation or a Xi is in some other idea. What can you say about the coefficients. These are the hardest things to pick out. You have some general relation, but how can you more easily determine the coefficients and this is where I think analytic methods were incredibly powerful in problems like this. And what tight closure is done to some extent is allow us to replace analytic methods by these characteristic key methods in almost all cases. A question will quicken the chat. So, in your question there, are the R s is supposed to be or the R is supposed to be in itself or an I bar. R eyes. Oh, sorry. Thank you. I bar is I because this is I. Yes, thank you. But what I really meant to write is, is this right. But when you take the interval closure of an ideal if you do it twice you, nothing new is added. Thank you. So let's prove this, give a sketch. So proof. So we have this sum. I'm just going to do it with ice squared. Well, I guess it doesn't. And let me go back to the I'm going to have to go backwards I know in zoom that's a real pain to see these screens flashing by but I'm going to go backwards for a second or two anyway. So I'm going to use this fourth property of, of integral closures that there's a C, not in any minimal prime above in our such that C x to the n is if if x is integral. So that has the following consequence there exists a C not in the union of the minimal primes of our such that C times I squared bar to the end is in either the two. You can do it at an element in time but because the ring is the theory and they'll be a finite set of generators, you can do it for each one and take the product of the seas that work for each one and that will So I want to now assume we're in characteristic P. This is prime characteristic and I want to take the element. Why, which is this element. And I want to use that. See why did the end is in. I did the two and actually sorry. I did I the same mistake again. I better just erase. It is the statement was correct but I want to use the stronger statement that it's in this idea. And that's simply because I squared is integral over x one through x d square. And I want to do the same thing here. And I want to use this with n equals P to the. The nice thing working in characteristic P. As I'm sure all of you already know, is that when you, it's a endomorphism so called Frobenius so it has this property. Simply because all the binomial coefficients, if you raise it to the p power all the binomial coefficients are zero characters, when they operate on the grid. So when we do this to why we get the file. But now, if we apply this boxed property here. The fun, we get the much nicer expression like this. And now what have we achieved by this. So what we've achieved is we've now got an equation. It involves essentially the x, the integral closures disappear. They are asking if you can kindly show the previous page. One more. It's okay. Thanks. Yeah. It's all right. Yeah, thanks. You probably want to specify that this is for all. Here. Yeah. That would help. Thanks. Actually just large in is enough but in fact you can do it. Like this. Thank you. So, I want to make a side comment about these coefficients and why characteristic P is so useful, which is a more of a heuristic philosophical. You see, if, if you're in characteristic zero. You take an expression like this, and you expand it out. Of course, there's all these terms. Multinomials in the X's with mixed, you know, mixed terms with the RIs. And it's extremely hard to usefully analyze that maybe there's a new method out there at some point that will discover that allows you to do it. Sometimes, at least I don't know how to do it truly usefully. And that's where things like harmonic analysis and stuff really helped in my own estimation. But what characteristic P does is it removes all the static all the noise and allows you just to look at the P to the powers themselves and this is really incredibly useful. So, finally, we need a little exercise now to finish off. So the exercise, whoa, what did I just do. I don't know what I just did, but I didn't want to do that. I'll write it in red. So the exercise is to prove if effects one through XD is a regular sequence. And you have a sum that looks like this for any. The SI are in the nth power. So I know you're doing a lot of exercises here. If they were variables, this would be quite easy. They always behave like variables so you can do this any number of ways but try it and see, see how you do. So I want to apply this exercise up here now back to the proof. And the conclusion from this exercise and this equation is that hence for all P to the CRI to the P to the is in X1 through XD to the P to the. And if you recall the criterion for integral closure that was number four on my list that I used in this. This exactly implies that our eye is actually in the integral closure. This is for all. And that's the end of the proof. Now, there's no tight closure here. But this was the argument. That in some sense. It's not the only argument. There were arguments like this done by Hoxter and his amiable system of parameters paper, where he talks about big coin Macaulay algebras. There are arguments, somewhat like this in the Hoxter Roberts paper about rings of invariance or coin Macaulay. But maybe the first time it had been so explicitly used with with this particular type of multiplier here. So, I guess I should give you the definition of tight closure before, but I mean you'll see it in a second anyway. And then I want to share a screen. I'm just showing you some of our first thoughts on tight closure so the characteristic of ours. By the way, I should have said here too. This is a characteristic P proof but by reduction to characteristic P which I hope you'll learn a little bit about this, these lectures. This immediately proves the same theorem, as long as the ring contains a field. I do want to point out it to did it by other methods without any restriction at all. Create a professor sponsored window in the chat but upstairs when you were working with a closure. It should have been infinitely many and or equivalence with. Yes. Did I not write for all large and or something like that. I think you're a for all and Yeah, yeah for okay I won't go back to correct it right now but yes, that's absolutely course. So characteristic P. We say an I an ideal generated by a bunch of elements. We say X is in the tight closure of I, if and only if there exists to see. Minimal primes of our such that for all, he sufficiently large. C x p to the E. Now if I just wrote, I to the PD that would be integral closure. So this is tighter than that. And it says, so that's the original definition of tight closure. One of the most important aspects of tight closure which I think you learn about in the second series of lectures is that the seas here can actually be chosen independent of X and I, this is enormously important called, we called such things test elements. So I want to stop this share now and share one other screen but let me before I. Before there may be some more questions. Yeah, there's one more question here. I'm going to chat. After proving the resulting characteristics, he, how do you conclude it in terms of zero. So that's a whole process. Basically, you, you localized first. So this is a general process if you want to if you have some equation you've shown or has to happen. And you want to show it. So you start in characteristic p. So you start in characteristic zero containing the field. And you first get a local. If you're not local already localized to make it local. Then you complete. And then you have to use a very big theorem called the art and approximation theorem. You assume you have a counter example basically, and you're going to carry that counter example from characteristic zero to characteristic p. Given a counter example, use art and approximation to get a counter example in a finitely generated over a field over the some base field. Then you use something called generic flatness. Sorry, we get a lot of spam calls. Stop in a second. generic flatness and collect coefficients to get a counter example in a finitely generated algebra over a finitely generated Z algebra, or the coefficients are, which is inside the F. And finally you go mod maximal ideals. Let me call this a of a to get a counter example in characteristics. So it's a very complicated process. But it was worked out by Hoxter. So Pesky and Spiro did a similar thing, starting here in their paper, using characteristic p methods, and Hoxter realized you could use art and approximation to do things in general and his previous work. So this whole machine as it were had already been set up where you can almost automatically conclude in characteristic zero, equi-characteristic zero theorems you've proved in characteristic p. I don't know if this actually will be covered in these courses or not, but it's sort of a black box if you, you can take it. Any other questions before I stop. Craig, there is another question if this process for this machine is often called reduction to characteristic p. Yes, exactly. That's correct. So when somebody says reduction to characteristic p, they mean essentially you, you've gone through this, this whole process and just anything else. Thank you. Can I stop this here? Okay. I just want to show you, I've been going through my old notebooks. I, by the way, I, I've always kept notebooks with my, just my calculations or work or, I mean, most of it's junk, truthfully. But I would highly recommend you do this yourself for anybody starting out. You will be glad you did. Often you think to yourself as you get older, oh, I think I did something about that 20 years ago and you can't remember. Well, I can find it in my notebooks. So I've been going through these and putting them in PDFs actually. And luckily I was doing one just before this. And I want to show you what I found. This was in the first notebook in tight closure. I started working on it with Mel. So can you see this? So this was a list of problems and questions we had almost after that first week of magical week. So to start with, we called it sharp closed action. So the sharp you see there is really tight closure. You can see the first thing we noticed is that if every ideal is tightly closed, we could prove Cohen Macaulay. We wanted to give a definition in characteristic zero, which we eventually did. We wanted to understand what class of rings had the property that every ideal was tightly closed. That's one of the F singularities you'll learn about. We wanted to give a criterion for quasi homogeneous hyper surfaces for when they had this property. We knew it had a relationship to rational singularities but at that time we didn't know what it was. Property five is to give a good name. It didn't really have a name for a while. And we eventually chose tight because it. As you'll see it's it's a very tight fit to your original idea. We wanted to know about what happened when you completed or if you localized course localization became a big open problem for a long time until the counter example my runner and monski and there's still open questions about that. Well, we wanted to sharpen results in our original notes so we wanted to apply it to direct summons of regular rates you'll hear about that and some of these. We wanted to know what it meant in low dimension. We wanted to give criteria other criteria for what it meant to be in the tight closure, what it means. And then at the very bottom you'll see that some notes about every ideal and a regular ring colons were in the shark closure. These are all things you'll, you'll be seeing about in this. But I think right away we knew pretty well that this was going to be related to all these things and working this out has taken a lot of people over a lot of years to do. Many, many people have made substantial contributions to this. As you'll hear about. So that's really all I wanted to say. It's been a great fun for me working on this and I hope you all enjoy it. Maybe not as much but at least someone.