 Okay, so let's start my talk. So maybe. Before you start, I'll put it more in the center. First, I tell about my plan. So the title may be everything. So it's okay by this size. Okay. So F means, of course, Frobenius. And my plan is so tight closure, the answers could as well. And the rational rings. Second lecture, tight closure modules, F pure rings, and F rush. Yeah. Third, deduction module of P, rationality, and rational rings. So what lectures is Frobenius splitting? So when 50, four times 50 lectures, so if I go precisely through these topics, it will take more than hours. And instead, so I sent to Jukar as a resume of 35 pages to Jukar. And so perhaps he will make that my resume so that everyone can see. And so I'm sorry to say that the resume was completed only two, three hours ago. So that means it should contain many mistakes also. And so F singularity, what does F singularity mean? So this is a kind of, so, about phenomena. So perhaps someone knows that the technique of character P is very famous in the algebraic geometers working on minimal model theory. And so there are so low-term and low canonical rings very important in minimal model theory. And so the idea is these concepts can characterize using character escape P method. So such phenomena is called in the term F singularities. Of course, F means Frobenius, OK? And so there is so explicitly work on F singularities by Shunsuke Takagi and me. So and I should say that was Shunsuke to be here instead of me. So it's so very pity because he is much better mathematician than me. But unfortunately, the date of this workshop at school has changed. And so I'm talking nonsense. So this time in Japan is called Golden Week. So from starting tomorrow, so every day is holiday in Japan. And so that's OK. But so if you want to come from Japan to Europe, so one thing is the wall of last year. So we can't go over last year. And we should take two or three hours more than usual. And also, of course, you can suppose the cost of flight is at least 50% up. So in some sense, so sorry about taking that nonsense. So I should say that I'm old. So perhaps the oldest of you members here. And so fish makes, so all makes everything very forgettable. So I can, there are so insolence, there are many conditions. And so forgettable means so I forget some of the conditions. But fortunately, so there are many authorities of these singularities. And at least the other speakers are very reliable persons. So I ask you to check. OK, so that's the idea of my talk. And so I want to, so these are schedules. And so maybe the thing will not go as scheduled. But anyway, we will go to the next step in any case. And you could check, so in my resume. So that's my idea. And also, also the expository work with shinsuke will be so available, so in some sense, so Jukar. And so the literature in this expository work contains 154 items. So that's your, so this theory. So Hockster-Hunek's work on tight closure began perhaps 83 or something. And so one more thing that I want to apologize is I'm not familiar with new things. And this, the manuscript, this was written 10 years ago. I feel so not so very important things has occurred except the things Kevin is going to talk. And so this is a very good place to go to some new feature starting from here. OK, so already, so yeah, Koruga talked about tight closures. So there is already the link, netaria of character P. So character P means that contains some field of character P. Actually, in my talk, so I have some things that this link should be, so almost always I, at least, deduced and sometimes inter-domain, but anyway. And we write a circle as minimal prime. And now, so take IDL and define the tight closure of I. So something, some element in A is contained in tight closure if some element such that, so. So character P and I use more the letter Q to show some power of P. So I will always use Q in this way. And so yeah, and I forgot to define I bracket Q. And so you know that in character P and so A plus P times as Q equal AQ plus PQ. So this is true in character P. And so C times x to the Q is contained every big Q. Actually this part, so this part can be for every Q in some good condition. And so it is contained in my resume, so OK. So what the properties? The first, so yeah, I star is an IDL containing of course I. And so if there are two IDLs, so the inclusion is preserved. And so yeah, OK, so X and I star. This depends on for every minimal prime. I plus P in A mod P. And so for example, let A is y to the n, z to the n. And so yeah, take X square, then yeah, so sorry. I equal the parameter IDL by y, y is image of right y. And then so you can take that this is always, the square is always in the tight closure of I. So this is very easy so. But for the past time, so yeah, take Q equal, yeah, sorry. So very easy. So you know that if equal to, so this is already not in I, and sorry. If n is rather than 3, then so this means that I, the tight closure of I is strictly bigger than I. So you know it. So I missed to write maybe later, OK. And so and then so I start bigger. Then so there is like the important thing is the theorem of Kuhn's saying that A is a local, Neutralian local of. So in my talk, everything is Neutralian always. And then so I'll just repeat. Then the following conditions are equivalent. Following conditions, following are equivalent. First, A is lower. And second, the Frobenius sending, of course, yeah, is flat. And so yeah, for some Q, length of a maximal ideal power is Q to the t. So I contained some proof of this in my resume. So talking about my resume, so it is not quite new, so newly in pre-piaz, so and I wrote in the end of it. I have some book in Japanese. So one is Goto Watanabe, Comedic Linus. Chapter 10 of that book is devoted in this kind of theory. And another is, so I have just finished the manuscript of singularity theory, algebraic aspect of singularity theory. And so I wrote in the end of the book, singularity theory. And so in the last chapter, so I wrote about higher dimensional and log canonical log terminal, rational singularities are contained in. So for these lectures, I translated them into Japanese English. So that's why my resume is, OK, 34 nonsense. OK, so this is singularity, I think. If a is a regular, is a regular locale, then for every I, I cycle. So let's prove this assume cxq is in i black q. So of course c is contained in for any q. And so this is, since the Frobenius is flat, this is i from black q. Yeah, they are here, we use flatness. And if x is not in i, then this part is contained in a maximal idea and c is in. And so you know that the intersection of all q should be zero. But c should be in n naught. So contradicting the fact that this is a contradiction. So this contradiction means that, yeah. OK, so definition. So how long do I have more? So I don't mind. OK, so the definition is weakly if regular if for every idea i is tightly closed. So we say that i is tightly closed in this. And so we have blue showed that a regular locale is weakly if regular. So this is weakly if regular. Actually, so there is a famous localization problem. So let us be a multiple closed subset. Then from by the definition, this is very easy to show that this content is inverse a tight closure. And so the localization problem is that this always equal. So the only counter example I know is by Holger and Moschi. So I asked this morning him, do you have new examples? And so perhaps he has a possible example in mind, but not written at least. OK? So just now there is one counter example in character two. And yeah, that's yeah. Do you something to add art? So the localization means that OK. The counter example is written in my resume. So FA is F regular. The definition for every localization is weakly if regular. And later we will have another notion of strongly if regular. And so in many good rings, for example, finite regenerate of field are complete local with some good similarities. So known to be so. In those cases, strongly if regular and weak if regular is the same thing. So I think privacy. So we don't need to mind too much about. So anyway, OK. So the next come story of indirect closure. So for simplicity, if we talk about indirect closure, so we assume A is normal. So in this case, it's internal closed if and only if. So there are some discrete variations, variations such that and some positive numbers with the integers such that some for some X. So these are so these two by training such that VI of A is also I. X is in I if I don't leave. VI of X is rather J for I. And like the tight closure definition of tight closure for X is an indirect closure. So I means the indirect closure. So usually the definition of indirect closure is the other way. But here, so for example, every variation, discrete variation, so exist some C, sorry, such that C times X is N, the value of bigger than N times the value of I. So value of I means the minimal or value of elements in I for some so that infinite subset. So by this definition, this is very clear that the tight closure is included in indirect closure. So if you remember the definition of tight closure, so this is in that case this should be the powers of P. And so they are very different. So in the case of usually very different. So the name tight closure comes from this fact that tight closure is very tight compared to integral closure. For tight closure. So I means some ideal generated by N elements and for every value is contained in. So for example, so A is some D dimensional local and I be some N primary ideal. So we can take a minimal reduction. And this is that, so if A is F regular, so F regular should mean weak or strong words. And so yes, and so minimal reduction is generated by D elements. And then so in particular I to the D are for every, yeah. So if we take the W to be one, D, the integral closure is, so if we don't assume F regular, so we need star here. But it is, I think it is very nice to give explicit bound for such things. And actually some application to, I'd like to geometry. So some people says the notion of tight closure is not necessary. Only they need Frobenius splitting. But I think this theorem is very nice because the ordinary theorem for Briano-Skoda theorem is a compagant policy of N variables in a regular case. And the proof was very, very, very difficult. And so I remember I attended the conference of George Mason University in 1979. And there are folks that made 10 lectures. So the analytic theory of commutative algebra. So folks are lectures on a big book of Griffiths and Harris. And so he talked to some functional analysis. And so then at last then he could prove Briano-Skoda theorem. So compare this, so the theorem of Briano-Skoda is very, very simple. So elementary, high school mathematics kind of. And so that's very nice. But if we don't have the notion of tight closure, it was not easy. So I think I love the notion of tight closure very much. What time did I start? I think six minutes. OK, very good. OK. Sorry. OK. So we have the Butto's type theorem for F regular. So if it's a pure serving of B, pure means for every m, a module. So this inclusion induces p tensor m is injective. So this is the definition. And in particular, if you put m equal to a over i, then this injective means, so i, b intersection. So the theorem is if B is weakly a regular, then so is a. The proof is very, very simple. So if assume that x is in tight closure, so then this is contained in since. So I said i star of B is contained in i, b star. I forgot there. So in the case of interromance, they are sort of trivial. And then so the proof is finished. And so it is very, very nice to have such simple proof. And so Butto's theorem is for rational singularities. So this is a rational singularity, pure rational singularity. So rational singularity is something defined over singularities where the character is 0. And then A is again a rational singularity. And so this shows that this type of theorem is valid for regularities. And it is very natural. So there is some link called f rational. And natural to think about f rational. So pure subring of f rational links are f rational. But this has counter example. And so in some sense Butto's theorem is a property of character is 0. And I must call on capturing and the rational links. So I forgot to say that after the Briemann-Soskoda theorem. So in the case of principal ideas, it could be the same as intellectuals. And this will be important next. So another thing is to ask for Koyama-Kore. Koyama-Kore property is a quotient of B is Koyama-Kore. So this is Koyama-Kore always. So let X1 to XD, so local of dimension D. So there is a system parameters. So I write SOP system of parameters. So since there is some idea generated by M primary idea. And so far to the condition. And then for every i. So do we need some condition, a mixedness or maybe not. So if these make regular sequence, then we have Koyama-Kore. And this is always included in the tight closure of X1 to XI. And so this is great to bound. This is called Koyama-Capturing. So there are another good way. But since perhaps I have two or three minutes more. So OK. And then define B of this is Koyama-Kore local. And if this is a parameter idea. If the tight closure of this. Then we recall a flationary. And so this definition implies that a flationary implies. So normal. So normal is equivalent to say inter closure of principle idea is the same. And since this is equal to tight closure. And so for tight closure. So if we assume a flationary, this is the same. And Koyama-Kore. Of course these are equal. And this is just the condition for the exact sequence. OK. So the question naturally occurs is that if assume that a is a flationary. But not a flagellar. So there should be some ideal. And a primal ideal. So to say that some ring is weakly a weakly a flagellar. It is sufficient to say that all m primary ideal is tightly closed. So anyway. So take a canonical module. So some ideal. And then should be as if this is not a flagellar. Yes, not. So this is showed by injective envelope. OK, perhaps I have no time. Thank you very much. Sorry. My ES is Koyama-Kore property. What is it? Thank you. Other questions?