 So yesterday I talked about this example. I said that let me talk more about this kind. So the theory of something. So it is if the ring is standard grade, then it is obvious from the approach and ample shift. But not standard grade, not necessarily. And normal, yes, normal is important. And still we have approach. I have no time to explain about approach. And assume that there is some element, t in degree 1 in quotient fields. Namely t equal in degree. This is homogenous RL plus 1. And homogenous RL. So makes degree 1. And then exist some d, ample, and so forth. Divide that with two rational coefficients. Such that R equal. What does this mean? So this is the element of function field. Such that divides. So this means so every coefficients. This means every coefficients are non-negative. We have a very good example. So in this case actually this is written as RB1. So perhaps you know this. We can write that a rational function field of 1, remembering 1 by 1. So no, no, no, rational function field 1 by 1 with divisor of x minus a. So you know this has 0 at x equal a. And 4 at infinity. So the divisor is, so this is the point divisor. The point is of course a closed variety of co-dimension 1. So if, yeah. So consider a times d. So d is this divisor. And so Ld equal infinity plus other things. So this means divisor plus ad. Since this is, there is infinity here. Then this vanishes as, so we have purchasing asset. And so small one divisor. So x, this x is nothing from d. So this is y, x, sorry. This is compass. Yeah, sorry. So this is y, x, t, a appears here. Okay, any questions? Okay. Very simple. So you know that, so there is some, so problem is that this part is always integer. And if some rational thing appears, it will be cute until it is more than 1. So no integer part is cute. That's the problem. And so usually in the standard case, so R is the standard. And so dimension R is d. Then dimension approach. We like x, of course d minus 1. And so canonical motive, we can write in this form. So R equal for some impartial stuff. So this is a canonical motive. And then like this, in this, our settings, we can write, sorry, this here. Yeah, this is a canonical device plus md. But something should be necessary. So because of this integer problem. And so we write if e plus, so e minus. So this passive integral. And these are irreducible devices. Of course reduced. And then we write, so here comes, this different comes here. This is very important. And then, so impartial stuff, so inject the member of, yeah. So in this case, d minus 1. So this is my cell duality. x plus d prime. What is the circle? So consider degree 0 of this. So degree 0, n equals 0. This is hd minus 1. So here is 0. And if other things are integer, this is killed. And my cell duality, this is just h0 of x0x. And yeah, sorry. I forgot one condition. So I feel of equal k. Equal, yeah, field. So, and we are interested in Robinius. And this has, of course, one time in general. This is degree 1 element, I like. So what is this one? It is easily supposed that the direction is hd minus 1. So this is r plus by q, q times. So this determines, so we, yeah, yesterday that r is f pure. Robin's image is 0, non-zero. So we are now in p1. If the degree of this is positive, then the, so multiply sometimes. So h1 will be 0. So in the case of p1, you can say that c if and only if, degree kx. So you know that for p1, degree of kx is just minus 2. And of course, degree is given. So this is, yeah. So for non-zero, of course, and if it is 0, then the image is killed by every element of maximal idea. That's proof of this. Okay, any questions? So I talked about this. It is very powerful, at least for great dreams. And maybe not known for many people. Actually, I wrote the paper on this subject 40 years ago. So sorry about the direction model of p, something like this. So let me, in fact, introduce rational thing like this. So essentially of finite type. To say that, essentially of finite, that means localization of finite generated. And so now in the world of characteristics 0, thanks to Hironaka, this resolution of singularity. And so resolution is, yeah, logic of morphism and isomorphic of minus singularity. Singularity is set of primes where P is not regular. Fortunately, the singularity is known to be closed. And so definition is A is normal and then A is rational singularity. Where the, so, one definition is HIXOX for every positive. And so these are equivalent conditions. Yeah, it's square macro, right? And yeah, I write omega A and K in the same sense. And the first question is why these are equivalent? Equivalent. So to show we need gluttonic duality story. So gluttonic duality for F is, so take some complex over on X. So this is for, yeah, positive model X and Y. And so these are dualizing complexes. And so for direct summand, direct image as complex. And so the general duality says, so taking the first direct image and then dual is also more big too. So the other one is very simple. So first direct image and then dual is the same as first dual and then direct image. So one more important thing is vanishing theorem called gravity dimension like that. And yeah, what is this? So this is HI of moving X. Rising chip is 0.4. So we need X as smooth. And so this is some variant of Kuzhara vanishing theorem. There are count examples after dimension 3. So I mean this is always true for Kuzhara Kuzhara 0s, but not true. And Kuzhara 0 and dimension 2. And dimension 2 for this vanishing is true even for Kuzhara P. Not true for dimension. So we can say in this form. This is a different form. D-damage of normal work. So this is vibrational. And X is normal. So assume this is just ground dimension vanishing. Of course in general this is false, but for this fixed one, this is very... So usually true. So of course we have some count example, but maybe in major cases this is true. So in this case, this is equivalent. And so perhaps you know very well about important fact is 8 square Makore equivalent to say. Namely, so everything in between is 0. So dualizing thing. The cohomology group dualizing complex on the degree minus 1 and D. And of course... We are talking about no P1. Since we are in new section. Okay. So everything is in general this setting. Okay. So let's prove of equivalence. So this one, if we assume this, yeah. Actually so this is derived category. So perhaps it is safer to confess that I am not good at derived category. So maybe you must be careful. I think, yeah. So put O1 equal OX here. This comb over OX is to do nothing. So since we assumed X is equivalent. Coimagore means dualizing complex of X is X is Coimagore. Just one. Like this, just one thing. And so this is, we had, this is OX and then this is just for Y. So yeah. This part, sorry. This part is, and by, we assumed gravity measure vanishing. And so the other side. This is just A. Yeah. Sorry. So since this is the assumption, this assumption is just, this one is becomes just A. This is A. Home A is nothing. So this becomes omega A. And this is just his condition. Coimagore. And this means A is Coimagore. And perhaps I have only 10 minutes left. So better to leave the other side to leave the audience. And so, so you see that to say something is Coimagore. So we always so go through the concept of rationality. And since rationality fails from dimension three, so also Coimagoreness fails usually in dimension three. So we have the, how to example. So we are talking about a log terminal, something terminal. And so it's just some terminal thing, right? In dimension three, which are not Coimagore. And so that was very long time open question. But nowadays we have many counter examples. Examples. Of course, these are rational things like this. But in dimension B these good things like this and not Coimagore. And so I recorded some references in my resume. Okay, so now we come to one more thing. So to show rational, rational thing there is a notion very convenient notion of pseudo-rational definition. So okay. D-dimensional, normal. Coimagore, pseudo-rational boundary for it. And we write in person image of cross point by E. Yeah. So maybe this is not familiar with many people here. So local homology exact sequence. So long exact sequence and then come to come to HD-1 W-E W. It's a general theory of local homology. And so this is and so under some yeah this is essentially this is of course if we assume that for simplicity we can assume spec A minus maximum. So then you know spec A minus maximum open set just local homology. So this this map is come from this exact sequence. In general we are in situation a little more complicated but essentially this is. Now also come to the following equivalent. Then sorry. All right yes. A is a rational so there is a rational if and only if there is no proper F-stable that module HDMA. And the second result is a rational imply so do you think you complete necessary you think so. So I forgot to say that in defined field rational no time that in the limits of character zero I mean essentially of finite type over field of character zero. So this is equivalent should rational rational singularity. So in this case should rational has meaning. Yes. So if there let's assume that then taking matrix you are yeah this is well known that okay. So this inject so of course this is ocean free and rank one. So this should be ocean and we have kernel and then also some C that's C times N and so since N is table under section that means X is in tight closure of zero. So since we are we are assumed rational this should be zero. Of course the other side second. So if we had not injective and then we have a kernel and since we have from instruction on W2 from instruction very natural thing is a stable then should be zero otherwise should be total I think I must stop here. Thank you very much. Are there any questions? If not we'll thank the speaker. We'll be back at 10.