 This morning's session is Professor Duan Chun-Kuong. And the title is Remarks on the Macaulayfications of Northeerian Schemes. Please. OK. Thank you. So first, I would like to thank the organizer to organize it wonderful, our group softs, and to give me an opportunity to come back to ICTP after a very long time, seeing I was a PhD student. I always enjoyed the conference of our community. Today, I'd like to talk about the Macaulayfications of the Northeerian Schemes. So Macaulayfication is the type of weak resolution of singularity. So let me start first with the definitions. So let pie from Y to ICTP be a rational problem of the Northeerian Scheme. Then we say that pie is a Macaulayfication if Y is cohemically. And pie is a strong Macaulayfication if it is a Macaulayfication. And it induces an isomorphism over the cohemically locals. So cohemically locals mean the open subscheme contains cohemically poised in eggs. So there is a remark. Instead of cohemically property, we can use a grand style or regular or complete intercession than we have on the North. So for example, if we do regularities, then we have a definition of resolution of singularities. So for this type of resolution, there are three problems people are inserted in. First is the assistant, and then the anchorage and application. So in my talk, I would like to discuss about the assistant. So this is some information about the history of this problem. So in the late of 1970, Prokman and Fanting initiated the program to macalify the natural schemes. And Fanting obtained the first important reason. He contracted a Macaulayfication for a natural scheme of finite type over natural rings. So that's not to admit a dualizing complex. And the dimension of the non-cohemically local scheme is at mode one. So the non-cohemically local means the closed sub-scheme contains the non-cohemically poise of the schemes. And in around 2000s, Kawasaki developed further Fanting's idea. And he proved he contracts the Macaulayfication of any natural scheme over those natural rings, admitting a dualizing complex, and success of non-cohemically local any dimensions. So he removed the condition on the dimension of the non-cohemically local. Later, he proved his contracts the Macaulayfication for natural scheme. If the scheme is same, excellent. So later, I will give the beside definition of CM excellent. But you can even see that we have the definition of excellent rings. And we replace, instead of derivative, we just replace the cohemically properties. So we have the definition of CM excellent. In 2021, Sena Vistut proved a more general result. He proved that a CM quasi-excellent scheme admits a strong Macaulayfication. So third, he extend the construction to CM quasi-excellent. So CM quasi-excellent means we remove the condition on catenality. So it admits a strong Macaulayfication. So it's stronger. Yeah, much stronger. So I remind that the strong Macaulayfication means that it's an isomorphism over the cohemically local. So there's some remark. So the first remark is the construction of the Macaulayfication of Fanting Kavazaki of Sena Vistut. They rely on pulling up an affixed scheme whose center is formulated by using certain strong disciplines. So the notion of disciplines was given first by Professor Yuniki in the beginning of 1980. And the property of these disciplines are very important in this context to prove some property of the blowing up. And the second remark is the Macaulayfication constructed are positive. And it doesn't depend on characteristic. So it's different from resolution of singality. OK, so now we come to the definition of the CM quasi-excellent and CM-excellent ring. So if we have a natural and local ring, we denote by R-H, the R-D completion, then we have this flat morphism. And the fiber of this flat morphism is called the formal fiber of the rings. More precisely, this fibering is called the formal fiber of R-P. And then the natural ring is CM quasi-excellent if all the formal fiber of its localization are co-hemaclased. It's a condition on the formal fiber. And the second condition is co-hemaclate localization of any portion domain is open. So this is similar to the definition of quasi-excellent rings. But here we require the formal fiber to be co-hemaclased. And a natural local ring is CM-excellent if it is CM quasi-excellent and universally category. And from the work of Kawasaki, we know that a natural local ring is CM-excellent even if it is a portion of a co-hemaclate ring. So the condition is very simple. And a natural scheme is CM quasi-excellent. Respectively, CM-excellent is local ring as CM quasi-excellent. Respectively, CM-excellent. So the relation between the macro-veccation and quasi-excellent property is somehow it's not new. In the sense that there are also similar relations between resolution of seniority and excellent and quasi-excellent properties. So that is some result. First, in EZA-4, there is somehow it's not state precisely, but implicitly, a structure of code and think that a quasi-excellent scheme admits a resolution of seniority. Somehow the relation between the resolution of seniority and the formal fiber was a motivation for code and think to define the notion of excellent and quasi-excellent scheme. And in 1964, from the work of Hironaka, he proved that an integral scheme of finite type of quasi-excellent Q and Z-parad admits a resolution of seniority. And later, Liebmann proved that a two-dimensional excellent natural scheme admits a resolution of seniority. And recently, Temkin in 2008, he proved that a quasi-excellent natural Q scheme admits a resolution of seniority. I'm sorry. At the same time, from the work of Gabber, we can, it implies that a quasi-excellent natural scheme admits an alteration. So alteration is, later, I will give you a definition of Z-notion, but it is a weaker version of resolution of seniority. And so people can add a similar reason for the restyping property or complete intercession. But until now, there's no reason about this property. And now I come back to the macroeffications. So here, the reason of Sena Visus is a more general reason about the existence of macroeffication. Then there are a couple of questions. The first is, is a CM quasi-excellent weakest condition to guarantee the existence of a macroeffication? Or we can prove, we can construct a macroeffication for a more general scheme? That is the first question. And the second question is a relation between the existence of a macroeffication and of a strong macroeffications. Or a more direct question, is it possible to construct a strong macroeffication from a macroeffication? Yeah. Yeah. So in my talk, I will discuss a question for the two questions. And for this, I need first a base chain result. And first, I define the notion of CM entaurations. So it's an analog notion of entaurations. So I can write the natural scheme, and pi be a proper dominant morphism. And we suppose pi is generally finite, and y is co-hemicalized. Then we call the pi CM entaurations. So the difference between the CM entauration and macroeffication is the first condition. So for macroeffications, we require pi to be generally isomorphic. But for entaurations, we just require it to be finite. Because the scheme are notary or notary, and the morphisms are finite, so the conditions are equivalent to saying that there is an open sub-scheme in x, such that the restriction of pi to this sub-scheme is finite. So it's quite similar to our intuitions. Now comes a technical lemma about base chains. So if we consider a fiber product diagram of local hypnotism schemes, and if we take a point y-pram in y-pram, and then x-pram in x-pram, and y and x corresponding points in x, then if the homomorphism from this local ring at x to the local ring at x-pram, induce an isomorphism on the MRDIC combination. It means that the two local rings are an analytic isomorphics. Then there are the base chains. So the base chain preserves the analytic isomorphism. This lemma, due to prothendic, is written in EGA. I forgot to write the name here. Then we have a corollary of the lemma. So if r is a local ring, then we denote x as a speck of r and y-pram as a speck of the combination or the hand synchronization. I mean, here we can replace, we can take any flat extension of r, so that it is an analytic isomorphic to r. Then if we have a finite time morphism from y to x, and we denote by y-pram and pi-pram base chain, and if we take a close point in the speck of fiber of y-pram and denote by y the emits, then the MRDIC combination of the two local rings are isomorphics. So in our work, we need these isomorphics to transfer the coherent property of this ring to the other ring. So if pi is universally closed, then the scheme y is coherent, even only if so is the base chains. So it's a base chain of the coherent property of the morphisms. It preserves the coherent property of the morphisms. And for, we stated for macrolation of CM underation. So if r is a local ring, and it is speck of the combination of the hand sanitization, then even morphisms pi is a CM underation of x, then the base chains pi-pram is also a CM underation of x-pram. So the base chain also preserves the resolutions. We do this, and the second property we need is some result on flat descent. We need to define a subset of the coen-mocular locals of x. So we say that a natural scheme, x has a CM form of fiber at a point. If we, for any closed points, which is a generalization of x, then the form of fiber of the local ring at z is of the local ring of xz at x is coen-mocular. So there is an important remark saying that, I catch a CM form of fiber. I even only if the Zariski closed at z points, had a CM form of fiber at a generic point. So we, instead, if we are interested in the coen-mocular property of the form of fiber at a point, instead of look at the whole scheme, we can consider only the sub-scheme corresponding to the point. And then it is a generic point of this closed sub-scheme, and we have some way to control it. And we set CMF to be the subset of the coen-mocular locals of x, so that it has a CM form of fiber. Then with this set, we have the following flat descent results. If we have a universal closed morphism of finite type between natural scheme, pi. And we suppose there are two sub-schemes, two open sub-schemes in x and y, so that the research morphism is flat. Then we have the following property. First is if pi will map the points in CMF-wise to the subset in CMF-x, it means that if on the point, for example, if on the point in v, it has coen-mocular form of fiber, then it will map to some, then on the point in x we have a form of fiber. This design is important in our work because we are interested in the macro-velocations. So we have a macro-velocation from y to x, and y is coen-mocular. So the question is, how good is x? So this somehow it's going to transfer property from y to x by the inclusions. Okay, so I keep the other property. And there are some remarks. So in this proposition, we have seen that there are two open sub-schemes in x and y, so that there's a flat morphism between the open sub-schemes. Then the open sub-schemes appear in the following situations. So the first is, if x is reduced, and pi is universally called morphism, then by generic flatness theorem, then there is a dense open subset u and x, so that the b-chain is flat. The second situation is if pi from y to x is vibrational, then of course there's a sub-scheme in x, so that we have the isomorphism, and open sub-schemes in x and y, and we have this isomorphism by definition. Okay, so now come a corollary of the proposition 11. So let pi be a universal closed morphism of finite type natural scheme. And if y has a same form of fiber at a point, then it also has a same form of fiber as a corresponding point. So if a natural scheme admit a morphification, then it has a same form of fiber at its own point. This is the implication from the descent property. And similar for CM alterations. I have a remark about the subset CMF. We may expect that this is an open sub-scheme of x, but it's not all the way like that. So there is an example by Pratman and Rastow. They constructed a three-dimensional domain, so that the combination is a domain. And this ring has a unique and principal prime ideal, so that the form of fiber at its point is not co-handled. But the form of fiber at other points are co-handled, but only at different points. So the co-handled ring is the punctured spectrum of the rings. So in this case, the co-handled ring is open, but this subset is not open. So in general, this subset is not open. So for our descent property, it's necessary to consider this set instead of the co-handled ring. Now comes the main result. So this is the main result. Using the descent property, we are able to solve that. Laterally, we can admit a molecular equation, even only if it's CM quasi-extend. So here we have even only if result. To prove these theorems, the configuration was a Senna-Wishu's result. As I said, he saw that CM quasi-extend admits a strong molecular equation. And this is the hardest part of the theorems. This is much harder. And for the other direction, we need to use the descent property. And it means that if I get a natural scheme and we admit a molecular equation, then by the descent property, because why it's a co-handled place, then it implies why is co-handled place so it has a co-handled place for more fiber at any point. So because of this, using the descent result, it implies that the point of x has co-handled place for more fiber. And also using the descent result as the other property, as a statement, we can so that I can also certify the second requirement for the open need of the co-handled place. Then we can conclude that I guess CM quasi-extend. I skip the proof for the open need of the CM McLean, of course. The more important part is about the form of fiber, I can say. Then we have some corollary of the theorems. So let I get a natural scheme, then I admit a strong and projective molecular equation even only if it admit a molecular equation and even if it admit a CM alterations. So if we look at the condition, then of course the CM alteration seems to be more general. But the existence of CM alteration and even strong molecular equation are equivalent. We are not able to prove directly for example from C to A, but we know that if I admit a CM alteration even if it is a CM quasi-excellent, so yeah, on an equivalent. Another corollary, because the scheme of finite type of a CM quasi-excellent is also a CM quasi-excellent, then we can, so that's a scheme of finite type of a CM quasi-excellent scheme where we admit a strong molecular equation. So in particular, if a natural scheme admit a molecular equation then any close sub-scheme also admit a molecular equation. In the last three minutes I have some remark. The remark is about the local case. So if R is a natural scheme, then the condition for the CM quasi-excellent is similar for local rings. So if R is a natural scheme then if in a formal form of aqua-helm place then we can show that the co-helmically local sub-array is open. It means that the condition on the formal fiber implies the condition on the openness of the co-helmically local. But this is true only for local ring. It's not true for any natural rings. So it's a real reason why for any scheme we also require the condition on the openness of the co-helmically local. So by this observation we have the corollary 17th saying that the spectrum of the local ring admit a molecular equation even only if on the formal fiber of the ring co-helmically. So it's a little bit similar statement. And the second remark is about flattening the stratifications. So it's a rational quote-and-dict that if we have a projective morphism between the two natural scheme then there is a stratification of ex-suggestion in the morphisms. It's flat. People call it flattening stratifications. So in our work we have a feeling that there are some relations with this property. And we raise this question if we have a projective morphism is there a flattening stratification of the morphism so that every strata at eye is co-helmically. I want to remark that there could be many flattening stratifications. It's not unique. So whether there's one if it's one then we can have some application of this. So I finish my talk here. Thank you for your attention. Thank you. So are there any questions or comments? I have some comments on the name of the Koemakole Excellent. No, Koemakole C.M. Excellent or Koemakole Excellent which is Koemakole Excellent? Yes, C.M. Excellent. So maybe the Gorinstein Excellent is acceptable? Have you heard of it? Yes, acceptable. When Kawasaki announced his result at the conference and Professor Sharp was there and I asked him is there any similar name for it? Then he answered I once used the name Tolerable. But I haven't seen the name in the literature. I'm sorry. So are there any questions? So if not, let's thank the speaker again.