 All right. Well, welcome back to our second session, everybody. I'm very happy to introduce our next speaker, Professor Gennady Lebesna from the University of Minnesota. He has been a long time contributor to community of algebra in many, many deep and fascinating ways. And he is going to talk about our plus as a big cone Macaulay algebra and characteristic key. So, please. You have the floor. Thank you. I would like to thank the organizers for the invitation to speak. Well, let me start. So, I am going to talk about the fact that in characteristic P. R plus is going to Macaulay. Let me fix the notation. So, let R be a commutative Ethereum. Now K is the fraction field R plus the absolute closure is the integral closure. And that is the algebraic closure. Right now, Hofster and Hewneke celebrated the result in 1992. So, the result says that if R, excellent, Pan refuses to work. Just a minute. Let's see if I can fix it. I apologize if R is excellent. No, still doesn't. You can stop sharing and share it once again. Just a minute. Well, if this goes on. Doesn't work. I don't know what to do. This is very unfortunate. Let's see. Oh, now it started again. So, if R is excellent. Now, here it refuses to work on this page. It works. Here it works. But here it does. I have no idea what is going on. Let me try to do it on a different page. So, here is excellent. And the characteristic of the fraction field is P. Then R plus is Cohen McCoy. So, what does it mean that it's Cohen McCoy? That is, every system of parameters is a regular system. Since many of the listeners are graduate students, I understand. I need to explain everything. So, let me recall what an SOP is and what a regular sequence is. So, if D is the dimension of R, a sequence of D elements x1 through xD in R is a system of parameters. If and only if the dimension of R, the ideal they generate is equal to zero. That's number one. And number two, I just can't believe this. When I get to this part of the page, it doesn't write. It's very, very strange. It never happened before. So, this is very new to me. I don't know what to do. All right. I think I need to erase. I need to finish this on this page. So, I need to erase this thing and continue. I apologize. It's very strange. So, x1, xD is a regular sequence. And our module, xI, multiplication, xI. If multiplication by xI on M, x1 through xI minus 1, M is injective. The statement of the result of Hofstra and Unicare is that I need to erase, unfortunately, because I have a problem with the iPad, which is very surprising to me. So, the statement is that if R is a local commutative Ethereum excellent ring, then R plus has the property that it is Coint Macaulay. A Coint Macaulay R module and Coint Macaulay represents that every system of parameters is a regular sequence on R plus, on the module. So, that's the statement. Now, why is this resultant? Because if the existence of a Coint Macaulay module over a ring is that the ring has many nice properties. And the statement of the Hofstra-Unicare result is that every domain, every local domain as long as it is excellent has such a Coint Macaulay module and moreover it is fairly simple and easy to understand. It's R plus, the integral closure of the ring inside the algebraic closure of the fraction key. Unfortunately, the bottom part of every page seems to be unusable. That's very strange. So, the proof, the original proof was quite long and complicated. However, in the year 2007, Heunek and Lubesnik published a fairly short proof of a very similar result. Not exactly the same, but a very similar result. So, they published a considerably shorter proof. The entire paper is only six pages long. The entire paper. And if you consider that the introduction and the list of references take up about two pages, then the proof itself only takes four pages. So, I would recommend to the young people who are listening to read the paper because it's a very short paper and with really a minimum of effort you can get a lot of understanding of how local cohomology works for proving a major result because it is a major result, in fact, that our class is quite difficult. So, since I have a problem with using the bottom of the page, I don't know if I can fit this into a single page, but we'll see. So, let me state the main result of the Heunek and Lubesnik paper. If our prime is an r-algebra and r-double prime is an r-prime algebra, then the map, the inclusion or the map, r-prime to r-double-1, prime induces a map on local cohomology. So, the map is from h i sub m of r-prime to h i sub m of r-double-prime. Now, what is m where m is the maximum ideal? So, here I am assuming that this is it, but not allow me to write. Very strange. I don't know what to say. So, the inclusion or the map of r-algebras or more generally, a map of r-modules induces a map on local cohomology with support in any ideal, of course, but in particular with support in the maximum ideal of r. If you assume that r is a local ring. All right. So, now I need to go to a new page because of this very peculiar problem. So, theorem. You'll make it a Desnick 2007. Assume that the characteristic of the fraction field is p and r, r is a surjective image, a Gorenstein local ring a. Let r-prime be an arse of algebra, k-bar is, I hope you remember, it's the algebraic closure of the fraction field k of r. So, in other words, r sits inside r-prime and r-prime sits inside k-bar. So, r-prime is an arse of algebra of k-bar, finite as an r-module over r. Let smaller than the dimension of r be an integer. Then there is. So, there is an r-prime sub-algebra, k-bar, that is, r sits inside r-prime and this sits inside k-bar, so that the inclusion of r-prime into r-prime induces the map. So, that's the map h i sub m of r-prime to h i sub m of r-prime is 0. All right? The paper... Oh, there is something in the chat. All right, so this must be about the paper. All right. So, the statement is that if you have a finite extension of r inside the algebraic closure of the fraction field, then there is a bigger finite extension of that r-prime with the property that all intermediate local cohomology with support on the maximal ideal vanish as you go to r-prime. That's under the assumption that r is a local ring, a local domain. The characteristic of the fraction field is p and r is a surjective image of a Goronstein local ring. So, you can see that here the assumption on the ring r is slightly different than in the result of Hoxter-Hunigate. Hoxter-Hunigate assumed that, in their result, assumed that the characteristic was p, r was local, and it was a local commutative netherian, of course, and it was excellent. So, here the assumption that r is excellent is replaced by the assumption that r is a surjective image of a Goronstein local ring. And the two assumptions are not equivalent. So, strictly speaking, these are two different theorems because they are proven under two different assumptions. However, let's see, it still allows me to write Tham, oh, no, this is it. I can't write it anymore. All right, so, what? Just a minute. So, in 2016, Vietnamese mathematician Tham Hoan Kui proved the same thing under the assumption is r double prime finally generated. So, that's the question. Well, let's see. I thought as an r-algebra. All right. And there is an r-prime sub-algebra. Yeah, I didn't say it here. I should have said that r double prime is not only finally generated as an r sub-algebra. It is a finite extension of r. So, both r-prime and r-prime are finite extensions of r. All right? So, that's the statement. Unfortunately, it still does not allow me to write below the line. This is very, very strange. I don't know what to tell you why this is happening. Okay. So, Tham Hoan Kui proved that the same is true. The same holds under the assumption that r is a surjective image of a coin makoli. And it was known. It was known that every excellent ring, every excellent commutative material and so on, is a surjective image, a local. So, every excellent local... I'm omitting everything else. Local commutative, netherian, and so on. Ring is a surjective image of a local coin makoli ring. Local coin makoli commutative, netherian, and so on. So, therefore, the result of Tham Hoan Kui proves or unites the two theorems, the Hofstra-Hunike theorem and the Heunike-Lebesnik theorem in one result. The proof follows the lines of the paper by Heunike-Lebesnik and it provides, therefore, a much simpler proof of the original result of Hofstra-Hunike. All right, so now let's go back to the statement of the Heunike-Lebesnik result. And what does it mean about r plus? All right, so it says here that certain local homology vanishes in the image. The paper by what? There is a chat. The paper by Kui is the following. Okay, so is r double prime independent of just a minute. Is r double prime independent of the i? I think here is that there is an r prime so that is equal to zero. This map is equal to zero for every i smaller than the dimension of r. For every i smaller than the dimension of r. But even if I didn't say it, yeah, but if you have, finally, many r double primes, then you can take the compositor, that is the smallest suborgibor that contains all of them. And it will be the suborgibor in which local homology in every dimension, in every degree smaller than the dimension vanishes. All right, so you can certainly make r double prime independent of i as long as i is smaller than the dimension. And there will be many r double primes. So now the question is, what does this have to do with r plus? r plus is not mentioned in the theorem. So what does the theorem have to do with r plus? So here is the corollary of the theorem h i sub m of r plus is equal to zero for all i smaller than the dimension of r. All right, so why is this? Here is a proof. Well, r plus is the limit over all r primes that are finite extensions of r inside the algebraic closure k bar of k. Right, and therefore local homology commutes with direct limits, so h i sub m of r plus is the direct limit of h i sub m of r prime. And r prime, like I said, runs over all finite extensions of r that sits inside k bar. Now the theorem says that for every such r prime, there is another r prime that is a finite extension of r and sits inside k bar so that the image of the local homology module in that of this r prime and that other r double prime is zero. So this, therefore, every local homology module of every r prime in this limit is equal to zero. So that's a consequence of the theorem. Therefore, h i sub m of r plus is equal to zero for all i smaller than the dimension of r. And this looks like something that everybody knows is equivalent to the property that a module is going and it is indeed equivalent under the assumption that the module is finitely generated as an r module. Here, there is no such assumption. r plus is not finitely generated. So yet, it is still true that if the local homology of a module vanishes in degrees strictly smaller than the dimension of r, I should say, if the local homology modules of r plus vanish in degrees strictly smaller than the dimension of r, then r plus is quite Macaulay. So let me recall what Macaulay means. Macaulay is defined in terms of in terms of sequences. All right. So where did I state the no, I think I stated it here. I think I didn't state the theorem of Hoxter-Hunikey. So the theorem of Hoxter-Hunikey says that if r is an excellent local commutative Nazarian ring, then the r plus is quite Macaulay in the sense that every in the sense that every in the sense that every system of parameters is a regular system, it is a regular sequence. All right. Sorry. So how are you going to show that the vanishing of local homology below the dimension of the ring implies that every system of parameters is a regular sequence. So here is another corollary. This is something I can prove. H every system of no, no, it still works in this part of the screen. Every system of parameters of r is a regular sequence on r plus. Here is a proof. So let corollary basically says that r plus is a coin Macaulay r-algebra. So let x1 and so on through x d be a system of parameters. So d is the dimension of we are going to we will prove by induction on j that x1 through xj is a regular sequence. So if j is equal to 1 done since r is a chain and therefore multiplication by x1 on r plus is injective. So we assume j is bigger than 1 and x1 through xj-1 is a regular sequence. So we set it equal to x1 through xt for all t smaller than j9 what did I do for all t here is where it refuses to work. So I need to continue here. So for t it is x1 through xt for t smaller than or equal to j-1. So there is a short exact sequence. The short exact sequence 0 to r plus modulo i t-1 r plus to r plus modulo i t-1 r plus to r plus modulo i t r plus to 0 here is multiplication by xt so this induces or produces a long exact sequence of local co-homology with supporting m. Now we know that h i sub m of r plus is equal to 0 for i smaller than the dimension so hence this long exact sequence implies by induction on t that's h t sorry h i sub m of r plus sub m of r modulo x1 x t r plus is equal to 0 for q smaller than e minus t so hence h naught sub m of r plus modulo x1 x t plus 0 for so since 0 is smaller than d minus j minus 1 so hence oh gosh I cannot write this page this is very very annoying hence m so hence maximal ideal is not an associated modulo x j minus 1 and this implies that the only associated primes of r plus modulo x1 through x j minus 1 r plus are the minimal primes of hence the only associated primes of r plus modulo x1 through x j minus 1 plus are the minimal primes modulo x1 x j minus 1 because if there is an embedded p that is associated to r plus modulo x1 through x j minus 1 r plus then you can localize at that p and you will have a ring whose dimension is bigger than strictly bigger than j minus 1 and the maximal ideal of that ring would be associated to r plus modulo x1 through x j minus 1 r plus and that's not possible as I have just shown. So this shows that, like I said, the only associated primes of r plus modulo x1 through x j minus 1 r plus are the minimal primes of the ideal x1 through x j minus 1 and since since x j is not in any of the minimal primes it is regular on r plus doesn't allow me to write it's a regular on r plus modulo x1 through x j minus 1 r plus alright so this shows that x1 through x d is a regular sequence on r plus so this means that the main result of the Hewne-Kirou-Beslin paper indeed shows that r plus is going on calling now what about the proof of the main result the proof of the main result uses the following the proof of the proof of the main result of Hewne-Kirou-Beslin uses the following the main result of the Hewne-Kirou-Beslin paper is what I stated in the beginning that for every finite extension are prime of r there is a bigger extension of r still finite as an r module so that the maps on the local cohomology modules are all 0 assuming that the degrees of those local cohomology modules are smaller than the dimension so uses the following two facts number one the Frobenius section on local cohomology so r prime to r prime to r prime and the map says every element to the pth power induces an action f over star on local cohomology modules this is called the Frobenius section on local cohomology and this exists only in characteristic p in characteristic 0 it's unavailable and secondly local dology it says the following fm is a finitely generated r module and minus i is isomorphic to h i sub m so a now keep in mind that a is the the Goranstein ring that surjects m i no longer can write and m little n is the dimension of a this was mentioned in the statement of the result of uniculobesnic in the beginning that the hypothesis on the ring r is that it's a surjective image of a Goranstein local ring a now the star so the star over here is the Matlis dual model so my time is almost up I cannot say anything about the proof but like I said the proof itself is fairly short the paper is only six pages long and the proof itself is only four pages and with everything that I have said in the talk the proof of the remaining things are only three pages long I would strongly urge especially younger people who are interested in seeing how local go homologies use and the proofs of major results to read the paper because for a minimum effort it's a fairly short proof only three pages and you'll get a substantial result an understanding of a substantial result in commuter to Bolsheva and the last two minutes I'd like to mention an amazing an absolutely amazing very recent development by Bargav he is an Indian mathematician who lives in America brilliant Indian mathematician I should say and he has a question in the chat can any our prime be coin Macaulay can any other prime be coin Macaulay well in principle I don't see why not but I doubt because if there was if there was a coin Macaulay finitely generated extension finite extension of R then it would make the coin Macaulayness of R plus unnecessary so even if it exists it would be a rare case in most cases for most rings R there wouldn't be a finite extension that is coin Macaulay so Bargav proved the following flat R the domain local and commutative and etherean excellent and so on of of mixed characteristic residual characteristic the periodic completion R plus is coin is an absolutely amazing result that completes decades of work in this area the proof is not at all simple it uses major machinery developed by Barton Scholtze prismatic cohomology it's called and the idea of the proof as he writes in the paper is suggested by our paper with a unicorn so if you plan on ever reading his paper I would suggest that you start with reading very short paper of ours which I should repeat once again is only six pages long and that's all thank you everybody for listening thank you Gennady are there any questions or comments if not let's all thank Professor for an excellent talk an amazing result