 Yeah. Okay. So our next speaker of this natural series is Florian Inescu from Georgia State University. And Florian is a student of mile hawkster, and he graduated in 2001. And he, I think he was a postdoc at Utah and then he moved to Georgia State as a faculty, and he has written important papers on this tight closure of rational rings. He will cause multiplicity. And I've learned a lot from him. So it's my great pleasure to introduce him. So the title of the talk is of the title of the lecturers at rational rings and rational similarities. So, Florian, go ahead. Thank you very much for this very friendly introduction. And thank you for your lectures they were very nice. I'd like to start by thanking the organizing committee for putting together such a nice summer school and for inviting me to, to say a few words about rational rings and rational similarities. The purpose of the nose the way I thought about them is to introduce the students to the concept of rational similarities and try to give a fairly careful introduction in the main ideas and develop the bulk of the fundamental results about the rational rings, and then sort of go ahead and provide the sum of the major contributions the field which I can't prove all in three lectures. So, the first part of it will be more detailed in terms of proofs. And as we advance in the subject, I will indicate results but either sketch proofs or results at all due to lack of time. However, I will try to include a fairly comprehensive reference list so people can start there and look at many of the papers written on the, written on the subject, if they're interested in this concept. Every rational rings are really among the classes or in one of the classes of rings associated to take pleasure to read that tends to have very many beautiful results, connected to it and many of the so to speak that the results that one would hope that they, they work, they actually work for a rational ring so he's one of the sort of test cases if you have some ideas on things that relate to try closing somewhere another ever rational is provided a ground for testing to see if they work. And that being said, many, some of the results are very, very deep and beautiful, and even the theory itself requires some care to develop carefully so I will spend some time on that to try to explain some of the intricacies as much as I can. So let's start with that. So, I should, I should say before I move to defining this class of rings that I'm freely using the work of many experts who contributed to this. Dr. Hewney, Karen Smith, Federer, Watanabe, Aburbach, Velez, among others, and I will try to quote the results but I might not be able to do that all the time. That being said, we should all realize that this is done by by other people not by me I'm just presenting the work. All right, so let's start with some notations. I know that this has been done already but I just want to fix some of the notations that I'm going to be using for the remaining of the notes of the lectures. So, rings are going to be material for the first lecture at least characteristic is just P where P is prime greater than zero, and Q is always going to denote the power of P, the Frobenius map is denoted by capital F. And it's it iteration, if he is the sense an element of the ring R to R to power q. So once we have this iteration of Frobenius, the target map, the target ring I'm sorry it can be regarded as an R algebra via the iteration of Frobenius, and I'm going to use the notation R to power parentheses, E, which is was often denoted in the previous lectures by a floor star. E over. So once we have this our algebra if you tense up with it, a module and you can come up with a functor on our modules. And sometimes this is called the Frobenius functor the pesky spiro functor. We need this functor to be able to define for modules. I know it's defined by I just want to recall what the definition is that I'm going to use. So in order to define the type of for some module and in a module M, you need to have this concept of N, Paul record q in M, which is the image of the tensor product between R E tensor and into R E tensor M. So the image of this map is this module. And now with that tight closure is consist of, oh, I need one more notation. So do you know X power q to be the image of one tensor X in. Okay, so if X is an element of M. You can take this map and the image of this into here is X support. Okay. So with that in mind. Let me recall the definition of tight closure consists of all these elements X in M such that there exists C not in any minimal prime of R such that C X to power q belongs into this and power bracket q of M. Okay, so if we take M equal equals R and N equals I an ideal in R we get the tight closure of an idea. Okay, so this has been defined so this is the notation that I'm going to use. Let me recall what the parameter ideally in the ring. R is it will be an ideal generated by K elements where the height of the ideal is exactly K. Okay, and under mild conditions. You can think of parameter ideals elements with the parameter ideas as I do generated by elements that remain are part of a system of parameters when you localize the prime idea for example, okay. This has been mentioned many times before but system of parameters in a local ring are of course a number of a string of elements x1 through xd where these the dimension of the ring, such that the radical of the ideal is the maximum ideal of the local. All right, so one of the main results that we need we'll need a couple of results to be able to develop the notion of irrational rings. So the first one is the clone capturing the property. It was proven by Huxley and Kinequin and was stated before by I think the meal and I think each one as well. If you have a local equidimensional ring that is a homomorphic image of a color my coloring and you do know by x1 drinks the system of parameters. Then the failure of the ring of being coming I call it which is quantified by this colon ideal. Is contained in the tight closure. Maybe I should. I hear. And I plus one. And this is true for all I between zero and D minus one would be. Is the dimensional. Okay, so this I this theorem was stated for is stated for equidimensional rings homomorphic images of color my color rings, and these assumptions will be crucially the way I set up the period. I'll get back to this point later but for now. I'm going to use this result many times so this is the colon capturing property of assistive parameters. All right, so now I would like to recall some facts about excellent and a finite reason will be used. I know that some authors before me just dealt with a finite rings but I don't think we need to restrict ourselves to that and I'm not able to just include the thoughts about excellence in general. So, let me recall what an excellent ring is a material ring are is excellent if it's universally can turn a ring. If the formal fibers of every localization localize the P where P is a prime ideal are geometrically regular. For every find it regenerated our algebra. As the regular locus is open as a leisurely skittable. Okay, so this is a rather technical condition. However, the important thing about this is that this class of excellent rings. Is stable under the major operations in committee of algebra, like localization homomorphic image, taking a finite degenerate algebra and a complete local ring is excellent. So if you know all these things you don't really need to worry about the definition too much you know that if you have a complete local ring you have an excellent ring, and then if you do natural. operations of speaker on this you're going to preserve the excellence property in relation to discuss that there are two. Two results that I should mention, first I need to recall the what a finite me so a ring is a finite if the Frobenius map is a finite map meaning the target are is a finite degenerate module over the domain are. And this was covered the length in the previous lectures, including these results that I want to mention so. The first result is the result of coins. If you have a finite ring, you have an excellent ring. And the second one, which is useful at times is that if you have a finite ring. This result is due to cover, then it's a homomorphic image of a. A finite regularly. Okay. All right, so if you want you can. So basically just have just a finite rings but I will try to use excellence. When, when possible, just to explain how that comes into play because you will see that in the literature is good to have an idea, or why that it comes into play. And this is really the, the one of the main reasons, I think. The class of excellent rings is, is so important for the theory of tight closure so you. We know that test elements in type of your theory, which in test ideas have been covered and they are very, very important. And then one of the deepest results in, in the theory about the existence of test elements is due to Hoxley here again, and the theorem stays the following. So there are two conditions, either the ring is reduced and a finite, or is reduced and essentially a finite type of a local excellent ring. Okay. So these are the two conditions so then, if you have an element seen are not such that our localize it's C is not a finite I mean I meant to write. If regular in Gorenstein, then a power of C is a test for our. Okay, so this result actually is crucial in the development of this class of a rationally so the I will say that to be able to set up the theory we clearly need colon capturing in this result. Alright, so now. One question that will appear towards the end part of the lecture today is, can we have a result is, is there any result that provides test elements you see this result that I stated is stated, provides a power of an element C as a test element. It's not as convenient to know that the power is a test element you would like to actually have examples of test elements because if you try to check type closure relations, you need a test elements. Otherwise you will just have to guess some C that works and that might not be easy in particular cases. So, one result that it's also not easy to prove and provides examples of test elements is the following that if you have, you need some conditions for example you need are to be geometrically reduced, and finally generated algebra. You need a field care characteristic P. Okay, equidimensional. Then the Jacobian ideal of our over K, the elements of the Jacobian ideal that are not in any minimal prime are test elements. And the chain was show for example use this in in the previous set of exercises to produce this elements in his examples. The result of Hoxter, Hewney key. Based on work of Liebman and set. Okay, so that's what source of test elements for us have a ring like this. You can look at the Jacobian ideal, and I will do some examples where you will see how we get this but this will provide clear test element for us. Alright, so what's the definition of an irrational ring so first I'm going to define it for local rings. And I'm taking the point of view here of Hoxter so you will see various definitions in the literature. And the main debate I will say, I mean it actually depends who you ask some people don't refer at all to type closure when, when define a rationality. This is a little newer point of view and I think link, link your mind Thomas Paul strong like that perspective. And most people refer to type closure of system parameters or talking about irrational rings, and the main debate is whether or not to assume that the ring is already a homomorphic image of a chemical ring. And my point of view is influenced by Hoxter who's latest presentations on this assume in the definition and when you talk about irrationality the ring is already a homomorphic image of a chemical ring. And you will see that a rational ring is calling my calling. So it doesn't hurt to assume already that it's a homomorphic image of a chemical ring. In addition, most greens that appear in geometry, or naturally in investigations are homomorphic images of chemical ring so that's not really a super strong assumption. Because local ring is a rational. If it is a homomorphic image of a color Macaulay ring. And every ideal generated by the system of parameters is tightly closed. And then so that's the local definition so you need to check that every ideal generated by a system of parameters is tightly closed. And now if you have an arbitrary ring you still needed to be a homomorphic image of my calling. And when you localized at any maximum ideal and get an irrational. Alright, so we define the notion locally and then we say that if every localization of the maximum ideal is is called is a rational then that's the, that's what a rational idea. Alright, so let's start now the stating that the first results on this so let's say we have an irrational ring. And then the first observation be that in fact every parameter ideal is tightly close TC would mean tightly close. Okay, to abbreviate. So, how do we prove this. First, know that you can pass to pass to a localization. Because the tight closure if you something is in the cloud type closure from ideal is preserved when you go to a localization. And their parameters become just part of a system of parameters. So, we can assume that are these local and takes one XK is part of a system parameters. So, I'm reducing this problem to showing that an ideal generated by a part of a system of parameters is tightly closed. Okay, so what we're going to do is we are going to complete to a full system of parameters. So let's say we have x one to x K, K plus one x D, where these the dimension of the ring are. And then, for every non negative integer, we're greater than actually been zero than one. So, the type closure of it will be containing the type closure of this ideal that, but this idea is generated by a full system of parameters so we know from the definition that this is tightly closed. And therefore, the type closure of this ideal is containing the intersection over T of all this, which is extra tricks. So we get it, we get that every parameter ideal is tightly closed. All right, so now I like to talk a little bit about. Yes. Okay. Let's talk about what happens when you localize. So, for that we need the an exercise which is a sign in the notes. The exercises the following that if you have a material ring of characteristic P, and this is a multiplicative close set and you have an ideal I generated by an element that form a regular sequence. And then, there exists an element of the multiplicative set such that the union of all these colon ideals for over all the elements, w in the multiplicative set. So you can just choose one particular element in the multiplicative set such that you obtain the so the union of these color ideals is just this cool. We're, again, you see n plus one appears there and then is the length of the regular sequence. So this, this exercise requires a little bit of effort to prove it is not hard but it's not the two liner either. And then there is another exercise that is also the deceivingly ingenious the proof of it says that if you have an element in S inverse R or S is a multiplicative set if you want to look at the AC versus our zero meaning the couple of the minimal primes of this localization. Every element in there can be written in the form C over T where C is in our zero and T is in w. Okay. So, I know WS. Okay. All right, so we'll need these two exercises for our next result. The next result says the following. So you take the value of characteristic P, and you take an ideal eye that is generated by regular sequence, and W is a multiplicative close set, then tight closure commutes with localization, or taking fractions in w. So W inverse I star is W inverse. All right, so one direction is pretty clear if you take an element you have the form X over S where S in W and X in the type closure of I. Well it's right. What it means for X to be in the type closure of I, and we get this line. There exists a C, not in any mineral prime such that C times X to power q belongs like power bracket you for Q large enough. So, if you involve C S in there then you get that C times X over S to power q belongs to W inverse I power bracket q for large enough q. And that simply says that X over S is in the type closure of W inverse I. Okay, so now the more intricate part is the converse. So if you show that if you have an element in the type closure of W inverse I then the dead element comes from the type closure of I. Okay, so using the exercise. You can find C in our zero and T in W, you can such that C over T X over S to power to power q belongs to W inverse I power bracket q for Q large enough. Okay. All right, so now if you clear denominators for every such q you obtain some element in the multiplicative set, such that this relation happens for or q large enough. Okay. Well what happens so what do we have we have this right. And C X q belongs to this union that appear in the previous exercise which can be written in a more nicer form with a specific as zero to power and plus one to power q and zero is independent of q. Okay. So from here you get it is zero to power and plus one times X if you write down what this means. It belongs to I star. So, you can get from here the X over S is in type closure by localized or inverted with W. All right. So this proves that regular sequences behave nice respect to any localization. All right. So this was the only place we use that I was generated by regular sequence was to apply that that previous exercise. Yes, that's what they think. Thanks. Yeah. All right. So regular sequences will localize nicely respect I mean type closure of regular sequences locally as well. So another aspect or that would be is relevant to mention is like behavior on the completion what happens there. So, here, I'm going to stay the following exercise. If you have an excellent local ring of characteristic P, and you have I am primary ideal. I star expanded the completion. It was the type closure of the of I are complete. Okay, so I'm going to I'm going to skip this exercise. But I'm going to use it to show that every rationality is preserved and the completion for for excellent rings. Actually, I think I. Yes. Okay. Take a local excellent ring. I'm claiming that the rationality is preserved. It's a ring is a rational if and only if the completion is a rational. So one direction, you take an ideal generated by system parameters in our hat, the completion of our respect to the maximum ideal that ideal I is no to come from an ideal generated by system parameters now so you can write it in this form. J, our hat, what J is the system parameters ideal. So then, if you have an element X in. I star. That is J, our hat star. And by the exercise that's the same as J star, our hat. J star is J because we know that the ring is a rational. So we get that I started close. So an element generated by system parameters in our is slightly close and conversely if you know that the ring is our hat is a rational. Then you take an ideal generated by system parameters in our, and you expand that to our hat, you get an ideal generated by system parameters in our hat. Okay. And then, how about I star. I start can be computed due to the faith, faithfully finest property of completion as expanding it to our hat and contracted back to our, and then using the fact that here that our hat is a rational. Idea generated by system parameters are slightly close so you can easily get that this intersection equals are using again the faithful flatness of the completion. Okay. All right. So, okay, so I like to let's see. Yeah, let me. I like to stay the following. If you have a ring that is a rational. Then, our is normal class. And if F is a rational local. Then you get for free domain as well right. Because the normal local ring is domain. Okay. So, this was mentioned before but I like to provide a quick proof proof. So, we get that the principle ideal using the property the parameter ideals are tightly closed principle ideals that generate the height one prime will be tightly closed. And we know that this condition applies normality of our. Okay, so now, how do we check the coin McCoy property. McCoy property can be checked locally. Okay, at every maximum ideal. So, assume now, are is local. You obtain. Therefore, normal domain. So, equidimensional and therefore, because we know that every part of a system of parameters. We obtained the regular sequence property from the cold capture. All right, so now we know that every rational rings are normal and come a colleague and if you assume local there also domains. So, all right, so now, let's prove that this property of refresh a rationality. Localize it where not it's just every maximum ideal but every everything so if you have a homomorphic image of a comic coloring, then the ring is a rational if and only if the localization is a rational for every maximum ideal. If and only if every localization is a rational with every part. So, here the important part is how to prove the following thing that if you have a local ring. That is a rational. How does one prove that our localizer P is a rational. Okay. So now if you check, take X one through X K. And R such that they are part of SOP in the localization. Take the ideal J that they generate. So we know that J is J star because R is a rational and this is a parameter ideal. So, using the localization. The property that we prove before. We get that this ideal generates a tightly closed ideally not a place. All right, so now let's start to with. So this word like this little low hanging fruit so to speak from the definition and the column capturing property. We can approve a little bit. A little bit more involved. And says the following so let's say, again, we take an equidimensional homomorphic image of a chemical ring. We do that because we're going to use colon capturing. And we take a system of parameters for the ring R. And now we're going to assume that the ring R and miss test elements. Okay, so go back to the original result about the existence of test elements that I stated, this is where you would want to use the reduce the finite or that the rings are is reducing essentially a finite type over the next on the local ring. So, plainly split we just admit we have test elements for our. So then we have a number of statements. The first one says that the colon. The next one is gives you the tight closure of x one tracks. Okay. So this is for all K. Between one. Okay. And importantly, if you know that this sequence generate a tightly closed ideal, then every partial ideal generated by a substance x one tracks K is also tightly closed. And then three, if you have a effects on 3d generate a tightly closed ideal, then one gets that they form a regular sequence. And R is clear McCoy and for effects on 3xd generated tightly closed ideal, then we get that the ring is irrational. So the importance of this result is established is the fact that least towards the fact that to check the rational property you need to only use one system of parameters. And if you remember, the definition is for local rings is, it's formulated in checking that all ideal generated by system parameters would be that the close this will allow us to just check one system of parameters. Okay. And for that though you need test elements. Okay, so let me sketch this just making sure so. It's a 50 minute lecture right. It's from 10 to 1050 right. Yeah. Okay, thank you. Okay. So, let's start with one. All right, so we want to show that if z times x k plus one belongs to the tight closure. Okay, so that's x 1 to x k, then somehow z is already next one to x k. Okay, so that's the essential the first part. Okay. So, right on the definition of what this line means. So we can have a C that works for a large enough q. And therefore C z to power q belongs to this colon, which due to the colon capturing property applied to the cute powers of these elements were here in the tight closure of x 1 to power q x k to power q. And now we use this test element. This is the place where we needed to further multiply this relation. This is a test element so then we have DC z to power q now belongs to the times this, but by the test element property this is in just in the ideal. And then if you look at this relation. This establishes that Z. I'm sorry, the Z belongs to the tight closure of the agenda by x 1 to x k. So we needed the test element here to not make it depend on you essentially right. Right, that's all we need the test element. We need a test element that works for all this cues. Okay. So, yeah, that's it's an essential point. Okay. Part two says that if x 1 to x d gives you type tightly close ideal then every partial every subset x 1 to x k also produces a tightly close ideal. And this is done by reverse induction so basically we have the statement for k was to be because we just said x 1 to x d generated I think was ideal, and we want to descend to one less parameter so in other words. We can assume x 1 to x k plus one tightly closed and we want to show that x 1 to x k is tightly closed. Okay. So, let's take an element in the closure of x 1 to x k. This naturally is containing x 1 to x k plus one star. Which we know is tightly closed. So it can be written like this. Right. So, you can write this element Z as an element x, x coming from here. Okay. So when you do that, then you, you note that are x k plus one is Z minus x. Which belongs to the tight closure of x 1 to x k. In other words, if you restate this you get it are itself belongs to this column. That appeared in the part one of this theorem that we just proved so this is by what we proved in part one. So x 1 to x k star. Okay. Well, if you trace back what we just did, you realize that this says this the composition says that in fact, x 1 to x k star can be written as a portion for x 1 to x k plus some multiple of something in the tight closure of x 1 to x k. And now Nakayama lemma shows that this ideal equals. Okay, so we use part one to show this. All right. Okay. Now part two, if we have a tightly closed ideal x 1 to x d, we know that every subset will form a tightly closed ideal so by using the column capture and property you get the regular sequence property immediately as before. So what we do produces that proves that if you have an ideal generated by system of parameters that is tightly closed and the ring is currently, and then, finally, it remains to show that if you have that if you have an ideal generated by one system of parameters, then this can actually that the fact that that ideal is tightly closed can be transferred to every ideal generated by system of parameters. And it's a little bit more involved and I don't think I'd be able to finish in one minute. So, I guess I actually already over time. So I'm going to start to that part next time. So next time, the plan is to finish this and move to the connection between a rationality to local homology, do some examples and then Vieta was rationalizing this. Thank you for that. Thank you for his. Okay. Yeah, let's thank foreign. Are there any questions. Yeah, I have a question real quick. Back when you were doing back when you were localizing a fraction alley. And, you know, as far as the check at maximum ideals or at any prime is this does it. I don't know about this. If our as a rational is S inverse R rational for any multiplicatively close set S. Yeah, that's because you if I if I was a rational then I'll localize the people if you point. That immediate. Thanks. Other questions. So if, if no more questions, I think let's thank flooring again for the nice lecture. Thank you.