 All right, so this is the last lecture of Florence, go ahead. Thank you, Richard. All right, so let's continue with this now. I had time to check on the question that Lynch and raised. So only one direction is for the non-local case the easier direction strongly irrational implies irrational. That's global. However, he stated only for local case for local rings, the converse of rationality implies strongly rationality so I don't know if any work has been done. I'm going to answer your question in trying whether rationality in general implies strongly rationality. My guess would be that knowing the fact that the ring is strongly rational locally doesn't imply that it is totally rational globally. It's not no, I guess. It's not obvious. So, yeah, so that's hopefully that clarifies. Again, only in the local case is known as a theorem that rationality implies strongly rational. All right, and that's the proof that I sketched earlier. All right, so now let's let's move on to, to other aspects that are connected to local homology. We're going to veer off to a number of results that are due to Chris me that are fundamental for the understanding the concept of rationality. So for to understand those results. Let me introduce some more notations. So it's often useful to think of the local homology is as I mentioned earlier as a direct limit of the rings of the form are mod x1 to power txd to power D. So if you use that point of view that I used to in just now in lecture two. You can denote the element of the local homology in this fashion Z plus as a concept Z plus x1 to power dxd to power T for something. So let's track down what happens with the Frobenius map on the local homology module. Basically when you apply it at times. You are raising the elements e to power q where q is p to the e. And now the the corset is considered in x1 to power qt xd to power qt. So now, what does it mean for an element to be in the type closure of zero in the highest local homology module. If you track down the definition, it means that there exists C. In a zero such that C times a type of all q is zero for q large enough. Okay. All right so now I'm slightly pressed for time so I'm I prove this proposition in the nose but I'm not gonna sketch it here so how does type closure of parameter ideals and powers of parameter ideals connect to that type closure of zero. So if you take an element and you presented in the form Z divided by extra power s, then if Z is in the type closure of x1 to power s xd to power s. The eight are corresponding to it in the type in the, the local homology modules belongs to the type closure of zero. However, equality happens if the ring are amidst test elements. Okay, so you can go back and forth basically when we have test elements between the type closure of zero in the highest local homology modules modules and type closure of parameter parameter ideal and its powers by its individual powers x1 to power s xd to power s. Okay. All right, so let's remember that we have this natural action of F on the highest local homology module so that makes this a non commutative change over this non commutative RF or RF, you can think of R to which you adjoin F maybe I shouldn't use this diagonal brackets. So you think of F. Sorry for you there's a comment in the chat on asking to Yes, for a moment. Yeah. Okay. We can move on. Thank you. This is generated by this non commutative relation. Okay, so that's a non commutative ring, and the local homology module becomes a module over it. And we call a some module of the local homology module. If and only if F maps it back to itself. On the other hand, this is the same as saying, this is a some module of the local homology is R of F module. Okay, so we're going to look at stable some module of the local homology that might appear a little surprising, but the next result will tell you why. So we have the following theorem of Karen Smith. So we take R, a local neaterian dimension D, then zero, the type of zero in the highest local homology module is f stable is an f stable somewhat. Okay. But more, more importantly, if our is also excellent normal, or you just need excellent and completion a domain. Okay, let's just say excellent normal, then. This is the unique proper maximal if stable somewhat. Okay. So this is a remarkable property of the type closure of zero. Okay, the type closure of zero as I showed earlier, governs what happens to the type closure of the of a full system parameter ideal and its powers like external to power s x d to power s. Not only it's an f stable some module but is the unique proper maximal one every substance f stable some module will be contained in the cycle of zero. Okay, so that's a remarkable property of local homology, and was was proven by Karen Smith so let me sketch a proof on this. Okay, so the f stable property is no difficult. Let's take an element in there, we want that if you apply for Venus to it, you're still in there. Okay. So by the definition we have an element in our not such that, and then so for each large enough, but you, you can rewrite this like this, this will happen also for large enough. So that means that this thing just by the definition. Okay. That wasn't hard. So the second part is the hardest part. Let's say we have excellent normal. And we want to show that the type of zero is the unique maximal proper f stable someone's of the highest of homology. So, for the second part, I forgot to, are we recording I forgot if we actually. Oh, okay, great. Thank you. Okay, so, for the second part we can reduce to the case where are is complete and domain. Okay. So, we can use methods. So what, how do we do this. So we're going to assume that there is an f stable some module, noting here okay. So that means that there exists. Such that the span of its Frobenus images is not contained in the type of zero. Okay. So, we're going to derive a contradiction from here. Okay. So, we have this let's call this span and. Okay. So this is a some module of the local homology. So we can put it in a short exact sequence of this. And then we can apply the method is reality, which is coming into E, where E is the injective hall. So when you do that, you preserve short exact sequences, and you get something like this. This is the canonical module. All right. So, this is torsion free module of wrong one. Okay. So, we're going to be a torsion free module. And because we have rank one, we have only two properties, either the fraction field tensor with the dual of C zero, or the other one is. Okay. Because the, the, the dual of C is torsion free. This is not possible. It's not possible to do this to be zero. Okay. But that means that there exists. C such that C times any zero. Okay. But when you apply duality. So if something kills an element it kills the dual. Okay. So that means that I'm sorry. C such that this happens implies C. The dual is that it is the dual of the dual, which means the C kills and. Okay, which means the C kills this for all P. Okay. And then from here we get that and is in the type of shorts. Okay. So this is a very neat argument based on the fact that this is actually torsion free of rank one. So we have a short exact sequence like that. It implies that either the, the, the when you talk to the total ring of fractions, either. The matter is dual of C zero or the matter is your own and zero. We get a matter is your own and is your after testing with the total fraction that the total ring of fractions of our, and we get an element in the order of N dual, which propagates to the two killer, the dual of the dual of N, but that's N by matters duality. So therefore C kills or the iterations under Frobenius of eight, therefore eight itself is in the type of zero. Okay, so that's that's a sketch of the proof of the fact that in an excellent normal ring, the type of zero zero is the unique. So that's the proper F stable subject. Okay. So now, with this in mind we obtain a very nice characterization of a rational rings. It's a corollary to the previous theorem, but I'm going to state it as theorem because it's very important. The excellent chemical ring of dimension B. R is a rational if and only if there is no, no zero F stable proper some modules in the highest local homology of R. Okay. So this characterization made the transition so nowadays is sort of the entire literature on results that or studies of F stable some of the highs of homologies of this this type of this result of who cares me open up an entire area of research that looks at what happens to the action of obedience on the local homology modules. And I just because of lack of time I don't, I can really mention almost anything about this but there's the entire literature on the various classes of rings and what can happen. Okay. All right, so I think we can move now to a different aspect of the theory which will make the transition towards connections to rationalizing varieties which is one of the things that I wanted to be able to cover. And the first thing to keep in mind would be to understand sort of the this subject, you need to think about where type closure comes when you have a parameter idea. Okay, so there are a few obvious places. And I like to talk a little bit about the results of this type. So I'm going to have two type of rings in this section either local rings or and graded rings, where the zero pieces of field of courtesy. Okay, and when I'm going to look at parameters I mean either parameters in the local case or homogeneous parameters in the graded case. So there are three parts of the type closure that appear naturally so I'm going to do you know but by I lean this union of all these columns. The unit is over as greater than one x12 power s x2 power s colon of the product x to power s minus one so. Okay, so basically this kind of quantifies the failure of x of being a regular sequence in a in a way so that's why. And it's quantifies it for all s. So, this is the definition of lin closure. There is this definition of I germ which I refer from here okay which is basically the sum of the type closure of ideas where you remove one of the parameters. Okay, so naturally that belongs to the type closure of the original. ideal, and you also have the Frobenius closure of I which we know, you know, it's, it's part of I start so the Frobenius closure would be x in our such that there is a power q. Such that extra power q is in a power bracket. So, we have a result that tells you how do this pieces contribute. So the, in the local case, if you have an equidimensional homomorphic image of chemical ring and a system of parameters. The integral closure of it to power D plus the I germ closure was the limb closure, plus the Frobenius closure is part of the type closure. Okay. So that's far first part is the integral closure of I to power D comes from the Briansson square the theorem, and he and Albert Bark is going to talk about that. So I'm not going to mention too much about this. In the, in the graded case, you have a version similar the only difference is that if you have a homogeneous system of parameters. You have their degrees, and you take D to be the sum of their these degrees, then the part coming from the Bianson square the theorem is the elements of non negative degree. Plus, I germ, plus, I lean. Plus I f is instant. Okay. So you can look at this theorem and see okay so type closure. The, the, some natural parts that belong to type closure would be the stuff that comes from the Bianson score the, the germ closure which comes from smaller ideals, basically ideas obtained by removing one of the parameters. The link closure which comes from the column capturing and the Frobenius closure which is naturally in the type of. In fact, one can do the short argument I don't have time to prove it. It's in the, it's in the notes. When you know that the, the, the homogeneous system parameters are actually also test elements which happens often enough in examples. And the ring is calling my colleague to begin with a short neat argument based on degrees can show that in fact the type closure of x one to x d is just the sum between the Frobenius closure and the part coming from the Bianson score that is the non negative degree part. Okay, but I'm going to, I'm going to skip the proof of this result because of I'm running short on time. Okay. All right, so. So what other. What else can be in the type closure for an ideal journey by a system of parameters and this comes to us. The important result due to Karen Smith, which connects type closure for an ideal journey by a system of parameters to the notion of plus closure. Okay, so let me remind you what it is. Let's assume that we have a domain and Q of R is a fraction field of R. And I'm going to take our plus to be the interior closure of our in the algebraic closure of cure. Okay, so this is going to be a non net in a ring. In general, and there is a big theorem about this that says that if you have a local excellent ring of characteristic PR plus is become a colleague so this is a resulting characteristic P divorce and always are but important. Okay, so. So this means show that in fact, the, the, the, this be chemical algebra can be used to characterize the type closure of parameters in in general in an excellent domain so that the result is as follows you take R to be local excellent domain and take I to be a parameter. Okay. Then, the statement is that I star is obtained from going to I to R plus and contracted back to R. And this thing here is, I'm sorry, this thing here is called plus closer. Okay. Oh, I guess it doesn't show on the screen. I'm sorry. Okay. So plus plus is called plus closure. So what's plus closure, you take I expanded to our plus you go to none of the ring and you contract back to R. Okay. So basically, they are elements that are in the expansion of I to a module finite extension and contracted to our but you don't fix the module finance station can take as big as you want. So plus closure is, and this fundamental result shows that I star equals this type of course. Okay, the proof is sketched in my notes. It's actually pretty difficult. It's based on a creative, so to speak, induction argument and math is duality and a few other things that you can see in the notes. Now, one thing that you can see in the notes, and I presented this in the lecture notes is the following green reinterpret this result in the following way, you can use it to show that if you take our plus which we know it's a Bitcoin McCoy P, then if you take an ideal I generated by a system of parameters in our. And if you look at what you'd mean for basically you try to compute each type closure in our plus, then you can prove using this result that you get I so in some, you know, vague way if you want this result can be rephrased as saying that our plus is not only become a call but it's also a rational in some way. Okay, so that's that's one way of thinking of this. Okay. All right, so I'd like to move on now and try to see a few things about rationalizing where it is. All right, so in the connection to a rational ring so let's let's move now to characteristic zero so let's take a local normal ring that is essential finite type of a few key of characteristic zero. And let's take the resolution of singularities of x. All right, so we say that are for x is rational singularities. If the direct image of a floor star. The right functor. RJF low star of OZ is zero for J greater than zero. Okay. And now we say that our has rational singularities in general if the localization has rational singularities at every point. Sorry, you wrote x equals x, do you mean x equals spec r. I'm sorry. And the, yes, thank you, thank you. Yes. So the derive functor RJ is described in harsh on so unfortunately, some of the setup for the rational singularities uses quite a bit of machinery from algebraic geometry that they don't have the possibility to review. So harsh on for for the most I all of it is actually harsh on. And what I'll try to do I'll try to sort of unravel some of these results to make them more concrete in terms of what it means concretely when we have examples. Okay, so that's the definition. A few things to mention here. If this definition is independent of the choice of resolution of singularities Z. If our is an isolated singularity, then RJF low star of OZ is the same as the homology so that indicates sort of the connection between irrational rings and rational similar to go through this local homology point of view for all J between one and dimension minus two and another observation be the structure she feels this homology of the structure. All right, so another thing that is known in the literature is a rational singularity is chemically a normal. Okay. All right, so how does one check that you have a rational singularity fast. Okay, so I like to mention that it's a result by friend and what another that is useful. So that result uses this concept of invariance so let me describe what an invariant is. We're talking now about n graded rings with m homogeneous maximal ideal over a field. K, it's a finite the journey to K algebra where K is the zero degree piece of R. Okay. When you have a graded a grading I'm sorry on our this grading is propagates naturally to a grading on the highs of a homology. Okay. So the highest the lock the highest of homology module is graded as well. Okay, is if I see great. The invariant will be the maximum integer and such that the degree and piece of the local homology module is non zero. Okay. All right so is the highest integer and such that the local homology module in degree and is non zero. Okay, you should remember here that this definition makes sense because the ring. Sorry, the local homology module is our opinion. All right. So, the following result by friend and what an habit provides a very convenient way of checking that you have a rational similarity. So let's say you have this setup where you have an n graded ring. The zero piece is a field care of statistic zero it's a finite the genetic algebra and is the homogeneous maximal ideal. Then our has a rational singularity if and only if the ring has to be core, my calling and normal. To you need to have a rational singularity at all P. And then again, and three, the invariant has to be less than zero. Okay. All right, so the first property usually checked easily so you not easily but you must be able to show this if you hope to show that have rational singularity. You know it can be challenging so most often this this result is very easy to apply when you have an isolated singularity and you can check that your ring has an isolated singularity by using the Jacobian criteria of regularity, which is a somewhat standard chapter of the algebra. So, if you're lucky enough and that applies and you have an isolated singularity then it all remains to check that the invariant is less than zero. And how does one do that well, luckily for complete intersections the invariant can be compute easily. Let's say we have, I'm going to say how that goes. Let's say we have a ring like this, where each fi are homogeneous of degree di. So if you want to show more regular sequence, regular sequence on our, then the invariant is the sum of the degrees of fi minus the sum of the degrees of exercise. Okay, so it's a simple computation, if you have essentially complete intersection. And this is not to R. This is K. Okay. This is our. Okay. Let's take a look at the example that I had earlier in the example later earlier how do you make this homogeneous so this is what we call a quasi homogeneous hyper surface because we can give us degrees to the variables XYZ to make it homogeneous. So for example if you take the degree of X to be 15. The degree of Y to be 10. And the degree of Z to be six. Okay, then F would have degree 30 right. So the X square plus Y group as if suddenly it's homogeneous of the 30. So the invariant in this situation would be what is 30 that because it's only one equation that where I can and I just realized that somehow the bottom of my slide doesn't actually Definitely showing up for me. Oh it doesn't show up on my computer. Oh I see maybe I don't have full screen. Yeah, anyways, so yeah but it's confusing for me so thank you though. So be the degree of every 30 minus the sum of the degrees right which is 15 plus 10 plus six. So it's 30 minus 31. So we get negative one. Okay. The invariant is negative for this. You can check is an isolated singularity using the Jacobi material regret you have only one equation so it's fairly easy to check this. And it's quite my quality because it's the polynomial ring divided mod out by a non zero device. Okay, a normality can also be checked. So, so this beautiful result of Flannery Watanabe shows you for example that in characteristic zero this ring is a rational singular. Okay. So, Yeah, so I like to stay the counterpart of this result in characteristic P. So, we're going towards some results that will show that there is a strong connection between a rational rings and rational singularities. Okay. And there is the final result of Hussein Hewney can suggest that this connection so that basically they have a resulting characteristic P that is the counterpart of the Flannery Watanabe result. It's very useful in practice to check every rationality so I would like to state it. So, as before assume that you have an engraved the ring now assume that the ring is for my colleague to begin with. Finality generated K algebra are zero is K as before, and we have an extra assumption which is a, you know, non trivial but we could use perhaps Jacobian elements to satisfy it. So assume it's homogeneous max ideal M has the property that every power is a test element. So I know every sorry. I want to say that. Every element, element in here has a power that is a test element. Okay, so you have this condition so with results from the theory of test elements, this is often satisfied. Okay. The following are equivalent. One, all ideals generated by homogeneous is system parameters are tightly closed. To when you localize that the homogenous maximum ideal you get an irrational three. The invariant is less than zero. And there exists, I homogenous system parameters generated by system parameters that is for being as close. And then for our reservation. Okay. Part three is the part that I want to emphasize. So basically, in under nice conditions regarding test elements. You can check that the Koyama Koyan graded ring is a rational, simply by being able to compute the invariant and being able to show that there is a homogenous system parameter days for being as close. Okay, so this is the new layer here in statistic p that he says that you, it's about the invariant and it's about for being as closure for system parameters. And luckily the part about for being as closure can be checked using feathers criteria, oftentimes. Okay, so they talks about the rings being f pure so I don't have time to do a little more on that but I just want to say that that part, oftentimes can be can be checked as well with simple computations. Okay, so this is a very strong result and this is the counterpart in statistic p of the Flannery what another result. All right. So, let's see. Okay. So, was the connection between every rational rings the rational singularities. So the first observation that we need to make is, we define rational singularities in characteristic zero. Okay, so we need the definition. The definition characteristic p. The reason we need this is because we don't know yet that there exists resolution of singularities in characteristic p right so we need to to get around that. Okay. Luckily, Libman and Tessier provided a characterization in arbitrary characteristic of rational singularities. And those singularities are called pseudo rational, and they overlap with rational singularities in characteristic zero so specifically, what is that what does it mean for a singularity to be pseudo rational. It means that for any desingularization or resolution of singularities from Z to the speck of R with Z normal and denoted the close fiber. The pre-image of the maximum idea in the speck of R. The canonical map for any desingularization the canonical map into HD with supported E of O of Z is injective. Okay. So this characterization exists in characteristic zero of rational singularities and this part here is by Libman and Tessier the definition of pseudo rational so this is what is taken as the definition of pseudo rational it recovers rational singularities in characteristic zero. Okay, so it boils out to the injectivity of a map between local comologies. All right, a canonical map between local comologies. So, Keras meet use this characterization to prove the following that if you have a local ring. It is excellent of characteristic P. Then are a rational implies our pseudo rational. So, how did. How was she able to do this so if you look at here the discretization pseudo rationality, it can be interpreted in the injectivity of a map between local comology. Modules basically. So what happens is, if you want to show that this map is injective, you will remember that every rationality was characterized as the fact that the local comology doesn't have any f stable some modules. That are proper. And you prove that the kernel of this map is such a someone so it automatically must be zero. So that's in a in a short essence of her argument. Okay. And the kernel of this map and you show it's a stable. So then for every rational rings, her original result about the, the highest of homology forces, this to be zero. So you get injectivity for free so it's a beautiful connection between that perspective of local comology and pseudo rational rings. Okay, so, sorry. What happens in characteristic zero. Okay, so this is in characteristic P. All right, so, in short in characteristic zeros. I'm basically out of time so I'm just going to write the two results and start here. Somebody I in fact asked in the previous lectures, how you go about in study typology in characteristic zero. Okay, so the last part of my lecture notes mentions a bit. So let's go from characteristic P to characteristic zero so using this idea as you can talk about rings of a rational type over a field of characteristic zero. Okay, and there is a descent to characteristic P that I sketch in my lecture notes. So, when we talk about a rational type, roughly speaking we say that when you go modular P in a convenient way. For most P, you have a rational ring so let's say you know you can imagine that I give you the equation x square plus y q plus if it's okay so that's in characteristic zero. So you can take that equation in regarded in Curtis in over the integers and then you go can go modular P. So a rational type would mean that the given equation is a rational for most P. Okay. Of course, the distance to characteristic P from characteristic zero can be a bit more involved and it is more involved but I'm sketching it in the in the notes. So what a rational type I mean by the center characteristic P for most P, I have a rational rings. So, using that point of view. It's a simple reduction to show that the result of Smith shows the following that if you have a scheme of finite type over K of characteristic zero over a field of characteristic zero, then X is of a rational type. It implies that X is rational singularities. Alright, so every rational type so essentially, if you have a rational for most P when you do center characteristic P then you have rational singularities so that's one. That's the result that was proven like you can submit and shortly after Hara was able to use work of an ideas of the linear Z vanishing theorems. I'm going to show you the Z vanishing theorem of Akizuki Kodaira Nakano, plus some pioneering point of view of federal Watanabe to show that if you have, if R is a finitely generated K algebra over a field K of characteristic zero. A speck of R is rational singularities, then R is selfies of rational. Alright, so in the so we can think of rational singularities as being the same of as rings of F rational type because of results by Smith and her. Okay. Alright, so I think I'm already four minutes over so I'm going to stop here. Okay, thank you, Lauren, Lauren. Let's talk and are there any questions. So are there any questions from the chat. Yeah, I don't, I don't think I don't see it. Okay. So, if no questions, let's thank flooring again for the nice lecture. Thank you. Thank you and I'll see you guys on Monday, when I'm going to introduce Professor Trivedi. I'm going to be a tutoring session with Kyle and then the second part of the day on Monday would be new lectures from Professor Trivedi.