 Let's continue with Manoj Khumini who is going to tell us about afrationality of RIS algebra. I would like to thank the organizers for this opportunity and so this is about some conditions for RIS algebras to be afrational and so this is motivated by some earlier work of Hiri and Hara, Watanabe and Yoshida and the newer results that I will discuss and at least sketch a little bit where obtained with Mitra Kohli it's I mean that's a slightly older work and even newer work with Nirmal Kohli. So I'll mention the method. Let's begin afrite. So this is the context. We have a Niterian local ring we will just assume that they are normal domains for more I mean we could of course raise these questions over anything but results will be in this context okay and this is of a prime characteristic and we consider a filtration R which is just which we just call I0 then I1 of M primary ideals and by this script R we will write the RIS algebra and the question the thing is find sufficient and or necessary conditions for this reasoning to be so we'll make a restriction immediately which is that which is which will be implicit in the discussion throughout we will just assume that the RIS algebra is our priority so we will assume I mean when we try to find conditions we will we'll have a prior assumption that this is both Kohli and Makore and normal okay so in fact therefore we are not discussing an arbitrary filtration we are this filtration I is therefore is the integral close I n is the integral closure of I1 I1 to the end therefore and therefore we will just change but therefore we should keep this I mean we keep this in mind so we'll write I as the first ideal and we will also assume that I has a reduction yeah these are in primary ideals has a reduction J generated by system of parameters these are not very restricted I mean once you make this assumption these are not realistic and just one more thing maybe I'll just put it here we will we'll also occasionally need to look at the extended RIS algebra for this filtration which is just where for negative indices I n is okay so what is the motivation for this question so there are characteristic zero results results of here and so let me state one there are many results in that that paper of here let me just take one that's sort of representative and sort of will motivate our results assume that R is a regular local ring the basing is a regular local ring and then if this is a rational single the RIS algebra is a rational singularity implies that the canonical module of home of the RIS algebra in degree n is the multiplier ideal sorry this is you do not know how to write that J or whatever symbol is okay this is the this is a this is a result in this there are other results but okay so okay so this is the multiplier ideal and as a corollary of this same same hypothesis thing and let us further denote by D the dimension of R okay and so the notation so there's I here which is the first ideal and then J is a system of parameter J is a reduction generated by a system of parameters then this this ideal the is Jn colon I to the D minus 1 bar so this is a this is a I mean this is really not a consequence of of this part it is really the consequence of this so maybe I should quickly sketch the argument because that is the part that is sort of motivated somehow this thinking about this problem okay so let me just quickly sketch this has nothing I mean essentially this is just a calculation of omega r in degree n it doesn't have to do with just with multiplier ideals so let a denote the extended RIS algebra for the reduction this is Gordon Stein r is Gordon Stein so this is Gordon Stein and omega a is as a graded module it is minus d plus 1 a shifted by minus d plus 1 and in fact in fact R this RIS algebra R R the extended RIS algebra R prime is the normalization of it when it's required the chalk doesn't break but and then we can calculate omega r bar as a graded module by this omega a and omega a is a a shifted by this and therefore this is a conductor and this is that's where this expression comes it is the conductor in graded in certain pieces that's where this expression comes okay so this is yeah so actually I was thinking at some point that such formulas might be useful in computing multiplier ideals computing meaning in my colleague to but now I'm not entirely sure of that thing it's not clear to me which is easier so okay so I sort of abandoned that part of the thought but there is this because one would think that there are existing algorithms to compute integral closure and colons let us say relatively easy to compute so we could do this and compute this one but I do not have any sort of intuition about the extreme cases in which where this works okay so I okay so now we have some characteristic P motivation okay so I mean substantial part of the characteristic P motivation comes from two three papers of Hara, Vatanabi and Yoshida okay so this is a paper which discussed a rationality of these algebras and then the the famous paper Hara and Yoshida defining test ideals and there is another paper of Vatanabi and Yoshida which does some other I mean I forget the the full content of this thing but it has similar looking colons and things that involve test ideals so let me quickly motivate I mean let me just quickly say some of these things okay so first so this is the paper which is described a rationality and more importantly it also has a good collection of interesting questions and problems and conjectures so it was a nice paper to start so let me just in this context in this context of this theorem and the corollary there let me quote a result from Hara and Yoshida this is a test ideals paper okay so if you assume R is Gorenstein oh sorry I didn't actually explicitly say anyway for example I used here that this is this is finite over A but so I am assuming the rings are all rings that we consider are excellent I didn't oh so I so all these rings that we are discussing are excellent our Gorenstein and a rational then sorry if the Ries algebra is is a rational this the same formula there which is in in degree n is the test ideal of this is yeah sorry this is yeah both of these quantity this is for all n positive n because this is anyway zero in in non-positive degrees and this is the test ideal okay so so there therefore there is a formula similar to that which involves involves powers of J and integral closure of ID minus one which will compute a test ideal as a colon but again as I said I was at this point I'm not sure I'm trying to compute test ideals that way is more efficient than anything any other known algorithm okay so this is so that's one result and yeah so and this is this paper of at Nabe and Yoshida also has similar formulas involving integral closure test ideals and and the reduction so there is okay so this is the sort of motivation for for to start thinking about these problems so let me now get into some new results that are already published so results and I'll also touch upon further new results okay so maybe I should at this point okay so I said this but all rings that week all rings schemes we don't need a lot of schemes but we need proj of our are excellent in this so that is and let me just quickly define a rationality although I stated things have without actually saying what it is so we'll assume since these are all excellent and we'll also further assume that they are homomorphic images of Cohen Macauley rings so we'll just take this restrictive definition but which is equivalent to the general definition is a rational there exist an ideal generated by system of parameters a system of parameters that is tightly closed and for a new theory and ring a new theory and ring slash scheme is a rational if it is so close points maximal ideals I mean we are all the things that we discuss their aspect localization so it's enough to check at maximum ideals and the main thing that we use to prove these results is the tight closure of zero in the top local homology module so this is a theorem of Karen Smith suppose that this is called Macauley then a fractional if and only if the tight closure of zero in the top local homology module of zero where D here is the dimension throughout for the base ring are I will write D for its dimension and so we are to slightly we rewrite this in a graded situation and so what we will do to prove is that so similarly to prove that the reese algebra is a fractional we will show that it is Cohen Macauley or sometimes we part of the hypothesis and to the the tight closure of zero in the top local homology module of the reese algebra is zero okay so let me not write the subscript it is the homogeneous maximal here we are right at once because I just introducing more notation when I write like this for the reese algebra I always mean with respect to the homogeneous maximal this it's a local ring so there's a unique one so this is essentially studying this is the all these new results that I'm going to present and so there was a question in that paper of paper of Hara Watnabi and Yoshida which asked what is the relation between a rationality of sorry sort of question it was actually conjecture that put one direction what is the what is the relation between their rationality of the of the reese algebra and the rationality of the extended reese algebra okay so that is so that is theorem okay so let me state the theorem and one direction was already known the following are equivalent the extended reese algebra is a rational okay so this is the x the basering is a rational basering and so the idea behind this okay so I should clear two implies one had already been proved by Hara Watnabi and Yoshida and they had asked for their conjecture that the other statement is true and so one implies two was proved by Mitra and myself and also this argument is different from me so this argument will give a different proof of the other direction also and what is I mean what is the what's the underlying idea if you take zero so if you take this top local cohomology module of the reese algebra this is so at some point you would reduce to some Cohen Macaulay some lemmas to prove that yeah either under under either of these hypothesis some Cohen Macaulayness will be there so so let me just write like this it is actually the negative part of the top local cohomology module of the extended reese algebra these are different maximal ideals but let me just be sloppy in my notation in degree j this is this is not a very difficult and I mean a new observation was that the tight closure of zero in this module tight closure of zero in this module here is has the same behavior which is the negative part of the tight closure of zero of this one and using this one can and and also what's it where does this come in in this hypothesis that in degree zero of the extended reese algebra this lives only in negative degrees but extended reese algebra top local cohomology lives in all degrees and in degree zero this is the homology of I mean these are not okay so that's where now you just one one looks at what happens to the tight closure of zero that theorem will follow yeah so that is the so now I would like to state another result this again came from the paper of the question came from the paper of Harawath Nabi and Yoshida see if the reese algebra is a fractional then so would its proge which is the blow-up of the so maybe I so let me call x the proge of the reese algebra and there is a map to speck are and this is the blow-up map of spec blow-up of spec are along spec are more I so that's this that's this map so if this is a fractional if the reese algebra is a fractional x would also be a fractional and so they were theorems of in the paper in characteristic zero of herey and in Harawath Nabi Yoshida in characteristic P where they ask questions or prove theorems related to the fact related I mean of the similar sort the sort of hypothesis it says some hypothesis on our and x is a fractional would that imply that the reese algebra is a fractional there were some so the next result that I want to list is related to that so I think there is a okay at least for me there is a motivation for this question which is I think the first example where of a non-affrational reasoning was by Anurag Singh and the example that he constructs the proge is not a fractional so the non-affrational points of the proge therefore also show up in the spec and therefore no it is okay so therefore I for somebody I was curious and they're actually there are many quite a few results which say that if x is a fractional and nice hypothesis on our should force the reese algebra to be a fractional so let me state the next theorem which I will quickly go through because it's it's a so let's assume that this is a let this be a three-dimensional rational singularity in characteristics P okay so by this at least the way we have defined this to mean that there exist a resolution of singularities with the usual vanishing property okay I quickly say where the three-dimension is coming is going to come in okay and again as usual we are taking in primary ideals assume that x is a fractional for all n sufficiently large the verinice subring of the reese algebra not that I mean we do not know the reese algebra as such is a fractional or not but if you look at the verinice subring of the nth power okay so this is a this is just skip every degree other than those are divisible by n and such results again at least in discussing whether reese algebras are corn macaulay or not there are older results of Craig and Hakaba and various people if you under suitable hypothesis on the nose the reese algebra may not be corn macaulay but a verinice subring are corn macaulay so this is maybe it's a result it's okay maybe in the interest of time I so maybe I'll just say okay so the point is that so the way the hypothesis here it means at least for us we took that as a definition of rational singularity as that there exist a resolution of singularities which which for which the usual vanishing of homology applies for a resolution singularities of our but so but where does three dimension three come into play that is because we would like we use the result of Kausar which said that in dimension three you can do so then we make we can make that resolution singularities factor through this x this morphism and then it is just a and then some some argument to prove that if you do like this it'll become corn macaulay then that here like this to say that if you further replace the n by larger ones we can get this also and that is the that's the idea of the proof okay so dimension three is to say that the resolution of singularity would factor through this x which I do not may exist in other cases but I I don't know much okay so I that's where we used yeah and also I think the sum of the spectral sequence arguments are quicker and we'll we'll finish our fast if it's in dimension three they may become too big if they're higher dimension okay so I okay so now further newer results which are not so these things are these this is still sort of in preparation okay so this is so I'll so there are the newer results are somewhat more technical to state and they were there in Nirmal's post I don't know the posters are still there so the full technical details are there I will just give one result which also looks similar to this so I think similar hypothesis RM three dimensional Gorenstein complete f finite Gorenstein because you have to do some duality in the middle okay so just calling my call I may not be calling my call I may be enough but we haven't really and if you instead of working with the with this yeah so that maybe we have to look at not test ideals but sub modules of omega in that case but so that we have not done complete is for some technical reasons finite are all for some using other results which are sort of they require that in the hypothesis yeah assume that the resalgebra for the maximal ideal is coin McCaulay and that the proge of the resalgebra of the maximal ideal is a rational then the resalgebra itself is so we have this only for the maximal ideal and yeah and we have I mean syntactically we have this extra hypothesis that you have to assume prior a priori that this is going my colleague is not required here but of course then you have to pass to a larger thing to get okay so this is the okay and yeah I yeah so the remaining results are technical any questions yes can you say something about the proof if it's not to that oh it is I can we'll upload the thing in a week or so hopefully so then yeah sorry the other questions we don't I mean I do not know if it's true or but we don't have a proof so it looked like earlier in your talk you were looking at the filtration by the integral closures of the powers of the ideal yeah under this hypothesis powers of M would be integrally closed but I mean in the early in the result that you're in thank you this I need okay thank you any other last round for questions okay let's take tomorrow's again