 Just an overview sort of of today's talk. I want you to have the impression in your mind that now we have done all this work to build up the basic objects that I need to start anew. So what we showed last time, or we sketched the proof of Andre's theorem that BCM algebras exist for any complete local domain. And so in some sense, the point of the today's talk is to talk about the ways in which we can now use them to try and mimic things from positive characteristic in mixed characteristic and sort of look at F singularities from this point of view. So we'll fix the following setup for the rest of today's talk. So let RMK will be a complete local domain. If I forget to say it, usually dimension D. So I'm going to specify again that P is inside the maximal ideal. So there are two cases, either the characteristic P case, meaning equal characteristic P, where the characteristic of R is also P, and the mixed characteristic case where now the characteristic of R is 0. So the first set of singularities I want to tell you about are what are known as BCM regular singularities. And these were introduced by Ma and Schwede. All right, so here we go. So let's say that fix first BCM algebra B. Everything's been perfectoid, let me say that. So we say that R is BCM regular with respect to B, if it's normal. And as Keiichi discussed this morning, also Kuhl-Gorenstein. And the main condition, the map from R to B splits. So there is an R module retraction here splitting off of B. And of course, since R is assumed to be complete, that's the same thing as asking this map here is pure. OK, all right, great. So there's my definition of BCM regular with respect to B. From this, and this is a pretty common trick here, so I can cook up an ideal which will detect whether this holds. And that ideal I want to think of is the test ideal with respect to B. So we'll denote by tau B of R, just the image of the evaluation at one map. All right, so, and this here, now, OK. So I want to think of as the BCM, or what was it, the B test ideal. And why I say that and the motivation behind that is something that will come up later in the talk, all right? So of course, this notation with the tau B, I haven't really used anything about BCM anywhere. So if I write tau sub S of R for any R algebra S, so this is, some people call this the trace of S as well, that also notation is still well-defined, OK? So, but note just from the definition here, we have that, right, of course, this ideal detects whether you have something which is BCM regular with respect to B. All right, so that's what you do if you have a fixed BCM R plus algebra B, right? And then, of course, we say that R is BCM regular, right? Without any qualifications, right? If, well, implicitly again, normal and Q-gorenstein, and it's BCM regular with respect to all perfectoid BCM R plus algebras. So again, I can cook up an ideal that would detect this condition just by intersecting over all of the B test ideals for all B, right? So you might write, right? So tau script B, right? To be the intersection test ideals for all Bs, OK? And we've already seen something which is maybe not so easy to get just directly by looking at this. This intersection here, in fact, stabilizes, right? So the intersection over all of these will be equal to tau B of R for all sufficiently large B because of the domination trick we saw or we asserted about BCM algebras before. So, right, so take any element, right? Or any set of BCM algebras, if you will. I can always find another one so that all of the maps, the original set, factor through to that guy. And when I do this, the traces can only get smaller, right? So it's pretty easy to convince yourself that that will give you some B for which you have a quality. And once you do any larger B, anything bigger than that is obviously also going to be something which has to give you a quality, right? So this is this domination trick of Manjui, right? Is everyone happy with this? Okay, so this is my first sort of class of singularities that you could define. And we'll see in a second why this is interesting, right? Or what the relationship is to the positive characteristic versions but I want to do something now here at the board which, again, maybe in a class is not so great to do if you're trying to take notes but the notes are all online. So I'm gonna just tweak this to give a second set of definitions, okay? So second topic, BCM rational instead of BCM regular, all right? So we say that if I have a fixed B, R is BCM rational with respect to B. If, well, instead of normal and Q-gorenstein we just have that it's comacoli. And I'm gonna tweak this definition here and say not that it splits but that the following map, well, let's do it in two ways, right? So first is maybe the traditional way, the way it was first defined. The induced map on the top local homology, that should be injective. And of course, equivalently, right? So when I state it like that it doesn't really look like it's something that's all that close to BCM regular but if I take the matless dual of what I have over there, what do I get? Well, I can look at the homset, hom B omega R and that comes together again with some sort of trace map here, just evaluation at one, right? And that map is dual to the induced map on local homology, right? So equivalently, this evaluation at one map from hom B omega R back to omega R should be surjective. Continuing with my tweak, I can obviously define an ideal or not ideal but test sub module, analogous again, as we'll see to the parameter test sub module if you will, right? By looking at the image of that evaluation map and that gives me at least some sub module of omega that detects BCM rational. And again, BCM rational in general means BCM rational with respect to all big comacoli algebras if you will, right? And the test sub module with respect to all the bees is the intersection and is the same as the test sub module for all sufficiently large bees, okay? Is everyone happy with my gloriously ornishly decorated tweak? Yeah, thank you, I didn't quite finish my tweak, right? So that's the implicit. Great, so what are some of the easy implications here? And maybe this will start to justify some of the terminology to you already. Well, if you're BCM regular, then in fact, you must be BCM rational. Why? Well, if you're BCM regular then all maps from R to B split, and what I can do is just take the splitting and hom it into omega R and that gives me that the induced map over here is surjective. Moreover, of course, if R is equal to omega R, I am Gorenstein, right? Then a surjection like this obviously has to split off. All right, so if you're Gorenstein then BCM rational is the same as BCM regular. All right, another easy one. Of course BCM regular implies that you are R plus regular. Now I'm gonna be a little sloppy with my notation here. I've only defined this assuming that I put in a B which is perfectoid, right? And all my perfectoids had better be P complete, right? So I'm gonna just omit the P completion from my notation. Let me note that that doesn't really affect anything because I can slip the P completion into local comology or to any of the hams because everything can be P completed, so, right? And so of course R plus regular, so this is a variant of BCM regular with respect to a fixed big comacoli algebra, if you will, is the same as saying you're a splinter that's also Gorenstein, all right? So this is just definitional, okay? So the interesting thing here is you already, even with these implications, run into something which we don't know and is a very important open question in some sense, which is the converse, right? So this is an open conjecture. So but let's just take this, or I'll revisit this in a second so after we've done the next set of discussions, right? So I could ask, right? In positive characteristic we'll see that this is a version of asking that F regular and Gorenstein is the same thing as a splinter which we know is true with the Gorenstein assumption, okay? So my assertion is that that is true in positive characteristic, right? Which is why I wanna make it a conjecture and mix characteristic. All right, but any questions before we move on from the definitions? That's the conjecture. So the conjecture is that asking your R plus regular gives you that your BCMB regular with respect to all Bs, right? And so, yeah. All right, so, okay. So, and again, I'm trying to be as slick as possible to write as many terms as I can on the board. So I've compiled a compositum of as many things as possible to tell you about what these things look like in positive characteristic that really motivate how they fit into the framework of this workshop. So, but without saying some massive list of names, right? So there are at least, you know, I don't know, too many to list here, right? But they include certainly Hoxer, Heuneky, Moshweed, Smith, and the list, Lubesnik, myself, many, many, Bargav, who knows, many people, right? On this list, okay? So, in characteristic P, right? So, assume we're in equal characteristic P, right? So, the first is that BCM regular is the same as strongly of regular, right? And moreover, the test ideal, one of the, this one with respect to all of the BCM are algebras, is just the usual test ideal of R, which, of course, I can write in many, many different ways. So maybe, because we're Kugorenstein, it doesn't matter how you think about which flavor of F regularity you'd like in the same way which flavor of test ideal is known to all be the same. So look at the collection of all the test elements. And of course, it's also known in the setting that you get many other things, right? So, the test ideal with respect to all the B's is the same as the usual test ideal of R, which is the same as the test ideal for R+, which is the same as the intersection of the trace ideals for all finite extensions, which is the same as the trace ideals for all sufficiently large extensions. So R, I guess I should, yeah, whatever. All right, so, and similarly, BCM rational is the same as F rational, and I have a similar string of equivalences for the parameter test submodule, if you like. So tau B omega R is the same as tau R+, omega R is the same as the usual parameter test submodule of R. So, now I can't quite give the same tight closure characterization here, because if I stick in all parameter ideals here, I get the parameter test ideal, which is something slightly different, right? But what I get is the matless dual of, say, the HTMLR module of the tight closure, right? So this is now a submodule of omega, okay? And again, this is the intersection over all of these sort of traces with respect to all finite extensions, or indeed after a sufficiently large extension. And so maybe in this framework here, this is a result of myself, Manuel Blocla and Carl Schwede, but really is a variant on the generalization of the original Equational Lama due to Uniki and Lubesnik, all right? So, so let me start my table over here. So at this point, what this means in some sense is I have now a bunch of notions, which I can sort of match up one by one, right? I guess maybe another thing I should say before I erase everything over here on the left is that many of these equivalences in things like this on these two boards are discussed in my lectures from last summer on test ideals, right? And the set of notes that came out of that as well, right? So, so, right? So we have sort of paired up these notions and see that, well, the definitions of these guys over here reflect at least one of the definitions or characterizations of the guys on the right, okay? And of course, you can see, I tried to pair this up. There's a split, there's a crack in the board right here. Okay, so, oh, what's your question, Holger? Everything, if I use a BCM regular, so, all right, so maybe here up here at the top, all of these are Q. Gorenstein, normal, and Comacali, so I run into no problems. So let's not get into any of the difficult cases, okay? So Holger points out that I am, so the obvious question, Holger's question is, what do I do if I have something which is not normal Q. Gorenstein or Comacali, and the answer is you go to work, okay? So, so I've paired these things up, all right? So, but the point here is, when I have something in mixed characteristic, all right, I can travel the characteristic zero just by inverting P, all right? So, since I showed it right, so, or at least equal characteristic zero, right? So where I'm working now over a characteristic zero field, okay? And the point here is, part of the package of results from Mon Schwede is that if I look at BCM regular or rational, all right? So, I get log terminal and rational, right, in characteristic zero. So, I have a characteristic zero singularity notion which matches up with the usual reduction mod P dictionary and many of the topics being discussed in Keiichi's lecture this morning. Of course, you're gonna ask if the same thing holds. The test ideals and what you would expect to get is, well, the multiplier ideal, I won't say it precisely, and say the multiplier submodule, right? Or grout room inch knight or sheaf, if you will, right? In characteristic zero, right? So, and at this point, these do not exist in literature, right? But, let me say that we're working on it. So, just as a quick idea, I can never remember how the order of the letters is supposed to go, right? So, in work with Batma, Pathakvali, Schwede, Wittechek and Waldron, I said the last two in the wrong order again, right? So, in progress, essentially what we can show is some localization results. This tau B, namely when you invert P again, what you get is exactly this multiplier submodule. And again, these results, without the question mark, so in particular BCM rational and inverting P gives you something which is rational, was referenced earlier this morning, at least implicitly in Keiichi's talks, right? So, this is the essential ingredients together with what, knowing what happens when you go mod P, so the adjunction type statement, that will allow you to take something which is F rational and a single characteristic and then deduce that it is, deduce something in characteristic zero. Okay, right? Any questions? So, the only bad thing about everything I've done so far, we just gave a whole bunch of definitions, right? Is much in the same way that writing down a BCM algebra took us a fair bit of time and actually proving that result. I've also been very impressed by the other lecture series speakers and many of the other speakers about how they remind us that it's important to be explicit and to give examples, right? Of many of the things. So, maybe it's not so easy to exhibit things and show that they are BCM rational, BCM regular. In fact, this is proving statements that look like the direct sum end theorem, of course, took quite a lot of time and machinery, right? So, let me give you at least some examples, right? So, at least of BCM regular, in many cases, these will be Gorenstein's, you know, it's the same thing for BCM rational and of course, these are the stronger ones anyway, so all of these will be BCM rational, right? So, here's the first one, right? So, look at the Fermat in mixed characteristic, if you will. So, look at ZP, power series of joint Y2 up to YN and kill the sum of the mth powers of the parameters, okay? And the claim is that that is BCM regular, if P is sufficiently large and M is less than, M is less than N, right, inside of this, right? So, and if this looks somewhat familiar, right? How do you show such a thing? Well, modulo P, these conditions produced the Fermat mod P in one fewer variable, right? And then, I can use reduction mod P, right? If you like, or Federer or whatever you would like to do to show the corresponding statement that mod P is F regular and then I have to lift this using some sort of a junction result, right? And so, this was done in a paper with Ma and Schwede with Chek and Waldron, all right? All right, so some sort of general statement that assuming everything inside is Gorinstein, for example. If you get, what you get mod P is F regular, you can lift that back up, right? So, there's one example, right? So, here's another one, right? So, ZP join XZ mod X squared plus P squared Z plus Z cubed, all right? So, again, to many of you, this may look like a familiar equation. This is, well, if P happened to be a variable instead, this is, I think, a D4 singularity, right? So, assuming P is at least five, or P is at least seven, right, bigger than five, right? Then, this thing here is, again, BCM regular, right? So, and this as well as a detailed analysis of things for all the rational double points was done in a paper with Javier Kovar-Horohas, Lingchua Ma, Thomas Poulstra, Karl Schwed and myself, right? So, more generally, we showed that if you take a rational double point in mixed characteristic with mixed characteristic zero P, at least, P bigger than five, then what you get is in fact a direct sum end of a regular ring, which is another easy way to check that something is BCM, rational or regular. So, the more general statement here is, let's say you have a Q-Gorenstein direct sum end, and of course I could keep going on. So, again, maybe I haven't put an attribution down here, but this requires some actual thinking, right? So, I also have some sort of Toric-like examples or what have been known as log-regular rings. So, these are in mixed characteristic, something which looks like a WIT ring, a join, completed, semi-normal, strongly convex saturated normal monoid, right? Modulism equations, right? So, something that essentially looks Toric, right? So, and I can tell you more about that later, if you'd like. Okay, so, but, and again, this appears in work of myself with Hanlin Kai, Sangsu Lee, Lingxuan Ma, Karl Schwed, right? So, which I'll talk about later in today's talk, but also generalizes work of Gabber and Romero, okay? Right, so, but the point is, is that even in some of these examples, it requires some real machinery to find explicit examples of BCM regular things at this point, okay? Any questions about this? So, with that, let me change gears, but I want to finish to tell you a little bit about it. So, hopefully you can see how BCM algebras at this point fit into the framework of F singularities from these kinds of tables, right? This related a fair bit more to K.H.E.'s talks, but you could ask about if there's something I can do that will put you in the setting of Ilya's talks. So, is there a mixed characteristic analog of F signature and Hilbert Kuhns? And that's the next piece of the puzzle that I want to tell you about for the remaining lecture, okay? So, let me add to my setup a little bit, right? So, additional setup that we'll hold. So, everything earlier plus apply the Cohen-Gabber theorem to get a nice northenormalization, right? So, find a regular subring so that the extension A to R is modulo finite and on fraction fields this is separable. So, this again came up in Ilya's talks already, right? With, let me write down A very explicitly here. So, we're in mixed characteristic zero P or in characteristic P. So, in characteristic P, then A is just the power series ring. If A D variable is X one up to X D and in mixed characteristic, right? So, A has to be the weight ring on my perfect field K, right? A joint X to up to X D and for this just to keep the notation easy to write down, let's set X one equal P so that X one up to X D are still a system of parameters, right? And the point here is somehow that sort of to tell you what the obstruction that has been around for a while to try to fill in this gap here of what the Hilbert-Kuhn's and F's signature are, right? Everything we've done up here before involved taking bigger and bigger comacole algebras B, okay? And in particular, even the smallest one that seems to work, say in positive characteristic is R plus and my assertion essentially is that we think at this point that it's just too big. Even R plus is just massive in comparison to R, okay? So, and maybe I learned about this from an example of Craig at some point. So certainly R plus is not an aetherian but it's even worse, it behaves in a lot of ways in a very non-geometric way. The sum of primes is prime and has other very strange behavior, right? So maybe the right thing to do or what we'll need to do is to do something which goes sort of in the other direction and I wanna think about the smallest perfectoid I could use. Instead of using bigger and bigger B's let's go the other direction, right? So let's set A infinity to be, right? So this ring right, join all the infinite p-th roots of this is sort of parameters and then complete with respect to p and of course this is the same thing as A perf in characteristic p and the other one I wanna introduce is what I'll denote by R A perf, perfed, right, which is, right? So well, if I look at A infinity we know that that thing is perfectoid, right? If I base change up to R what I get is now, well, certainly finite over A infinity so I can apply this perfed functor of botan shoulder to get the smallest perfectoid guy over this tensor product that both R and A infinity map to over A, right? So and again, maybe it's worth saying this is equal to R perf, right? So point here is that it's an analog of R perf which is somehow much closer and smaller and close to R than R plus was and even in characteristic p. So in order to tell you about this mixed characteristic F signature and Helberg-Kuhn's multiplicity the notion I need to first tell you about is normalized length which is originally due to faultings, right? So and so what he did is he says, well, let's take a module over A infinity and assume that it happens to be MA power torsion, right? So in practice, most of the cases the modules I'm gonna be looking at M kills it. So it could be MA torsion if you like, okay? And to such a module I can associate a number called the normalized length and you wanna do it in such a way that it should sort of mimic what you want length to be, but use the explicit structure for A infinity that I have. So first, we're gonna do this in steps. And so the first step, if M is finally presented, so this is pretty much the only case I know how to do this very explicitly. So we define if M is finally presented, in fact, since this A infinity really comes from adjoining p to the eth roots for all E there's in fact some E so that in order to define M I didn't really need to go all the way to infinity. I could have just joined p to the eth roots for all the system parameters. IE M is isomorphic to or is equal to a base change ME tends to over AE with A infinity for some ME. And in this case, you can set the normalized length to be well one over p to the ED times the length of this module ME that you base change to get your original guy. And the point here is that, well up to the p completion this A infinity is given as the co limit over all the AEs, but more than that the map from AE to AE plus one where I just joined adjoined roots of each of the system of parameters is flat, well locally is free, right? Of rank equal to p to the ED, all right? So once I know that it's easy to see how the length scale if I pass from AE to AE plus one and the scaling factor will give me a number that's independent of how I chose this presentation over here event, okay? All right, so I want to define normalized length. I've told you how to do it for every finite presented module. So the second thing I do is define it for a finitely generated infinity module, okay? And in that case, I just build it up from inside. So, well every finitely generated A infinity module is a co limit of the finitely presented quotients right of this thing. So I'm infinity M, I just set to be the infimum of the normalized length of all of the quotients from a finite presented finitely presented module onto M. All right, so certainly if I was dealing with usual length and I take a surjection then the length of this guy has to be bigger and I force that property to hold. And then three, all right? So if now I know how to do it for all finitely generated modules, well an arbitrary module is the co limit of its finitely generated sub modules. So if M is arbitrary, I set lambda infinity of M to be the supremum over the normalized length of all finitely generated sub modules. And again, if I have a sub module of another module I expect the length to be smaller. So this is capturing what that is. So once I've done this, one checks, right? So, well, this is in fact well defined. The crucial step still comes right down here in checking that this is independent of the presentation in the finitely presented case that you chose. But more than that, what you get is an additive untrue to exact sequences. It really does behave like a length function in some way. So just a couple of examples, although the warning in some senses they're not so easy to come by. So one, if you take an ideal I inside of the regular ring A, all right, then, well, with finite co-length, then the normalized length of A infinity mod I, A infinity, is still just the normalized length, or is still just the length of A mod I, simply because A infinity essentially is flat over A. So all of the AEs are flat over A and I can calculate what happens along all those base changes, right? So it doesn't really do anything if I look at ideals inside of A, right? Two, second example, let's say that you take some non-zero element F inside of A, right? And just for simplicity here, or let's say that the characteristic of A is P, so that A infinity is equal to A perf, then you can check that the normalized length of, well, A perf, module O. Well, now we're in characteristic P, so my element F in A perf has infinite P throats. Oh, do I wanna do still just A, right? So take A perf and kill the ideal generated by those infinite P throats as well as, well, I want all my modules to be M power torsion, so I might as well kill M as well, right? So the claim is if you do that, this normalized length is zero, right? And one of the reasons I write this example down is there are many other properties of normalized length I'm not telling you, right? But one particularly important observation here is, well, to put this in the framework of the last talk, this module I stuck in here is so called F almost zero, right? F and all of its P throats kill this guy. And one of the points of introducing normalized length is it gives you here some ways to capture that sort of in a heuristic way, things that are almost zero should have zero normalized length, all right? So I'd love to tell you that that statement was actually true, but the, so, and it is, but which notion of almost you mean there factors into the statement, all right? So let me tell you the theorem here that really puts F signature and Hilbert Kuhn's on that table over there, right? So again, still say you're in characteristic P, all right? So as I said, if you wanna do something where you add to that table and put something in mixed characteristic, often what you have to do is come up with a new interpretation of how you used to do everything in positive characteristic. And so that's what this theorem is gonna do, right? So the first thing is to say, well, if you take ideal I inside of R with finite co-length then the normalized length of R infinite, oh, okay, R perf, mod I R perf equals the Hilbert Kuhn's multiplicity, right? So, and, right, so before I write the second statement, I just wanna point out that this is a statement purely characteristic P. It doesn't use anything that you would didn't have 15 years ago to talk about, right? So, or much longer. So I have R perf, I, Hilbert Kuhn's multiplicity, whatever, all right? So in some sense, this is just a reinterpretation of the definition of Hilbert Kuhn's multiplicity, right? But it is one that allows you to get an idea of what to put on that table over there. So, and in the same vein, let's let I infinity be the non-splitting ideal of R perf, right? So look at the set of elements of R perf so that if you map one to those elements which you get does not split, okay? So then, again, the normalized length of R perf mod I infinity R perf, in fact, equals the F signature, right? So for the experts in the room, I challenge you, go ahead and prove that theorem for yourself. It doesn't involve essentially any of the perfect way of mathematics that I've actually done in the earlier lectures. Exactly, so I just erased it, but you could ask, where's the limit? Where's the convergence? Where's anything? And somehow, right, it's hidden inside of the definition of normalized length. So there is content to this, right? So you have to do some fair work, but it means that the end result is something that's not visible as a limit. It's just a length in some generalized fashion, okay? Not that I know of, right? So I'll say something about the relationship with the usual multiplicity here in a second, but that's the thing I'm going to say in a second. So. And again, so if you go ahead and try to prove this for yourself, I just have a caution for you, right? So our perf is not finitely generated over a perf, right? So you have to really work with the definition to try and figure out what this means. And in fact, tracing through back many of Mellon Craig's proof, come down to finding ways to get around this fact, right? So finding, for instance, some element C so that C times our perf is contained in our joint A perf in the existence of test elements all deal with exactly this problem in some other way, right? So, and maybe I'll say more. I'll put a sketch of this in the notes as well if you'd like to see it afterwards. All right, so okay, great. So we're in the setup for the actual definition of factoid Hilbert-Kuhl's multiplicity and signature. All right, so again, in either mixed characteristic or characteristic P, right? Take a finite Kuhl length ideal and I just set the perfectoid Hilbert-Kuhl's multiplicity. Oh my goodness. EX perfed, not HK, right? To be the normalized length of RA perfed, mod I, RA perfed, and the signature, well, the perfectoid signature of R to be the normalized length of RA perfed, mod I infinity, where again, I infinity defined is defined as to the left replacing RA perfed with R, or replacing our perf with RA perfed, right? So, to justify some of the notation, you'll see in comparison to usual Hilbert-Kuhl's multiplicity RF signature, you see these little X's up here at the top, right? And one of the reasons they put that there is my definitions depend on the explicit system of parameters I chose, right? So, and they depend explicitly on the presentations of A infinity that I wrote down, okay? So I joined all the infinite P's roots of the X's, but not some other system of parameters as you saw, right? If I wanna go and add more, that takes extra work, but then often will make my ring big enough but I don't know what to do, all right? So at this point, we don't know independence of X, right? So let me just finish the lecture series by giving you some of the key results on these guys, get a slightly bigger piece of chalk, right? So, well, and again, the point of these results, a lot of the things I'm gonna write down are hard to show, the paper is quite long, 70 pages or something like that, right? But you'll see that that really starts to mimic what you think of for Hilbert-Kuhl's multiplicity RF signature. So for instance, ex-perft of R, which of course means ex-perft of the maximal ideal, is bigger than or equal to one with equality if and only if R is regular, right? Now I've assumed R as a domain, so, you know, don't get too excited about doing the, about associativity formula on mix and this and things and all that else. I've been just doing the domain case here, okay? All right, so and similarly, Sx-perft less than or equal to one with equality if and only if R is regular, okay? And moreover, the relationship between Sx and Hilbert-Kuhl's multiplicity also carries over, right? So the perfectoid signature is the infimum over the relative perfectoid differences, right? For containment of ideals, right? So this is the conjecture of Watanabe Yoshida, that Thomas and I proved, right? This is the Watanabe Yoshida theorem that Ilya spent some time talking about. So you see the shadows of all the things from positive characteristic. Great, so what's another thing that we know Ilya used today in his lectures? Well, we also know that Hilbert-Kuhl's multiplicity, for example, uses those Hilbert-Kuhl's differences in order to detect tight closure. So if I have a containment of ideals in R or finite co-length, all right? So again, the result is that it's easy to show from the definitions that the perfectoid signature or perfectoid Hilbert-Kuhl's of J is less than the perfectoid multiplicity of I, all right? But the interesting part is that you get equality, again, if and only if the extended full plus closure of I equals the extended full plus closure of J. So this, right? So it was really flushed out in the work of Moniteman, at least the closure operation, not the statement on the perfectoid signature, right? But just to have the definition on the board, right? So X is in this ideal IEPF, if and only if something that looks like a tweak of the tight closure definition holds. So there exists some non-zero element C with the property that C, the one over P to the E, let's say, is in the ideal I, P to the N, R plus for all E and N. Bigger than zero, right? So instead of just looking at IR plus, you add in the powers of P for all N. R is a domain, everything is a complete domain, whole lecture, so yes, it's definitely there. C one over P to the E is in the ideal I, P to the N, R plus times that thing, thank you, right? So C one over P to the E multiplies X into the ideal I, P to the N, R plus, yeah, right? And this is what Ilya referred to before. If Y went up to YD as a system of parameters, right? Perfectoid multiplicity of Y, with respect to the X's, is equal to the standard Euler characteristic just as you would expect for multiplicity or hypocons, right? So, but the warning here is that this is not easy. We have to go to some great lengths in this paper to show this and that's partly because the Y's may have no relationship to these X's over here and it's not so easy to switch them, okay? In particular, let me just advertise some basic open question. You could take any statement about Hilbert Kuhns at this point and try and mimic it for perfectoid signature and we don't really know the answer, right? So, here's something that's open. Well, you'd expect, we know from this statement just by taking a minimal reduction, right? And applying this formula that the perfectoid signature is bounded above by the regular multiplicity or the perfectoid multiplicity is bounded by the regular multiplicity. But I don't know something I do in positive characteristic that this is bigger than one of our defectorial times E of I because this uses something that's really about the Frobenius powers of the ideals, right? So, uses something that I don't really have access to in positive character and mixed characteristics still at this point, right? So, this inequality is open. All right, so, and sort of the last thing I'll tell you, right, so, if in addition R happens to be Kuhnstein, well, again, we know that F signature is positive if and only if the ring is strong enough regular. So, here I would expect the perfectoid signature is positive if and only if R is BCM regular. So, analyzes that as well, right? Okay, so I have a lot more information in the notes, right? So, if you're really interested in particular, let me say, you can also do a tweak on this. You can do a dual F signature variance, so, which is something I have investigated quite a bit to get a perfectoid relative rational signature that will, whose positivity will detect BCM rational instead of F regular, all right? And, again, maybe I should say that one of the big applications of F signature also comes out of the paper. So, we get transformation rules for perfectoid signature under finite quasi-etal extensions. So, in particular, we recover sort of well-known results at this point about with applications of the finiteness of local fundamental groups and finite torsion in class groups. And, in particular, we recover results on computing the perfectoid signature for finite quotient singularities, which in many ways date back to, again, the work of Watanabe and Yoshida and Hunikileski for relative Hilbert-Kunz and F signature, respectively. Okay? All right, so I mentioned the big open question with independence of the X's earlier, but I'll end here with another, again, very closely related to Ilya's talks, right? Pretty much anything in mixed characteristic that has to do with localization at this point still is very hard, right? So, this is one of the reasons why it's taking so many of us to just invert P in those kinds of statements, okay? But if you knew the independence, it'd be very natural to ask for localization compatibility, so I expect the F signature to only go up under localization and Hilbert-Kunz to only go down, right? And then I could ask that this determines the semi-continuous function on spec R, either upper or lower in the two cases, right? And again, at this point, these are pretty much completely out of reach for all of us, right? So, there still is a lot of stuff to do in order to make everything match over in this table, right? All right, well, thank you very much. Joe, we did something, I think there may be some inequalities inside of there, mimicking some of these things, but I have to look again before I can answer. So, great question, right? So, as I said, one of the ideas here is that R plus, or these big comacola algebras that we really have our hands on in some way, are huge, right? So, if those are the things that are making things look like F regular, right? So, versus, so F regular is asking that bigger and bigger B split, you might try to weaken this as much as possible. And so, of course, in positive characteristic, F split is the same thing as saying the map from R to R perf, splits, right? Not R plus, but R, right? So, forget, but R perf, okay? At which point, so the natural thing to ask is, well, can you just ask that you are split with respect to some BCM algebra, right? Or maybe that the map to R, just R perfs in some absolute sense splits, right? So, but at this point, there are a group of us that are working on some of those things and we can show a couple of results, but that's not fleshed out yet, but it's a great idea, right? So, F split characterization, F injective, F nopotent, all these other things, I don't have a way to put them here on this thing. In some sense, these are the interior cases where things should be easy, right? That as I go to the boundary towards log canonical or F split and other things, we really are still kind of in no man's land. Yep, very natural idea, which is the question was, if it's not independent X, what do you do, right? And I don't know, right? So, obviously, it is at least in one case where I use this as sort of parameters, right? But that's the only evidence I really have one way or the other, right? And so, the idea then is that, well, maybe I just take an inf, for instance, okay? So, of course, if I want to characterize positivity, I have to be a little careful, right? Because an inf of positive numbers, unfortunately, need not be positive, okay? So, maybe I have to take a soup instead and then I'm not sure, right? So...