 Time to start the next lecture. All right, thank you for setting the recording. It's my pleasure to introduce Thomas Poulstra. I give the next set of lectures applications a big cone McCauley modules and algebras. I first met Thomas when he was a starry eyed undergraduate looking at graduate programs. And I'm happy to report he came to Missouri and was even my student. And in spite of that, he's had a lot of success. So Thomas is currently finishing up an MSRI postdoc and will be starting a position at the University of Alabama. They'll finally, for those of us in the United States, Alabama will finally have something besides their football team to be happy about. Yeah, it's a very good school. We're very glad that we'll open up a community of algebra area there. And Thomas has been working in characteristic P in a variety of areas related to to tight closure and Hilbert Kuhn's multiplicity and many other related areas so thank you Thomas and we're happy to have you here. All right, well, wonderful. So thank you for such a kind introduction. And thank you to the organizers for inviting me to speak and and also invited me to speak on such a wonderful topic. One of the more rewarding and interesting topics and community of algebra to dive into. So, and, and I'm fully convinced this topic of big coma colleague models and algebras will always be in the future of community of algebra and study. So it's a remarkably important topic and a subject and it's a great responsibility to talk about it to you all. So, so yeah so yes this is the summer school on high closure and I get the pleasure of giving a couple of lectures where most of it in prime characteristic. We are gone. So what I will be doing in my two lectures is I just want to talk about the applications of they call Macaulay modules and algebras, as it relates to specifically the homological conjectures, at least some of them. But I also that we won't get to this so much today but definitely on Wednesday. I also want to talk about how big calling Macaulay modules and algebras are used or specifically algebras are used to define and study this is this is talk about work at that but so we'll talk about the homological conjectures and how to study singularities that resemble tight closure singularities but in a way that's independent of characteristic, and it's allowed us to So, so, so, so yes are will be commutative no theory ring, not necessarily a prime characteristic I'll be careful to say when we do need prime characteristic or when it doesn't matter. All right, we're going to be talking about big calling Macaulay models, particularly these are modules that do not have to enjoy the property being finally generated so we'll be careful about specifying when something is finally generated or not. So the definition to start with so let's start off the local ring curl dimension of the ring is a d take a system of parameters x1 through xd, then a module is going to be called say big Cohen Macaulay with respect to this system of parameters if couple of two things happen. One, we want that this sub module of the ideal generated by the parameters times m is not the entire module itself. And so right so the module m doesn't have to be finally generated so this is a non trivial condition to impose a module is non zero if it's not finally generated. And two is we want that x bar the sequence is a regular sequence. Right, so what I mean by this. I that Xi is regular is a non zero divisor on him mod x1 through Xi in or a different way to say this so we don't have to work with quotients but we can instead just talk about colon sub modules condition to is the same thing as saying that. This should be an Xi minus one down here. It's going to be the same thing is that x1. My screen froze. Excuse I don't know if y'all can hear me I'm having cool issues on my end. Yeah, you're audible. I don't know do you want to try on sharing or sharing. Well no that's the thing my entire screen is just frozen. I don't even know the mouse or anything. Yeah, I mean your audio is coming through find your videos also. Okay, thank you. This is my computer. Okay, I hope I can, can you, can I be heard. Okay, I see some thumbs up. All right, yeah. All right, I think I think the writing program I was using is going to be trouble so we're going to use something else. Okay, so we'll just go this online. So I believe, right so we talked about what a big Cole Macaulay module is with respect to system parameters so keep our ring local for a little bit, or keep our ring local so we're going to say them is balanced. Okay, or what you'll, what I'll just call this or just, I just like calling these big Cole Macaulay, or you'll just see the abbreviate this even further as just BCM. If it is Cole Macaulay with respect to every system of parameters. And so, a little bit of history here, some history. So, so the study of these things, and their importance and relevance was due to Hoxster's work in the 70s, which I realized the importance of these things in the subject of commutative algebra and really preach about these. So, but what Hoxster was able to establish is that if so just to make life easy, RMK is a complete local domain. So if, if R contains a field. So, either a field a character CP if we're in the character CP setup or a copy of the rational numbers if we're in characters to zero. Then are in that's a big Cole Macaulay module, a balance. So if it's characteristic at this point, then, let's see, some other very significant advances around this so Hoxster unit key. So this would be in the tight closures as this 90s. This would be that if, if characteristic of ours prime, then are plus is a big Cole Macaulay algebra. Our plus so our plus is the absolute integral closure of our, and a choice of the algebraic closure of its fraction field. But, am I going out with this right so big Cole Macaulay algebra is our algebra that is big Cole Macaulay is a module. So but, but the, the amazing thing here is that if your characteristic P there's this canonical natural given. Big Cole Macaulay algebra that always exists and then. And then along these same lines so something much more recent, but so I believe this is, this is recent, this is 2022 21 or no no excuse me maybe this time when this first appeared on the archive for 21. So if, if our is mixed characteristic, say, zero P, then the patic completion. So, our plus is big Cole Macaulay. And then I'll remark in the echo characters zero case our plus is never calling the colleague in dimension three and larger, and that's actually part of the exercises. But in the characteristic P in the mix characteristic setup there's is really canonical choices and so so a couple remarks about these these theorems. So the the Hoxley unit keys result. This was after their result came out a simplified proof of this appeared by work of unity and loop as Nick and. We'll be talking about this in a couple of lectures after I finish up my lectures can I will talk about our plus being Cole Macaulay in case he and exactly what they're able to achieve using relatively simple techniques. So, but the thought result is cannot be described as a simple result at all it's depending fundamentally on the theory of perfectoid spaces of shoal say that, you know, arguably aren't him fields metal. It's depending on the development of prismatic homology and so forth. So there's a comment in the chat. Yes, when I say big Cole Macaulay point out, it always is going to mean balance. I don't want to talk about on balance calling the colleague modules. But there are, but there's exercises on on on how to construct modules that are big Cole Macaulay but not balanced. But from here on out, we only care about the balanced ones because those are what are important for proving these conjectures, most of the time. And the more serious ones. But the greatest achievement and all of this and the existence of big Cole Macaulay modules and algebras hands down goes to Andre. So, so the most powerful theorem at our disposal, and arguably the greatest, one of the best contributions to the subject to commutative algebra up to this point is the following theorem. And I believe this is published in jams in 2019. So Andre team. So if say our to s is a map of say complete local domains. You can do much lighter than this but let's just keep it simple for the time being. And then there exists a diagram to be our to us. We are. Yes, such that are to be our S to be so best are BCM algebras over RS. And in fact, Andre does more than this is it's given a map artist and there exists these BR and BS is so for example characteristic P. You could always put our plus and s plus that would be fine if aren't as a characteristic P there's always a diagram with our plus and s plus so we'll just put it up there. You can just take our plus s plus. So this would work. Characteristic bars P. But I still want to emphasize that this map right here does not have to be unique. And so, so the terminology that gets used here is, is what Andre has proven is there's the existence of what are called weekly functorial big coma college algebras. And this theorem is the greatest achievement of this and we'll be able to prove some interesting things with it. So. Okay. So, so the first one, say let's focus on this focus on, I mean, let's focus on tight closure first. So some applications is just these theorems are just going to be fundamental tools for proving these theorems I'm not to discuss. So, so, so the first one, say let's focus on this focus on, I mean, let's focus on tight closure first. For a little bit so some applications to tight closure theory. There's a couple of theorems here so theorem. And so once again let's just take RMK complete domain and characters. So theorem. If let's say X bar. Hi, is a parameter ideal. Nice, I believe flooring and rescue would have talked about this. And the tight closure this parameter ideal can be realized by extending the ideal I to our plus and then contracting it back to our. But of course when it comes to proving theorems are very powerful and interesting statements using the techniques and what's going on in the background is way more useful and powerful and interesting as things to extract. So that's going on in the background of this theorem. So let's just call it another theorem this is rightfully due to Karen Smith, which will motivate an analogous definition of a rational and rings at art of prime characteristics so which will get to more on the second lecture so theorem. Following our equivalents. One, or is that rational to for all BCM our algebras say are to be the top map of what we call model biology modules is injective three. HDM art. HDM for us is injected. But the key one to focus on here is number two that ends up being a useful way to define what went one long Carl Schrie call what are called BCM rational singularities and these are going to end up mimicking f rational singularities and characters. Thomas. Yes. Sorry, this is a little bit poorly phrased but can this property being a big code Macaulay I'll probably be detected just from the co kernel of RTV. Certainly no. Well, I mean, certainly the co kernel had that D minus one but I mean that's that's a lot to ask. Right. I mean it's equivalent. Yes. Yeah, sure. Yeah. Thanks. We said, actually hold on maybe I'm just understood your question to ask it again. Well, your answer about the depth of the controls is pretty much what I was. So, so those are just some applications. All right, and then let's see, and then I want to mention a conjecture here, I believe this talked about an inch on Mars lectures. So conjecture is the following our equivalent on our is strongly a regular to our to our plus is here. But it is I just want to write this conjecture out this is conjecture. A different way to say this is that every ring of prime characteristic that a splinter should be strongly a regular and you've seen in the lectures at every strongly a regular radius splinter. And, you know, the hope is is this conjecture is true, and we give more meaningful insight on how to extend the notions of strong F regularity in the mixed characteristic world and just studying say splinter rings maybe. But, but this, but I do want to mention that mention on my cross we do have ways of extending the notion of strong the F regular to the mixed characteristics setup. In terms of big column of collie algebras. But it's a, you know, it's a little bit of work to be done to really understand what's going on there, but, but the point I want to make here is if you. I wholeheartedly believe that if you want to study this conjecture of splinter implies strong or even the weaker conjectures of, you know, every weekly F regularity is F regular like the property of week F regularity passes to localization. So I've become increasingly convinced that the study of that problem is equivalent to studying properties of big column of collie algebras over our. So, and it looks that's what's going to be necessary to solve some of these big open problems. So the some other applications so the homological conjectures which will we will be focusing most of our time on. Let me just directly show up here so the ones we will prove are the menomial conjecture and then bit as a corollary effectively we can derive the direct summer and conjecture. Amazingly enough is completely trivial and characteristic zero. Three. The vanishing map of tour conductor. These are all theorems now. And then one of my favorite things. Direct summands. Regular rings are calling college. And so hopefully I believe for has shown up in the lectures, at least in prime characteristic right if you are a direct summon of a regular ring, or even just a weekly F regular ring, then you actually have to enjoy the property of being weekly F regular. And then that's enough to imply Colin McCauley. And so the proof that direct summands are regular rings are Colin McCauley is a relatively, or thing in prime characteristic and equity characteristic zero and especially the mixed characteristic case. There's a much different beast and it requires these, this theorem of Andre on the weekly functorial big Colin McCauley algebras will use the true force of the homological or excuse me the of Andre's work to get number four. But for one and two. We will, we will not need nearly as much. So, and then the other application that we won't spend any time on, but the, the study of big Colin McCauley algebras shows up very significantly in the study of test ideals or multiplier ideals. So the Lynch one line Carl Schreed have done these mixed characteristic test ideals, and they developed this theory in an appropriate manner to, to, to establish the uniform symbolic apology property and regular things. So, what does this theorem say, so it says that if, if RMK is an excellent regular local ring. And this is for all R so no singularities. Then of say to crawl dimension D, D, then for all P and spec R, the D symbolic in power is contained in regular and power, and this is for all natural numbers. So we won't be going into this but we say where this comes from so I'm not sure Smith and the equity characteristics zero case, and they use the theory of so called multiplier ideals to make this happen. Which ended up being the inspiration to finish off the proof and mixed characteristic. Yeah. So you have Hoxton Heaney key around the same time as the ELS result. Prove the same thing in character CP and actually recover the iron laser fell Smith theorem by a reduction to prime characteristic. And then, then my street in the mixed characteristic case. But I will say that the ma street stuff was certainly inspired by the multiplier ideal theory of iron laser spiff or analogously the test ideal theory of work to Kagi to prove the Hoxton unity results and characteristic P in a novel way. And should also mention there's, there's an analogous version of this property for not necessarily prime ideals or non reduced ideals and that solution in the in the mixed characteristic case was settled by a to Kumi Moriyama somewhat recently. All right. Okay, that's just a maybe very partial and incomplete list of reasons to care about this subject. So that being said, I want to just start diving into some math. So, so we're going to be you. We're like I said, it's just, we're just assuming that they call Macaulay modules exist. Balance they call Macaulay alters exist and we're going to assume weekly functorial they call Macaulay alters exist as well. We're going to assume the whole thing and derive some application so subject one. So applications to the homological connectors. All right, so the first one will go for is the monomial construction. All right, so it's going to say the following. So we're going to take a local ring of curl dimension D. Let's make positive dimension. So it's an interesting statement. Let's say X one through XD, a system of parameters. And for every natural number and we're going to have that X one times XD raised to the power has no chance of belonging to the monomial ideal generated by X one to the N one. And let me put a remark here like what a more useful way to maybe a very useful think about what this is saying. So what is the same. So, so I recall that we can identify the top local module of R as a direct limit system involving these parameters. And so how's that natural identification. So the top local co homology module of our is isomorphic to a direct limit. Let's say over the natural numbers index by cities of well it's going to be the teeth piece will be our mod X one to the T, XD to the T. So that's the teeth piece of the direct limit system. The T plus one piece would then be X one to the T plus one X D to the T plus one. And then we're going to multiply by the product of the excess here. And so, all we're saying so. So notice. So let's do a converse of the statement so X one through XD to the end not being inside this monomial ideal. Okay, so what let's suppose that element wasn't. So notice X one dot XD to the end is an X one to the N plus one, XD to the N plus one is equivalent to. So what is that if and only if if and only if, well, not an if and only if, but it in particular, it implies that X one dot XD to the end, plus the N plus one piece here of the direct limit system. So I'm looking at the image of the element X one times XD to the end. And the N plus one piece of the direct limit system up here is zero and HDR. Okay. But this this element right here is right if you go backwards in the direct limit system you see that this is naturally the image of one plus X one dot XD. Right, I'm saying right. Right, these two out these, these two elements are the same because their representatives are of this, you know, hit each other in the direct limit system. And so, and so what I'm saying here is so therefore monomial conjecture is if and only if. Monomial conjecture is true if and only if one plus X one, XD, are does not equal zero as an element of the top local. Right, so what the monomial conjecture is doing is it's just pointing out particular elements of top local homology that have to be non zero points amount. Okay, so let's do a proof. Let's do the proof. There's going to be a couple of limits so in the notes I put a lot more in the notes that I'm going to put on this. In particular is a couple of limits and I believe have been covered in other lectures but if not, please just go to the type notes that you should have available to you to read the proofs and in fact the proofs that I've written out will help out with some of the exercises. So in the notes, there's a lima 3.4 that we're going to use. And so what is it going to say. So, okay, local, local arts, a map. Oh, wait, I'm getting ahead of myself. We'll need this for the direct summing gesture I'm getting ahead of myself. Excuse me, we're ready for the proof of the monomial conjecture. Proof, we'll use limit 3.4 for the next year. Proof of the monomial conjecture so. So suppose for contradiction, we can multiply these parameters. Raise it to the power that we can land inside x1 to the n plus one. Okay. This is going to be if and only if the R linear map are to our mod x1 to the n plus one xd to the n plus one, sending the element one to x1 xd to the end is the zero. Let him be a balanced big cone McCauley module, our module. And we're going to tensor with him. Okay. So that's going to tell you that the map him to ill mod x1 xd to the n plus one and plus one is the zero. But, okay, so what is this really saying so this says give me any element of them multiplied by x1 through xd to the end, and I have to land inside this sub module x1 to the n plus one up to xd to the n plus one expanded down. So this is if and only if using like these colon ideal language. Now, it's equal to x1 to the n plus one up to xd colon. All again. But there's going to be an exercise so let's see that exercise number is two. So using the fact that m is calling McCauley. It's not a big call McCauley module. It's not difficult to prove that that colon ideal gets quite trivialized. Just using the fact that x1 through xd is a regular sequence. And all of its permutations are regular sequences on him, right that's important that is balanced big call McCauley here. But this is a contradiction. Okay, this is exercise two. But this is a, because it's not equal to the sum. All right now, let's move on to the infamous direct sum and conductor so. And arguably, the reason, I mean, from what I can tell from context is that the direct sum and conjecture is the motivation behind this, this all the study was Mel's motivations was the main focus, in terms of all these homologic injectors, direct sum and conjecture about to talk about was the main focus. Common in chat. And he just said, yes, yes, you have to reduce to the complete case. So yes, let's back up just a minute. Yes, so there's a comment here I should have made some reductions to use that there is a big call McCauley module I need to get to ours a complete domain, and I didn't do that reduction so so yes. So, suppose you knew that, you know, we're going to prove by calculation. So if x one through xd to the end was inside that in primary ideal. Then you can that same containment has to hold in the completion. And then, and then if your completion doesn't happen to be a domain, you just pick a minimal prime P, so that x one through xd is a system of parameters on that prime P. Choose a piece of the dimension of our is dimension of our P, but are my P. And you can just replace your ring with that. So yes, you can reduce the complete domain case and so therefore you can use the call McCauley modules. Thank you for the comment. So we're going to get to direct summoning conductor and to do that, we need a couple lemma so so we're going to be talking about things splitting so we're going to need some criterion for splitting. And this is lima 3.4 in the notes so arm KB local me finally generate a module. And R to him a map, or linear map so send them the one to some element, the module. The following are equivalent. One, R to M splits. And then there's a direct summon of them by this map. And then to is that I take the injective whole of the rescue fields and tensor with this map. Objective, where the RK was injected whole right so splitting is a property that can just be checked after tensoring with just one module. So we're mapping to a funny, funny. And then lima 3.5 will also use is that if we have very explicit descriptions of injective holes for certain classes of freeing so. So if arm K is a Gornstein, and we'll use, for example, if your ring is regular. For example, regular. Then the the injective whole the residue field has a very explicit description. Namely, is a topical co homology module, which of course we have this very nice way to think about these is certain direct systems, and then so one of the exercises. Here so I did so if you haven't seen how something like lima 3.5 is proven. And there's an exercise right up in the notes. And there's an exercise immediately following this lima about how to describe the the injective whole as a similar looking direct limit system that looks like local co homology. But isn't. It's a case that are is not going steam but only assumed to be calling McCauley and has a canonical ideal. It's a very useful exercise. Right. You take a look up. All right, so that being said, we can do the direct something conjecture. So I'm going to say the following so let a be a regular ring. A to R a finite map. A to R splits. And let's do the proof by the monomial conjecture so So the so there's a lima. I didn't write out. So by lima 3.2. You may assume a is local regular in which case s is going to be a finite extension array. Or excuse me ours a finite extension. And then let's see so. Okay, so we're a is a regular local ring so alright so that means this maximal ideal is minimally generated by the dimension elements. If D equals to mention our or dimension a, which will be the mention of our Let's say the maximal ideal of a is generated by a regular system of parameters. Then the injective hole by limit 3.4 The injective hole of a k is isomorphic to top local homology model of a which can be realized as the direct limit system with this with respect to the regular system of parameters x12 xd. Moreover, the the soccer k isomorphic so k say is a my maximal ideal of a I can realize as the stuff inside the top local homology module. Annihilated by the maximal ideal, which is then generated principally by the class generated by this element. Right so this this single element one plus x12 xd of a principally generates the soccer. We also know that the right so this is the injective hole the residue fields so the way k sits inside this injective module, it's essential is what that means so zero colon hdm. To this in this local homology module is essential. And so what does that mean so that's if which just means that if we have a non zero sub module of the top local homology module, then the soccer actually has to be contained inside of it. So in conclusion, so by based on the dilemma 3.4 a to R splits. If and only if HDM a to HDM a are so this is isomorphic to our tensor. So if this map is one to one, but that map is one to one, if right so the point is is that the kernel of that map is a sub module of HDMA of a. And so that map is not one to one if only if the kernel is a non trivial, but that's if it only if that sub module contains the soccer. But that soccer is principally generated. And so that's going to be if and only if the image of one plus X1, XD. So, let me just spell it out very carefully here. So that's if and only if, well, if and only if one plus X1, XD. Okay, so the soccer generator, which would then be mapped to one plus X1, XD are right so you're thinking this would be inside the direct limit system. Hey, X1 to the T, XD to the T. You're thinking about this element inside the direct limit same exact looking direct limit system but with our array and say, and right so what we're crying is is that map is one to one if and only if the image of the soccer was non zero. That's what lima 3.4 is going to buy the fact that splitting can be just checked after tensoring with the injected hole. So, meaning we only have to check that the soccer the injected holes map to something on zero. But we can very explicitly point out what that soccer element is, and where it's going to inside this map and the monomial conjecture says precisely that that element can never be zero. And yeah, that's, that's 950 so. So next time, we'll talk about the bandishing maps of tour conjecture. We'll talk about a very nice application of that which is one of my favorite theorems is that direct some ends of regular rings. They are always Colin McCauley. And then, after that we'll talk about these so called big Colin McCauley rational singularities as they were introduced by Lynch one month Carl Schweed in a fairly recent publication. So, thank you. Thank you Thomas. Do we have any questions or comments. I do have a question. So, it is always true in a coin McCauley ring that every system of parameters is a regular sequence. So what's the difference between coin McCauley and the coin McCauley I think I'm missing some very big Colin McCauley, big Colin McCauley means it doesn't have to be finitely generated. Oh, I see. Okay. Yeah, yes. So yes, so the joke is right. All right, so the definition you'll see is that a module is called a small Colin McCauley module if it's finally generated and big Colin McCauley. Right so every small Colin McCauley module big. It doesn't sound right but big is not small. Small is big, big not small. Sounds good. Thank you. Do we do we have any other questions. All right, well let's thank Thomas for his fine talk. And I guess we will reconvene on Wednesday at whatever the appropriate time is for your time zone. Great.