 Welcome back. Welcome back. The first lecture in the series of 30 lectures on type closure will be given by Neil. He will give three lectures and conduct one tutorial. So Neil is from George Mason, Mason University, which was mentioned in the lecture here for a conference where a hoaxter had given his talk. So Neil, welcome and you may start your lectures. All right, well thank you very much to the organizers for making this this whole thing go. I hope this is good for students and thanks very much to Craig he and he for giving this intro talk it's. Every time I hear him even though even when I hear stuff that's familiar always learn new stuff and new ways to think about things so so thanks there as well. Okay, so I'll say that that my lectures. So, there will be some overlap with what he did because. But I'm. Well, we didn't we didn't coordinate and and and so and I'm a student so that shouldn't be that surprising Apple doesn't fall far from the tree right. And so and so I would say that my introduction to tight closure theory is going to be pretty traditional. In particular that I'm going to concentrate on like, you know, as part of a set of a collection of closure operations. And, but I'm also going to put more. I'm going to talk about modules in a way that that he didn't I think they they they're important as sort of part of the theory. And I won't touch reduction to characteristic, so I won't touch the characteristics you're okay I don't know someone else's plan to. And I apologize in advance for my handwriting but you know that's why we have election. Hopefully it'll also be useful to have me write stuff down well. Okay, so. So that the notes that I'm putting up have eight sections I'm going to cover seven of them I guess the first three sections are in the first lecture and that's closure operations and the for Venus. The previous, I mean, like, not just the previous endomorphism but also for Venus closure and the previous factors. And then the second lecture which will be on Wednesday, beginning is tight closure and also module finite extensions. And, and then part three is going to be a section six and seven of my lecture notes, colon capturing test elements and persistence and you've heard a Craig talk about a number of those of those words already. I'm going to kind of work everything out from first principles, and kind of a nuts and bolts approach. So so that's certainly an idea of what I'm going to. Okay. So, let's start. Okay, so everyone sees the full screen right. So, the, so the first part. So as I said it's going to be based on on the idea of closure operations in general. And so, section one, what is a closure operation. So, definition. So, and this definition is due to you like each more from 1910. I don't know if it goes back further. But anyway, so. And so, let M be a set. And so for that, you want to think it's a ring. Or is an arm module, but his definition was on sets like he was interested in like more was interested in, in the analysis so this is just very, very general. And it's useful this to see what's going on really generally to get perspective on a lot of things. And let us be a collection of subsets of such that. You know that notation two to the m it's just to collect that that's the power set of them. And so, this is just going to be just a collection of sets of them, such to that. One, M is one of those sets, and to us is closed under arbitrary intersection. So if you have a bunch of sets and s. And then any intersection that's also an element that's fast. And so, and for us, let's think. And so, the sum of s is equal to all ideals, or all sub modules in these cases but it's actually useful to sometimes even think of it as just being the power set like all of the, of all subsets. So, um, then, um, so a closure operation on s is a function. See, when he was like super scripting station from s to s such that three things hold. So the first one's called the extensive or expansive property. Um, for any element of s. So I'm using L as the suggested notation think in terms of sub modules or like m like a module, but for any L and s. It's contained in its closure. So, so that's extensiveness. And then this is order preservation, which is that for any K and L and s, such that K contain an L, you have the K closure is contained in L closure. Okay. So there are properties that sort of are, you know, abstracted from the, from the notion of a courtesy closure operation in in topology, right. But not all the properties of it so. And then third one is item for any L and s, if you take the closure of L. Again, nothing else happens. So, um, Craig mentioned that this is the case for interval closure, it's this is actually just one of the axiomatic properties for the closure. So this is a closure operation. Um, and, um, and if L equals LC. Sorry about that. So, um, yeah, zoom has been acting up lately. I don't know why. Anyway, so if, so if L is equal to it's closer than we say it's close. So, um, so remark, the note the M and s, the superfluous convention that the intersection of the Z of the empty subset of s is just the full set. It's not a common convention set theory. Okay. Um, so, so just right away, let me give an example. If M is a ring, and yes, is is the set of all subsets of M of R. Then you can define, you know, the closure of X to be the ideal generated by X. If you think about that is a closure operation in this sense, any subset of a ring is the subset of the ideal that it generates. So if you have two subsets X and Y of a ring and X is contained in Y then the ideal generally X is contained in the ideal by Y, and the ideal generated by an ideal is just that same ideal. Right. You can do the, so anyway, so in some sense you've seen this this concept before. In terms of generation, you can talk about generation of sub rings generation of modules and so on. So this is that this is not usually called a closure operation, but, but it is. So, now let me give you an equivalent characterization of closure operations. So, proposition that M and S be of this above. So there is a one to one correspondence between on the one hand, all the closure operations on S. And this is by also by more. As I know, I'm the closure operations on S, on the one hand, and the set of sub collections of S that are likewise closed under arbitrary intersection. I like this is true of ideals and arbitrary intersection of ideals is an ideal. So, and if you know about like the radical is going to be a closure operation on the set of ideals and intersection of radical ideals is still radical so anyway so hopefully the examples you know already, you can see already do this. And here's the correspondence. So if you have a closure operation, you can see, you just take the closed elements of S to be your this sub collection so and it's easy to show from the axioms of the closure operation that any intersection of closed objects of S will be closed. So the other way, if you have a sub collection D of S that is closed under intersection. So then I have to define an operation so I'm going to say that the closure of L is the smallest element of D. That contains up so and so why is there a smallest one. It's because it's because D is closed under intersection so you just take all the elements of D that contain the contain L intersect them you're going to get an element of D. And then I'll be the smallest and it turns out that if you have a sub collection that's closed under intersection, then you this operation does satisfy the property closure and these are usually inverse operations. So, you know, sub collections, so you have a collection that's like, that's close an area section, you take a sub collection that's closed under intersection then then you get a closure operation. And in general, So indeed, for any subset of S. The assignment of just taking taking an element of S and taking it to the intersection of those K in E that contain L is going to be a closure operation. So, for instance, if S is instead of ideals of R and E is equal to the prime ideals. Neil, I'm not sure what happened with your sharing stuff. Yeah, apparently zoom is going to keep putting out for a while. Sorry about that. I don't know why it's been doing this lately it didn't do that this before the last update so Okay, so, so for instance, so where was I so if you have a collection of elements of S. And you say okay I want to intersect all the elements of either contain L, then that is going to be a closure operation. So for instance, if you have S the ideals of R and E is the prime ideals of R which is not closed and around under intersection generally, then the result enclosure operation is what is what I call the radical. What Craig was calling the nil radical. I think this is a perspective on closure operations that isn't really emphasized very often, but I but it's, you know it's it's useful, I think it's a useful thing and because. So these these notes. So usually the definition of closure operations just given as these three things which might seem like, you know, a laundry list like okay so you get bigger. It's the same assignment, and if you do it twice it's the same as doing it once, but in some sense that's not arbitrary it's just the same as saying, you know, okay I'm I'm taking a collection of elements of S, and I want to intersect them together closer. So, and so but of course for these lectures. So typically, and it's going to be a module. And in the above is a module or a ring. And then S is going to be all ideals, respectively sub modules. And you're going to say that C is going to be a closure closure operation on M. So closure operation module I mean the closure operation on the collection of sub modules of that module, and a closure operation on a ring is the closure operation on the on the ideals that. So not the generation, but I would just want to point out that generating, you know, generation is itself a closure operation on the sets. So, but okay so so that's the more so. So, here's a definition by Petro. So, we want our operations to be to be well behaved in terms of location, and I think Craig mentioned something like this for the closure. So a closure operation C on a ring. R is called semi prime. Historical reasons doesn't have anything to do with that prime ideals for it for all ideals, I and J. J times the closure of I is contained in the closure of J. And hence, J closure times I closure is also contained in J. I closure, just by changing the roles and the rings community. And in particular, if I'm going to skip that one, but I'll just say that good things come from sending crime this that I'm going to say if you see is a semi prime operation are and if I is C closed, then so is I call a J for all ideals J and hence the minimal primary components of, of, of I are also going to be closed. You know, by a team at Donald or whatever those are those are always colors. So the proof is they is is just this J times the closure of I J is contained in J times the times I call J closure, which by definition Michael and J is just I so I call J closure is contained in I closure color J. But, but that's I. It looks like the zoom was quitting out well, hopefully, the professor has seen what we're doing right away. Oh sorry. So more generally you want to find a closure operation on on a whole collection of modules, category of our modules so closure operation on a category. And there you want to use the notation closure in M. Yeah, but you often want to be for every module you define a closure operation on it but you want those to be well related and and so the thing we have is called factorial. And that means that for all our linear G from M to N, you know, in the category. And for all some modules L of M. If you take the closure of L in M, and you apply G. You can find side to closure of G of L in. So, so lemma, any factorial closure M, which is either the category of finally generated our modules or just our modules is going to minimize what I closure times L closure and M is contained in the closure of I L. For all ideals, I, for all. And so hence, if you just let M also equal R this shows that it's sent prime. And all the closure operations that we're going to deal with are going to be. This is the proof. I mean I keeps putting out a case in point. So maybe I'll skip that proof but it's elementary. It's elementary. So, and your first examples of closure operations are the identity closure operation, which just sends every module to itself. And the radical, which we're only defining on ideals. Although you could define it on sub modules, though, using those maps to fields that Craig mentioned is a paper bike. But anyway, the previous endomorphism for genius. Okay. So, from now on all rings are commutative. The theory in any characteristic. Consider the, that just sends an element to its power. And when I, and let's call this the freshman stream. Right. In calculus classes everybody, like a lot of people often think that a plus the quantity squared has to be a square plus B squared. And that's usually false but in characteristic to its true. So, and, yeah, Craig mentioned this, let Q be a power of P, and X and Y be in R, then X plus Y to the Q power is equal to X to the Q power plus Y to the Q power. And of course my proof is the, you know, the proof. The proof that you alluded to, which was. Yeah, sure. Sorry. So I'm going to prove that this because it's so fundamental to what what follows. So, it's enough to I claim it's enough to show it for equal one. It's enough to be because it otherwise just follows by a very quick induction, you know, X plus Y to the P to the E is equal to X plus Y to the P to the P minus one, which is then X to P plus Y to the P to the P to the E minus one and then by induction that's X to Q plus So, so let's prove it for equals one. Now we all theorem X plus Y to the P. What is it it's X of P plus Y to P plus the sum from I equals one to P minus one of P choose I a X minus X to the I Y to the P minus. Right. Everybody goes that. For each I between one and P minus one. Let's look at P choose I choose I is P factorial divided by I factorial times P minus I factorial. And he divides numerator, but he does not divide. And that's because all the factors in the denominator are less than P and he's a prime number and there's not, and there's no way to multiply things that are less than P and get a multiple P because he's a prime number. So, thus, he divides choose I, let's say that P choose I is equal to P times and so I. So then the I summoned in the above becomes N sub I times P times X to the I wider P minus I, but because the ring is characteristic zero. Sorry, because the ring isn't characteristic zero because the ring is characteristic P. This is N sub I times zero. And so all these summands are all zero. And so what's left is X to the P plus wider P. So now let's talk about the functors, if the upper star and if the lower star. So, in general, right. Let G from R to S be any ring holism. Then you have then you have two functors. So, one obtains the extension of scalars under G upper star from our mod to S mod. Where what you do, at least our modules is tensor with S and the restriction of scalars under lower star S mod to our mod. And what you do there is that G lower star of N is an R module by you take the same a billion group structure. R times G lower star of an element X and N is just going to be. R times X is going to be G of R times X which makes sense because N is an X mod and is a S module and G of R is an element of S. And so we all can write R times G lower star of facts, and then G lower star of G of R X. And so, for any literate F E of the Frobenius functor, you then obtain the upper star and lower star from our mod. So that seems very abstract. So let me tell you concretely what at the upper star is if you happen to have a presentation of your model. So, so if given the presentation, or to the T, or to the S and zero of a final generative module by in your house this matrix, a i j of elements of R this S by T matrix. Then, at the upper star of M is just the co kernel of that you take the elements of that matrix to the power. So that's that's that's sort of how to think very concretely if you have a presentation of your matrix and so in particular that tells you what goes on for a month for an ideal. So, in particular, if I is an ideal. Let's say it's generated by X one to X T. Then that gives you this presentation RT to R S is equal to one here. So F E upper star of our mind is our mind generated by X one to the E through X, and what's going to be useful later. Just one comment. When you are writing here on equality this is not that equality this is actually an isomorphism natural isomorphism, even if you want to be precise. Sure, natural isomorphism. Good. So here's a change of rings, lemma. Let S be a ring. And remember, all of our rings are prime characteristics and theory and I be an ideal are equal as my eye. And I will prove this I'm just, I mean there's the proof is in the notes but we'll use this as an upgrade principle. So if you want to take any R module. M. Well, sorry, that's type of S module M. And my I am is an R module. And if you want to take the upper star of it. It's the same as you with respect to ours the same as taking up the upper star with respect to S of M and modding out by I times. And star. That's going to be useful later. So it makes sense to really meditate on this for genius. stuff. So let's meditate some more. So let's talk about the cute root extension. So, suppose R is reduced. So we get K. The product of the outbreak closers. There is our mod script B. Or P minimal times. Then what you have is that, so that product of the arm of keys is the total ring of fractions of our, and so it contains our, so then our embeds into K. For any power of P. Right, R one over Q to be those elements of K, such that their two power lands back in R. Then here's a fact. R one over Q is abstractly isomorphic. Basically, since every element of R has a unique. Q through. Okay, so it has a cute through it in K because of you know, algebraic closure and stuff. The uniqueness is basically because K is reduced. When beta are in K and alpha the Q equals beta the Q, then well, it's reduced and also freshman street. What do you have alpha minus beta to the Q is equal to alpha Q minus beta Q by freshman stream, which is then zero. So since K is reduced. So alpha is beta. Okay, so, so right so we can just sort of so so. So but but of course the usual way to consider R and R one over Q to be related is to think of the inclusion between them. R is it, you know, it sits naturally as a sub as a sub. Rank of R. So, so let's be from R one over Q to R be the isomorphism that sends alpha to alpha Q. And I, R to R one over Q, the natural inclusion. Then, what do you have the compose I. So R is the power of the previous. So, you can think of the previous. So up to isomorphism. FB can be thought is like inclusion. And that's often a very useful way to think about about it. There's a question in the chat. Yes. Yes, the question says, if I was a greater structure will R one over Q also have a great instruction. Yes, absolutely. But you got to be careful it's so if it's graded by the group G, then in some sense, the R one over Q has the is great by the group one over Q times G. So like, like that's sort of the natural, the natural way to do it. But yeah, yeah, absolutely. So right to the Q, basically the truth root of any element is of say degree D is going to have degree D over Q. Yeah, and that's, that's a useful thing for a lot of applications. So yeah, good question. Okay. So, now let's talk about racket powers of ideals and submodules. So, so if I was an ideal in Q is power of the accused bracket power of I is defined in any of the following equivalent ways. Right. Craig already said some of this. I to the back of Q is generated by the set a to the Q. It's not usually equal to that sex that sets not usually an ideal, but it's the ideal generated by it. So suppose you have a generating set for I, then I said I had to just be the ideal generated by the set that was one question just let me ask you one question, one of the participants is also also has the same question. What do you mean by an abstract isomorphism in this case, because before because of course you can in some books like harsh on is just like an abstract it means that you choose a basis. Oh, no, I just literally mean that they're isomorphic. And that's it. But our models as a billion groups or like rings, rings. Okay. Okay. As rings. Yeah, they're isomorphic as rings. Okay. Thank you. So, um, so as I said before, basically, I record q is the annihilator of the upper star of arm. Which is a cyclic arm module. And so, as I said before, if the upper star of our eyes, basically our right. And if ours reduced, then if you take I and you extend it to our one of our q thought of as a as the, you know, an extension of our. It's the same thing is what happens if you take I bracket q, and you take the q th roots of all the elements. Those are the equivalent ways to define bracket powers of an ideal. Now for a sub module. We're generally let L, L and M be our modules. Then, and let I be the inclusion. Then F the upper star of I from F the upper star of L to F the upper star of M is reduced. And then L bracket q in M is defined to be the image. And note, if M equals R, then the two definitions coincide, identified. The upper star of R. So, so now I have my first exercise. And I won't state it here I'll just say, our powers have nice algebraic properties. So now, let's define the Frobenius closure of an ideal or a module. So, if X is an R, and I is an ideal. We say that X is in the Frobenius closure of I, if by definition, there's power q of P, such that X, the q is an I bracket q. And more generally, if you have our modules and X, then X is in the Frobenius closure of L in M means that there's some power q with X q in M. So what do I mean by that. So notation, if you have inclusion from L, L minus M and X and equal R X in M, let's have this. And X q M is at the upper star of I. But by the algebra of bracket powers, which you'll do as an exercise it is also kind of works nicely. And so lemma is Frobenius closure is a functorial closure operation on the category of all. So I guess I have what four minutes left before questions. So I wanted to state one more theorem and that has to do with regularity. But maybe I should do that next time and instead I'll prove this. I'll prove this and I'll get to the regularity. So let L be a sub module of M. You want to show that the Frobenius closure of M is in fact a sub module. So let X and Y be in Frobenius closure of L in M and let R be in R. And what do you have so X is in some L QM, Y is in. Sorry, X and Q is an L QM, Y to the Q prime is an L to prime M. And then what do you have X plus R Y and Q, Q prime by the algebra of these powers is X and Q, Q prime M plus R to Q, Q prime, Y to Q, Q prime M. Which is X Q and Q prime and something plus R to the Q, Q prime times Y to the Q prime and M to the Q and something. And then that's in L QM, Q prime plus L, Q prime M, Q and then that's L, Q, Q prime M. So it's a module. And it contains L by setting Q equal one. Containment preserving is a claim clear. Basically because KQM is contained in L QM, very Q. So let's just do item potents and functoriality. So let X, let Z be an L, Frobenius closure of M, Frobenius closure of M. And then that means that there is a Q such that Z QM is in, is in the Q is back to power of the closure of M. So that means that there exists Y1 through YT in the Frobenius closure of M and R1 through RT in R with Z QM is equal to some RI, YI, QM. And each YI admits some Q prime, some QI with YI, QI and M is in L QM because these are in the Frobenius closure of L and M. Should be a QI. Okay, so let Q prime be the maximum of QIs and also Q. So just take the maximum of all those things and call it Q prime. Then Z to the Q prime is equal to the sum from I equals one to the T, RI to the Q prime over Q, YI, Q prime. Q prime is a multiple Q, so that makes sense. But each of these are in L, Q, I, M to the Q prime over Q, I, which is then in L, Q, M, L, Q prime. So it's in the Frobenius closure of L. And let's just do functoriality. So let M be an R module, L and M, then G are linear, and let Z be in the Frobenius closure of L and M. So there's a Q with Z, Q is in L, Q, M. And again, by the algebra Frobenius powers, G of Z, Q is going to be in F E star of G of Z, Q, which is in, that's equal to it. So it's in F E star of G of L, Q, M. And then by the exercise, this is contained in G of L. That's G of Z is in G of L. So it's a nice functorial closure operation. I'm sorry I had to rush it. I'm sorry I did rush it yet. I won't say I had to. All right. Questions, we started a little late so. I have a question real quick. So, for starters, I like this phrase meditate on Frobenius I think that should have been my meditations on Frobenius should have been the title of my dissertation or something. Anyway, I did have a question so as you were defining kind of these these abstract closures. So I have a question of semi primality for ideal closures, and functoriality for module closures. So I really, I can obviously see why functoriality is the right thing to look at for module closures. It makes a lot of sense that that's sort of very natural to run into. But for semi primality it seems a little less obvious as to why that's the natural thing to look at to me. I don't have any insight on why semi primality might be natural to look at here. Well, I mean, if you care about primary decomposition, and has nice. I would say semi primality is like the poor man's version of functoriality. It's, I mean, it ends up being equivalent saying that any R linear map from R to R is preserves the closure in this sense. Yeah, I think it's actually equivalent because our linear maps from R to R the same thing is home of things right that the same thing is picking up the bar and you know the map is multiplied by that element. And so this semi prime thing says exactly that that's going to preserve that you know, I mean it basically says that element a times the closure of eyes contained in the closure of AI. And, you know, you can make sort of crazy crazy closure operations where that doesn't happen. But, for instance, that happens in all these star and semi star operations that the, you know, that's popular in the non appearing world and it's true in, you know, tight closure closure radical. I mean, I don't know it's not really good reason to say that it's that it's natural but I mean to me it's like functorial on that star. And not every closure operation has like the obvious way to define it on modules like for instance radical and even the interval closure takes a lot of work to find on modules. And, and so, you know, the semi primality so stand into that. Until it's functoriality on home of eighties. So, Neil, are you saying integral closure of modules is a semi prime operation. Um, I'm saying it's fun toriel. Okay. Yeah, yeah. I think there's a lot of definitions of closure of modules and it's like, but I think in all the, you know, the existing ones that they are actually fun toriel. Okay. The question in chat. Oh, the relation between for Venus closer and integral closure. Oh yeah, I'm going to. So, I was going to get into that later in the lectures for these closure is so pretty. The closure is bigger is the short answer. But in fact you have a whole sequence of closures. So, so, for any ideal I the previous closure lie is contained in the tight closure by which in turn is contained in the urban closure by which in turn is contained in the radical. And, and tight closure has characteristics zero and a lot does for Venus closer also have characteristics zero and a lot. Um, I don't think so. Um, I mean I've never seen it. I guess you could try to do reduction characteristic key I mean the natural thing would would be to say that in all the characteristic key models. I'm sorry that infinitely many characteristic key models, the, the, the, the sort of the, the, the characters, the version of the element would be in the previous closure of the characteristic version of the, of the ideal. But I mean. Yeah, I've never seen anybody do that. I mean, there's reasons like, I mean, for a long time people thought that that, you know, tight closure would have this property that, you know, during the tight closure and all in infinitely many character CP models that you don't get the tight closure and almost all of them. But then Brenner and Patsman have a counter example to that so that's not true. Yeah, but it was known from early on that that wasn't true with previous closure I have an example in my, in my lecture notes. So, but yeah, I guess you could make that definition. But in general for Venus closure is considered to be more poorly behaved in many ways than tight closure. So, segue into another question that I have and this I guess is more related to material from from Craig's talks so feel free to defer but I'm very interested personally in my own research, it's time for me as closure. And in particular I've been trying to develop the right, you know, and maybe it's it's out there, but the right map theoretic notion to get a closure of ideal or sorry for being as closure of ideals. There's a notion of, you know, maps to the right kinds of rings that when you can track back you're going to get the previous closure. Oh yeah, I mean, let me just be careful. Yeah, so there's this thing called the, I don't know if it's maps to them. But I will say there's this thing called the perfect closure of, of a rain which is the, you probably know about it, which is the, just all the all the basically all the art of the one over cues at once, like the union of the art of the one over the ring cake that I showed assuming the wings reduced. And, and so, and, and one way to find for the disclosure is extend to that ring and contract that. And similarly, you know, you look at the program module, if you have a module and you look at the national map from M to M tensors with it's called our perf. It used to be called our infinity that everyone says our perfume for perfect. So you look at the map and M to M to M tensor are perfect and you look at your sub module L and you look at the elements that that in that map land inside L tensor are perfect. And then that's the elements of the elements of the M that landed out to our curve. And you call and then that's that's the previous closure of Alabama. But I don't know if that's what you, what you're asking for a little bit I mean so are to our purpose as an example of a purely inseparable extension. So, I'm wondering if something like, you know, if you contract, I back from the recurrence of the extension of even a certain flavors that you'll get. But anyway, this is a research question that's not. That'll work. Okay. Yeah, maybe I'll follow up with you some other time. Thanks. Can I ask you something new. Of course. Yeah, this is not regarding what you covered but like, is this first part of your talk will broadly follow your paper on like closure operation in community value bro. Yeah, yeah, that's, that's, yeah, yeah, that's that's sort of where I, that's when I developed this, this approach is. Yeah, that's, that's in there. Some version of that, but I mean some version of that's in the selection. Yeah, you know, I wanted to do a survey and closure operations and I was thinking and I thought well how broadly defined are these things and like really brother they're even more broadly defined in this after more did his stuff. You know, a post that theorists said, okay, well you can sort of define this on any post that it doesn't have to be the, the post set of subsets of the given set can just be any partially or set and then you, you change your containment relations to you know the less than or equal to sign that exists in that post that you still, and then it's still, if it's a post that we're all arbitrary meets exist. Then, you know, then, then you still get this one to one correspondence that more pointed out for 112 years ago. And so yeah I mean it's very, it's very general. And, and that's, you know, they use that it's somehow a lattice theory in a way that I don't that I've never looked at. Any other questions. I'm sorry. But yeah, I'll happily defer your question to anybody that might know more about it than I did. Any more questions. I've looked at it seems like no one has written about about, I mean, much less is written about three years ago, which is where I'm trying to do this.