Pullbacks for the Rost-Schmid Complex
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Bachmann, Tom
Let k be a perfect field and M a strictly homotopy invariant sheaf of abelian groups on Sm_k. The cousin complex can be used to compute the cohomology of a smooth variety X over k with coefficients in M. However, if X -> Y is a morphism of smooth varieties, there is not in general an induced map on cousin complexes, so computing pullbacks of cohomology classes is difficult. In this talk I will explain how such pullbacks may nonetheless be computed, at least up to choosing a good enough cycle representing the cohomology class (which is always possible in principle, but may be difficult in practice). Time permitting, I will mention applications to the
 Well, I mean, thank you very much for the invitation to speak here and for organizing a summer school in my living room. It's super convenient. Excellent, guys. Okay, so the title is maybe a little bit cryptic, but I, right, so maybe I will convince you hopefully that the result is not so much. Okay, so let's try to get there. So first of all, of course, well, that's why I prepared the first few things, which I want to say because they are all obvious because you've been listening for more than five minutes. So I will, what I want to talk about is some results in motivic homotopy theory. So I'm interested in this category of motivic spaces. What's a motivic space? Well, it's a special kind of pre-sheaf valued in spaces. And so a pre-sheaf on the category of smooth varieties and such a pre-sheaf, which happens to be a sheaf. So I will always be dealing with misnavage sheaves. This is a typology. Well, it's some sort of topology, which is somehow relevant. It's the correct one for various reasons. It doesn't matter hugely right now what it is. And then of course, there's going to be some extra condition because it would be silly to have like two names for the same thing. And the extra condition is this so-called A1 invariance, right? So you just, you look at all those sheaves F such that if you evaluate it on some smooth variety X, or if you evaluate it on X times A1, you get the same thing. And this is the so-called category of motivic spaces. So there are of course, supposed to be lots of motivic spaces because supposedly studying this category will tell us good things about the world by which I mean polynomial equations. So for example, if I take any smooth scheme whatsoever, then I can try to, well, it will not usually as a pre-sheaf, of course it will live here, but it may not be here. But you can just sort of brutally move it into this category because this inclusion has a right adjoint which is called the motivic localization. And that's, that's, that's the obvious thing to do. And usually I will not even write this L mode and I will not write this unada embedding thing. I will just say view X as a, well, this is not what I was going to say. What I want to say is I want to view X as a motivic space. And this just means that you should do this localization, which in general is of course highly non-trivial. But I want you to please imagine doing it. And then there's, there's another class of examples, which is going to be very important for my talk, which is let's say you take some sheaf of a B in groups, right? So in this category here, you take some of the simplest possible objects. Well, of those which are, I mean, you do write some not sheaves of spaces, you now make them zero truncated, but you give them some extra structure. This is a B in group structure. So that seems like a pretty reasonable thing. And then you could ask, is it the case that this, that this sheaf actually lives in the category of motivic spaces? And I mean, that's not always true. There's a condition and I mean, it's just this condition here, but maybe for a sheaf of a B in groups, this looks a little bit more familiar. It just says that if you take F, you evaluate it on X or you evaluate it on X times A1. And you should just get the same a B in group. Okay. So this maybe was not very exciting. So what else can you do with a B in groups? Well, you can look at their Eilberg-McLean spaces or in this case, I can look at the Eilberg-McLean sheaf KNF, right? So this is going to define some sheaf of spaces by definition. And then I can ask again, does it live in this category of motivic spaces? And well, you need this homotopy in various property. And what does it mean to evaluate some, some Eilberg-McLean sheaf on a smooth variety? You get some space. The homotopy groups of which are the homology groups. So in this case, the condition is that this homology, HI of X times A1 with coefficients in F should be the same thing as the homology X with coefficients in F. And this should hold for all I less than or equal to. And so you see, in principle, as you make this N bigger, you get somehow a more interesting object and you get, get a more stringent condition. And then it's somehow natural to single out those pre-sheafs or those sheaves of a BN group so I can do this for all N at the same time. And those are called strictly A1 invariant. Right. So let's, let's just leave it at that. Okay. So now we come somehow to, to something which is close to the heart of the whole thing, which of course, classically, you have this notion of connectivity of space. I already implicitly used it by saying that like an Albert McLean space is somehow particularly simple. And what, what, what is, what this has to do with is that you some sort of sphere and you build how you can attach spheres to your spaces, for example. And more typically we somehow we have two spheres again. I'm sure you know this, right? So that's the usual sphere, usual circle if you want. Sir. Sir. Sir. And then there's somehow some algebraic circle, which is denoted GM. And I mean, it's just you take the complement of zero in the affine line and let's say you point at one and then these spheres, they define your point of motifs spaces, which, well, they're, they're a bit like spheres somehow. For example, I mean, the complex points, of course, of GM is just C minus the origin, which has the homotopy type of S1. So it looks like it's just the same thing. That somehow this is a very topological observation and algebraically, they're different. And then what this tells you is that you somehow get two notions of connectivity out of this whole thing, right? You can measure connectivity somehow in terms of S1 or you can measure it in terms of GM. And I mean, there's no reason for this, for these to agree in general. Now I just said there's no reason for them to agree. So let me contradict myself. So here's an important example. If I take a smooth scheme and I take some closed up scheme, but it doesn't have to be smooth. And but I will assume that it has a co-dimension at least D everywhere. And if you do this, well, I mean, whether you do this or not, you can look at sort of the formal tubular neighborhood of C and X, right? So I do this thing X mod X minus Z. Right? So X is a smooth scheme. So it defines a motivic homotopy type and X minus C is a smooth scheme defines a homotopy type. So I can take the co-fiber of the inclusion, just define some motivic homotopy type. And yeah, I mean, if Z was smooth, then you should think of this as some sort of algebraic version of a tubular neighborhood. And if Z is not smooth, then I know it's this algebraic version of a tubular neighborhood, but maybe a bit harder to imagine. And it turns out that this guy, I mean, if I view this as a pointed space, this is D minus one connected. D minus, that's not very good. D. And well, it's a D minus one connected in the S1 direction or in the GM direction. It turns out in both directions. Okay. So how do you see this? Well, first is let's say that the Z is smooth. And then there's something called the homotopy purity theorem, which basically says that this really does behave like a tubular neighborhood. And it tells you that it looks basically like a tomespace of some, I mean, like the tomespace of some vector bundle. And so it means that locally somehow it looks like this sort of thing, a1 mod GM, which is P1 and which is somehow, which is S1 smash GM. So the point is locally it looks like it's connected, or I mean, if it's dimension D, then it's going to be S1 to the, I mean, it's SD smash GM to the D. And so locally it looks connected in both directions. And then somehow the way the topology works, this is the name which topology works in the TLCMU theory is that this also implies that it's globally connected in the sense. And so what if Z is not smooth, it turns out that there's some sort of filtration argument and you find that this result is still true. Okay. So this will somehow play a big role in what is to follow. And now we can look again at our favorite strictly homotopy invariant sheaves, right? So abstractly homotopy invariant. Right. So then my F defines a particularly simple type of motivic space. And then what happens is that this space is discrete in the S1 direction. Okay. So it looks like the simplest possible thing which you could deal with in this direction, but it doesn't have to be in the GM direction. So somehow there's some, some, some structure going on, which is masquerading, but yeah, so it's not immediately obvious, but it turns out that there's some more things going on. Okay. But not in general in the GM direction. And I mean, I'm not sure that have anything. That's a question for you, Tom. Yes. I saw the question, is there a nice reference for this filtration argument when Z is not smooth? Probably, but not off the top of my head. So I mean, also when I make this claim, I'm saying that it's going to happen in the category of S1 spectra. If you know, like these sorts of details, I'm not claiming this in the sort of completely unstable sense if I want to be very precise. So I'm sure I can find you one. If you email me, I can find something, but not right now. If I clearly remember something is at the end of Röndig's erstware advances mass paper, but at the end of the paper, this filtration argument. Okay. Thank you. Yes. Okay. So I was, I was trying to explain why it's not always the case that these guys, right? So if I take one of these guys, why is it not always somehow connected in the GM direction? And I mean, I think just because there's no reason for it to happen, right? There are these two directions and you picked something which by definition was sort of discreet in the, in the one direction. Why would it be in the other direction also? So you can just work out some examples and you see that this happens. So let me introduce some notation. So in general, let's see sort of a spectrum. For example, it would be discreet if and only if it's loops vanish, right? So it seems important to study the GM loops of this guy. So if this is always zero, then we would learn that these guys are always somehow discreet in the GM direction, but it's not. So let's give you the name. This is called F minus one. That's why we have this contraction, construction. And the point is that this need not be zero. And they, right? So now let me just give you an example. So what you can do is you can work out, this is not totally trivial. The F minus, the discontent, this loop space is again just a, a sheaf. And it's section over, let's say K is given by F of A1 minus zero mod F of K. And I mean, you can put X instead of K in as an obvious modification of this formula. And then you can just, that there are some strictly homotopy invariant sheaves, which we know, for example, the witch heave. So you take the pre-sheaf, which assigns to some, let's say, smooth variety. It's a wit group or wit ring. And then you take the associated sheave in the, in the, in the, in the same topology. And this turns out to be strictly homotopy invariant, not obvious. And what you can compute, what one can manage to compute is that if you do this contraction business, you just get the wit guy again. And this is not zero. So, and also another very famous example is these KN millenowitz. So the unremifined millenowitz K theory. Also, if you contract it down, they're never zero and they form like this, this internet GM loop sheave. Okay. So this is just something which happens and which may be surprising at the beginning. But okay. So we got to live with this. Now I want to put on my, my topologist's hat for a little bit and think about, think about loops based theory. And so one slogan, which I have picked up is that somehow taking loops increases structure, let me just roll this out here. Increases structure. So what do I mean by this? Well, I mean, if X is a topological space, let's say pointed, right? And then I take the usual loop space. Now this has more structure, right? Because loops can be composed. So in other words, this is some sort of monite. And if you want to be fancy, this is an E1 monite or a infinity. Okay. So this is somehow the beginning of classical finite loops based theory. So if you take higher loops, right? So if I put an N here, then I make this an EN monite. And somehow the point is that in fact, this is all the, this extra structure recovers everything. So that's somehow the crux of it. Finite loops based theory. The most basic result, I'm sure there are very difficult things which one can study. So now what I want to claim is that if I do the same thing with my strictly homotopy invariant, chief F, and I do this in the GM direction, I still get more structure. So let's take F strictly homotopy invariant. Right now, let me look at the first loop. So I will denote it F minus one for reasons of tradition. And now what's clear, well, what's almost clear is that this is going to be a module over somehow stable maps from GM to GM. And the point is this thing has been computed by Morrell to be the growth and de-quittering of K. Okay. So what you learn is that this F, I mean, it was just some arbitrary, yeah, the F, it was just some strictly homotopy invariant chief, which someone has given you. You contract it down once. And now suddenly everything becomes a module over the growth and de-quittering, which I mean, can be a moderately complicated ring. It's just some extra structure which somehow pumps out of the sky. And maybe it will be useful. And there's this, there's also more. So there's also something called, well, I would call it monogeneic transfers. And I don't want to go into too much detail exactly what this is. So what it does is, if you have some fields, let's say, finitely generate over K, and then you have some finite monogeneic extension. Right, so this is finite extension. And you've chosen a generator, then this yields a transfer map, sort of tau x from F evaluated K of x goes to F evaluated K. Right, so this is somehow some fancy way of adding things together in some special way. And we will see later some other incarnation of this. But all I'm saying is that, right, so also in our motivic setting, it turns out that if you do take GM loops, you do find more structure. And so what I was saying here is that, classically, if you recover enough structure, then somehow this loops, then you can reverse this loop operation, right? So I do this omega n thing, and it goes, let's say, from n minus 1 connected spaces. And then I go to en monoids. Monoids, as I said. This is somehow some way of encoding the fact that the first loop space has a multiplication. The second loop space, the multiplication becomes commutative. And then higher loop spaces becomes more and more commutative. This is actually in equivalence. So if you give me only the en monoid, which was the loops on your space, then I can actually reconstruct this space. And I mean, you just gave me everything. OK. And so what this tells you, for example, is that, whoops, why is this not OK? Right, so if you have some, if x is n connected, maybe it's n minus 1 connected, then, and y is arbitrary, then if you do maps from x into y, well, first of all, it is only depends on the n minus 1 connected cover of x. Yeah, of y, only on x and, well, sort of the n minus 1 connected cover of y. But then, of course, I can, I've already learned that n minus 1 connected cover only depends on the loops. So I can put an omega n here and I can forget about the covering, whatever. Doesn't change anything as an en monoid. OK, so this is maybe a bit of an esoteric observation. But if you do make this observation, you could. I mean, right, so now we have a motivic homotopy theory and how does this work? Well, usually something which works classically has an analog, hopefully, emotively. And then sometimes, if you're very lucky, you can prove this analogous thing. And then it gives some interesting result about algebraic varieties. That's somehow secretly the plan of our motivic homotopy theory is supposed to be useful. So now let's try to think about this analog while you might guess the following thing based on that. So I put some motivic space here, which is highly connected. What do I take? Well, I take my x mod x minus z, right, from the first example. Core dimension of z greater than or equal to z. And now I'm going to map it into something else. And let me put k and f here. Right. And so now remember, this guy here is d, so it should be a d. So this guy here is d connected in both directions. So if you believe that this analog somehow should told, it tells me that this set of map, I mean, this homotopy classes of maps thing, it should only depend on the default loop space of this guy in both directions with its extra structure remembered. OK, so it only depends on. Well, if I take the default loops in the S1 direction here, I just get back f. And I remember that it's the chief of a billion groups. And then if I take the default loops in the GM direction, I get f minus d. F minus d with its extra structure. Oops. What did I do? OK, so we have to somehow figure out what we think is all the possible extra structure. And what we think is all the structure, i.e. the transverse plus GW module structure. I mean, of course, we don't know this. Maybe there's more, which nobody has discovered. But unless you provide me with more structure, which could possibly depend on, I will just guess maybe that's all there is. And it turns out that that's true. So we do not have motivic finite loops based theory. Certainly we do not have this. But I mean, of course, this is an extremely special case. And you might imagine texting this by other means. And you can. So this follows from a theorem of Morale. So before I do this, let me observe that this set of homotopic classes here, this actually has a more classically algebraic interpretation. Because it's just the same thing as taking cohomology. So mapping into this guy here is taking cohomology. So this is some hd of something with coefficients in f. And now I'm mapping out of x mod x minus c, which means I take cohomology of x with support in c. OK, so now this topological statement here, or this statement coming from topological intuition, it says that this cohomology group somehow only depends on the contraction of it. And when you get there and you know some things about motivic combotopic theory, you will easily see that this is true. So Morale, well, I mean, you will see this is true because of some difficult results. So Morale says or proves that the so-called Rothschmidt complex, and now we get to the title. So it looks a bit like this. This is some, you take your f and you view this as a sheet on the, this name which side of x, and then you can resolve it. So you see 0x at c1 of x. And it keeps going. And this computes cohomology. OK, so this is not super happy yet. I will try to answer your question in a second, Sean, because I mean, I need to tell you something about what these cis are. But so the point is, so this computes cohomology. And so roughly what are these cis? Roughly what I want to say is that the cd of xf, this is going to be the sum over all the points of co-dimension d and x of the default contraction evaluated at x. So this is not literally true, but it is morally true. And this is, right? So, and then there's a differential. And it uses transfers and stuff. But the point is, you can check explicitly, right? So now what does it mean to take cohomology with support in z? It just means that I have to sum here not over things which are of co-dimension d, but also inside z, right? And the effect is just that you chop off the first d minus 1 term. So this one goes away, this one goes away, and so on and so forth. And so you find some complex. And the terms, well, it starts with f minus d of something, and then you take f minus d minus 1 of something, and so on and so forth. And so you can easily convince yourself that also this differential, which has some explicit form, it says, well, you do something, you put it back to here, and you transfer, and you multiply, and it's a big mess. But you never use anything which is not already encoded in f minus d. And well, that's how you see that this topological guess here, it turns out to be true. So now I have a chance question. Can we view the extra structure? So I don't think so. So it's important that you do this with support. So it comes from the fact that somehow you're supported in high co-dimension. Or the extra structure on f minus d. I still don't see how to get this out of the fact that hd with coefficients in f is a group. I mean, it comes from the fact, no, I mean, it comes from the fact that GM somehow has special maps of GM, give you a special structure on the loop space. So I don't see how this is really related with them, with co-mology. So I feel like this would more give you extra structure in the sort of S1 direction, but we're dealing with sheaves of a billion groups, so it has already all the extra structure in the S1 direction. Okay, so now we have this topological guess here, and I explained to you that maybe for some esoteric reason, you want to convince yourself that this is true, and then Morale has proved that this is true. But I mean, categorically minded people that we are, of course, we expect all of this stuff to be factorial. Right, so if I change x to some other variety, then obviously this is more, I mean, then there's going to be some pullback map, and clearly this also only has to depend on the contraction. So that seems obvious enough. So this is sort of this obvious question, but what about pullback? Right, so let me amplify this, what do I mean? Right, so we let, sometimes I don't know, let f from y to x be some morphism of smooth schemes, and let z contained in x closed, closed, sorry, co-dimension, greater than or equal to d, and then I have to assume, of course, that the pullback also has co-dimension greater than or equal to d, this is not automatic. And I let my f be one strictly homotopy invariant sheaf. Okay, so then there's this pullback map. Okay, I mean, that's because it's comodity with support, it's factorial by definition, whatever abstract definition you use when like injective resolution, so whatever. And so this group here only depends on the contraction, right, on f minus. Yeah, obviously this map only depends on f minus d with extra structure. Okay, so this only depends on f minus d plus transfers, let me just write so this is just some shortcut for all this extra structure which we discovered. And, right, so when I, when I, when I came to this, and this little exercise about the topological expectations and I noticed that Fabian has already proved that I was very happy, because, well, this would allow me to solve some interesting problem, and I assumed that surely just souping it up a little bit whatever we've done so far, and it should give you should give you this. And, so I will tell you in a little bit what I was hoping to do, but it turned out that I spent many hours days and nights trying to do this obvious thing. And I could not do it, or it took me very, very, very long time so this turns out somehow, either I'm a bit dumb or this is much harder, or equally hard as proving this is not at all follow obviously. So let me note this theorem. And this is what this is the theorem which the title of the talk is about right, so the pullbacks for the rush mid complex exactly is supposed to say, Well, how do you want to write what you would want to think is that there's some some map, which you do on this rush mid complex, and that should tell you how to do the pullback, and then you should easily deduce this result here. And it's true. At least this is let's say to then there's somehow a map which you can write down on this partial rush mid complex which you think should be the pullback and it would have the desired property. But it's very difficult somehow to prove that this map which you do write down, indeed is the map which is supposed to write down. And that's what the theorem is about. Okay, so I hope the statement makes sense. So let me interject for a little bit. So who the hell cares. I feel like that that would be a very reasonable question. I mean, the extent of course, we're just testing the waters of a motivic finite loop space theory I think by itself this is a reasonable thing but maybe also, it would be fair to say it's a little esoteric but so let me, let me give you one corollary which one kind of came from this. So this is in joint work with Maria Yackelson. And it says the following so. So let's say that this is a perfect fute k. I should have said this from the beginning so case always perfect file and attribute results. Okay, so let me fix some motive space. I'm pointing motive space. Now what can I do what I can do is I can try to stabilize this guy right so spaces are hard. It's simpler and how do you make it simpler basically by somehow is by smashing with p1 right by smashing with both the directions of connectivity with s one smash yeah. So what I can do is I take my x. And then I map it to omega p1 sigma p1 x. In some sense, classically what happens is that this sort of Freudian tiles suspension theorem which tells you that somehow this map here, depending on maybe the connectivity of x it will induce nice morphism and some homotopy groups. And this is this sort of stabilization phenomenon and once you observe this and in fact what you do is you do this a bunch of times. Let's say this n plus one times here. And basically you want to take the core limit of the whole system right and then this is going to be the right to this core limit here this is going to be the stabilization and this this space at the end somehow. This is supposed to be the simpler version. And so the point of the Freudian tiles suspension theorem is that if you just do this right you're starting with x and then you're doing a bit more and a bit more. And these homotopy groups they will stabilize. So there is this nicer stable answer at the end in the infinite infinitely far, but actually you already reach it at the finite stage. So that's, that's I would say a very important result in classical topology, which unfortunately we do not have an equivalent of motivically. And so, so this is a terrific result here, it can be used to prove that this map here well for entire says that you should get an isomorphism on the first couple of homotopy groups depending on how high up you go. And so we can do this now on pi zero, which of course is much much weaker than the what you would hope for, but it says that this is an isomorphism. If n is greater than equal to three. Right, so you can imagine you have your space you have x and then you have loops sigma x and your loops squared sigma squared x and on to fourth. And then you reach the stabilization here, sigma infinity x somehow. And pi zero of Sigma infinity x is going to be the same thing as pi zero of Sigma three, Omega, no Omega three Sigma three X. This stabilization this for entire thing, which is supposed of course to happen for all homotopy sheeps somehow, at least on pi zero it does happen and it happens at three step. So I would I would like to believe that this is a, this is quite a nice result. And then hopefully it would justify expanding energy trying to prove this. So now, the rest of my time I would like to spend trying to indicate to you how to prove this theorem. I mean how to prove this theorem. So I will not indicate how you get right I won't do this, because well that would be another half hour, but so I will try to do this. And then at least for my taste. This is basically some sort of pretty, pretty hardcore algebraic geometry. But, okay, so let's let's try. So how do we prove the theorem. Well, I mean it's, it's, it's going to be, it's going to be a struggle. We have to fight. Now there, first of all, there's some easy cases. So there's an easy case, which is when F is smooth or more generally is flat, but somehow it doesn't. And somehow does not help. This is, I don't believe this is known to fail. I mean, maybe it's, it's known not to like work in the most optimal bounds which you could guess but I don't know I don't think anyone knows these things to fail. But we don't we we can prove it. So I was a question if this was if this stabilization result which I proved for pi zero. Will it also hold is it known to be false for higher home to be sheeps and I believe it's not known to be false I'm sure everyone believes it to be true, but we don't know how. The easy cases is if F is smooth and the reason is that in this case, there's a pullback map on the entire rostrum complex without support. It's not compatible with the pullback in F because I mean basically you build it to do that. And then the universal properties sort of what the resolution means immediately tells you that everything works so smooth maps super easy. There's another easy case, which I don't want to treat, which is if your F is what's called the homotopy module or an infinite loop sheet. And the reason is that in this case, I told you there's this fantasy formula which you want to write down, but it uses transfers, and then you have a problem because you don't have transfers on C zero so you cannot make this map. But if you have an infinite loop sheet and you have transfers on C zero. And again, you can then use this construction of Ross to write down the pullback map and you can check that this does everything that you want. And basically the problem in general is that you cannot somehow write down this map on the Russian with complex because there's some somehow degree zero you don't know what to do. But you can write it down in the higher degrees but then you have to somehow argue that it's the correct. So here's a key observation. And I'm just the fact about homotopy modules because somehow the whole point of this result is to prove it for us really homotopy and range of. And they're definitely not all homotopy modules. So the key one key observation is as follows. Very simple. So let's say I have my Z contained in X, what I mentioned greater than equal to D. And then what I do is I look at the generic points of Z. And I only look at those let's say of co dimension D on X. I mean every point of Z has co dimension at least D on X. So I'm looking at those which have like the smallest possible co dimension, not every generic point needs to do this because I mean, it could be stupid things like Z is a union of two things and one is much smaller so it has higher co dimension. Okay, so let's typically exit some smooth guy sees some close integral guy and then there's just one generic point and whatever. And then what you do is you look at, you look at this map HD with Z X F. And then I map it to this HD. And what I do is I hand it to lies my X in this in this generic point so that's it need not be a close point of exit also this is maybe some sort of slightly if the algebraic thing to do but okay so that's the beauty of algebraic geometry we can do some slightly things. And okay so here the support Z should maybe Z intersected with a HCI but I mean it's going to be annoying to write. The point is two fault one is that this map is an injection. Right and the reason is just that you look at the most mid complex. Here. Right so what if I do come on to the support in Z. It means I chop off the first couple of terms, namely first D minus one terms. So it's going to inject into this thing here. So I see why I see exactly those points of co dimension D which lie in Z, and I do the default contraction. And so, since the hands realization does not change the rest of your field. This map is in an injection. And the other point is that that this map from the hands realization to access basically pro et al. So, up to some pretension pretending it's it's it's all. That's basically one of the smooth maps which we understand. So pullbacks understood. Okay, so what this means is that in general whenever I have to somehow pull, I write I have to pull back along some arbitrary map, I can at least sort of shrink this target in some et al neighborhood of certain points. And this hopefully will allow me to make it a simpler. So that's what I want to say, or how I want to summarize is that the some of this problem is local in some specific sense. Okay. So now, now we come to the real thing. So I will want to explain. Yeah, so I want to sketch the proof the following lemma. And I will just show you the assesses follow so I assume that the field is characteristic zero. And I let y and x essentially smooth. So this is just some trickery which allows me to look at something like the hands realization it's not quite a smooth scheme but it's reasonably close to one. So I give myself some map from y to x which is in fact a closed immersion of co dimension one. And then of course I have to give myself a z content in x co dimension greater than equal to D. And I have to assume that if I intersected with y content in why still has co dimension greater than equal to D. So I give myself some F script the homotopy invariant. And then I have to prove and this is what I want to prove then this pullback map. I have a star HD was part in Z X F. And see in the sec why I suppose why F, and I will call this thing here W. This only depends on F minus D. Last transfer. So this is this is the special case of the main theorem. It's a special case a because I'm assuming that the characteristic is zero and be because I'm assuming that this map F is them is the regular immersion of co dimension close the version of co dimension one. So this, this is not a huge deal, because we've already dealt with smooth map so I can, I can reduce to write every map composers can be written as a composite of the smooth map and the regular immersion. So you have to deal with regular emotions, and the problem is also local. So I can factor it locally as a composite of the co dimension one immersions. So you reduce to this co dimension one case easy. So the characteristic zero assumption is somehow serious so I'm going to say some things and that they don't quite work in positive characteristic you have to argue more carefully but something along the same lines also works except you have to deal with annoying things like well regular not being the same as smooth and blah blah blah. This is, yeah, so I have only a finite amount of time and energy to explain things to you and I think focusing in character to zero somehow. It's already complicated enough to argument. Okay, so we're 15 minutes I hope that we can, that I will be able to convey some ideas to you. So what do we do. So the first step, we reduce to the case where access dimension D plus one, then of course why has dimension D and Z has dimension one. And of course then W has dimension zero and I want to assume that it's just one point. And it's a rational point. Okay, so how do you how do you do this. Somehow this is this is some sort of standard trickery. So what you do is, you replace X by the hands realization right so I mean what I do is, of course I pick. Right so I pick my W and W. Point of co dimension D and why, right because the problem was local somehow in why, right because of this business here. The problem is local on points of co dimension D in why and W. So I picked this point and I just have to look somehow locally around this point, and I replace X by the hands realization in this point. And of course I also replace why by the hands realization in this point and see by the hands realization in this point. Okay, and then it's a, it's a sort of standard fact that this this inclusion here, then it admits in retraction. Okay, so this is not immediately obvious but I mean, algebraic ways of seeing this a geometric ways of seeing this. Please believe me that this is the case. And now of course because in character six zero disguise regular and disguise any field whatsoever so this map here is going to be smooth. Well, essentially smooth. Okay. So now what have I done. Well, I have localized in this point so you will see that these dimensions things happen automatically. And I've, right and I replace. By of course the residue field of w. And so by this by this trickery, I have assumed that I can now assume that this w is indeed a rational point. It's just I mean, the statement is so general and it's, it's, it's going to be hard enough so we be, we make our lives a bit more reasonable. Okay, so now I still want some further reductions. So I can assume that actually acts and wires move. Right so in the assumptions and also because I did the centralization business here. I only said it's essentially smooth so it's some co-founded limit of smooth schemes. And so I mean, all of these things that are going to happen sort of at the finite stage in this co-founded limit. And so you can always, and then there's some continuity thing that the value on F also will be done the co limit. And so you can always things happen at some finite stage so you can, you can assume that X and Y actually prop us I mean, not not proper but like real bonafide as smooth schemes. And then I can assume this Z in principle could intersect. Why in many points but I can as well I mean I can just throw out all the others so I can assume that C intersect Y is just W. I guess it's already included here. And I can assume that these move away from right. Because I mean Z is a curve in character six zero it's only have gonna have finitely many singularities and I just throw out all of them except for W also then Z will be smooth. No problem. So now this Y into X is a regular margin of co dimension one so it's locally principles again by working locally around this point W. I can assume that why is the vanishing locus of a single function. And finally I can assume that X is fine. Again, all of this because it's somehow it's a local problem. Okay, so I will admit that maybe it's not quite clear where we're going. But let me let me try so I'm going to now make two claims which basically are the heart of the proof and which I'm not going to prove because, well, time attention span various reasons. So the first claim is the following. So that there exists a function you bar from Z to a one having a bunch of good properties. And so first of all I wanted to not be zero at W. And secondly, I want this product you bar F right there's also a function from Z to a one. And this is going to be finite. Okay, and also I want you bar not to have double roots. Okay, so now how do you do this it's basically a Riemann Rock argument. Plus using the fact right so you can use Riemann Rock somehow because it's mostly a smooth curve. And this allows you to basically get all of this and then how do you make sure it doesn't have double roots it uses the fact that it fields of characteristics zero are infinite right because I could always add some constant on it and then no double roots. Okay, so let's believe this for now. And then what I do is I put five one equal to this product you go. Well, so I do sorry. Right, so I use the fact that this this axis affine. So I pick some you from X to a one extending you bar. And then I put by one equals this you have from X to a one. Okay. And now here's the second claim. And the second claim is that there exists more functions that exist by two by three and so on up to five D plus one from X to a one. And then the following holds. So I write five for all of them together if I want to fight D plus one. No, no. And then I hold bunch of properties. So, fire w equals zero and fire set tall at all points of five one inverse of zero intersected with Z. And the thing is there exists an open neighborhood you contained in a one over D such that if I base change you to there and I map it via five to you times a one. This is going to be a closed immersion. And this maybe if you know about these things it looks like Gabbers lemma or like part of Gabbers lemma and indeed that's it's very got it from as you can prove this because of an infinite field by general projection argument. Okay, so now I'm sure you all remember all the 15 functions and schemes and everything that we had so far. So just in case that you don't and also so that I feel like I'm, I'm telling you something let me try to give you an artistic impression of what's going on. So I hope this is going to work and help. But let's try. So what had we done right so we had my we had our scheme X or smooth scheme X. We only have two dimensions so everything is only going to be two dimensional. And so X is just going to be some blob here. Okay, so that's X. Now inside there. We had, we had the Y right this was from smooth scheme of co dimension one. So it's going to be some sort of curvy thing here. So this is my Y. But that's it we have well, we had a Z right so we had some close up scheme also of whatever could I mention D but I mean I only the only choice I have is also going to be some one dimensional guy. But it could have singularities and it turns out that the problem mainly arises if the singularities of Z of meet why. So this is sort of the kind of thing which is going to happen. So this is going to be Z. And then we had the intersection with my which we call W. And that the big deal is that our W is this point here, which is the singularity of Z. So this is what makes everything difficult. Okay, so this is what we had started with. Okay, so what else did we do. Well the first thing which we did was right we cooked up this you. So let's see, let me draw this in here. So the zoo that the you did with this will give me some some other guy right so this is going to be the vanishing locals of you. And this white line is by the way the vanishing locals of. Okay, and then we had like lots of maps the five one up to five D plus one so there's some map which I call by equals five one up to five D plus one. And so this now goes to a n of course, a D plus one. So let me try to draw this. Again, I only have two dimensions. So this is somehow an a one here on the coordinate x one. And there's a one here on the coordinate x E plus one. And then there are some more coordinates in the middle of course but I cannot draw them. And so what what happened here well what happened is that. Right to the five one was the product of you and F. So this axis here is the pre image of that is going to be the union of the blue bits. Right so it's going to be the union of y and some extra thing. And so what else we had to see. Now the Z is going to do something. I'm going to try to draw this maybe like this something like this. Okay so what's going on here well so the W the W still here and still this still this singular point we couldn't do anything. And, but also what can happen is that this you write so this thing here this has some some could have some new intersections with Z right so for example here. So this is some new intersection. So this I cannot avoid. So this is some new intersection. But I had asked that it should not have double roots so this is some new transverse intersection. So at the beginning that the problem is basically when the Z has a singularity on the intersection. So basically what I'm telling you is there's a new intersection but it's, it's good we can understand it. Right and then also what we have said is that on some open neighborhood. This guy here. The Z should be a closed immersion right so now my Z what was it it was the sort of curve here, and it turned into that sort of curve here but now it has a new double point. So that's a problem here. So that's one of my bad points somehow. It's problem, but that's fine. Right so what this is saying here is that it should be a closed immersion locally in this coordinate. So I just throw out this point right so then I might you. I'm going to be a D, which in this case is a one, I suppose, and I'm removing this finitely many bad guys. No problem. And then away from that it's supposed to be a closed immersion so maybe, maybe that's that's what I'm doing. So, somehow morally what has what has happened here is that we managed to straighten out the right to the X and the Y. There were some arbitrary smooth schemes and now we've somehow managed to straighten them out into an into a full a one. That's somehow what the what the gap a lemma always does it gives it gives you somehow enough room to get the full a one. So it's the expense of having sort of this, this new guy here and these new guys here which we now have to deal with. But if you take nothing away from my talk and please somehow remember that that the point of Gabas lemma is to straighten out into an actual a one, whatever whatever that means. So now I have four minutes to try and finish the proof so what do I do. One thing which I do is I let you zero be the intersection of you with the thing where the first one is zero. So in our case, that's just this point here but generally that could be more dimensions. And then it turns out that there's some open subset here, which will be well chosen, and I will not tell you what it does the choice. And then the following things going to happen. So what do I do. Okay, so maybe. Yeah, so maybe I will abbreviate this. And what I will say is, so more steps. I will tell you it suffices to understand the pullback somehow like this HD, the U a one over you have those two HD, the U zero a one over you zero with confidence in it. So the whole exercise of this game was that I replace the situation on the left by the situation on the right. So how is this any better. So one thing which we can observe is that this Z. It is finite over you. So this is some proper morphism. So what I can do is zero was to you finite means that I can embed a one of course into P one, and then it will remain closed in P one. And so I can instead, I can instead let's say, do this. Right. I can instead do this. And it's also enough to understand this. And now, what can I do. Okay, so I have this HD, the U, P one over you with coefficients in F. Now I'm supposed to pull it back here to HD, C, zero, P one over you zero with coefficients in F. However, right, so this zero zero, this is basically just finitely many points. So this group here I can, I can work out using the wash me complex right. So that's just M minus D of W. So that's this point. And then maybe there's some stupid other points, which we have to deal with. Plus, so a whole bunch of points some of I am minus the of why I so these annoying new points. And now, now is where the transfer comes. So there is a transfer map here. And I mean, yes, let me write it. So it goes to HD P of C you by which I mean just the image under the projection in this direction. With coefficient, you with coefficients in F minus one. Right. So if you ignore, if you ignore the supports here, or that's of course I did it wrong. Right. So if you ignore the supports here, this is just saying what is HD P one over something coefficients in F. But you just use the fact that P one is S ones, my GM. So you should remove one from the D and you should remove one from the F. And that's how I got this. And you can check that you can play with the supports here. And this, it turns out is basically the nature of the transfer. So this is what the transfer is. And I can do the same thing. Then I can put back here, HD minus one P of C U zero, U zero, coefficients in F. And there's another transfer here. And this diagram commutes since very elementary. And I can also work out this thing here. So this is M minus D of W again because it's a rational point, plus maybe some other, some other things. So I'm out of time, but also I'm at the end. Right, because you had this. Right. So what does, what does the transfer do. I told you that this abstractly defined map, which has something to do with the transfer. And well, you can see here in this simple case, right. So this is some sum of M minus these evaluated some points, and then they map down to some other points. And so you what you need to do is that you need to find some kind of transfer. And I mean, that's this construction here gives you a transfer. And you can see what this is. But now the point is we had designed this to be right. So this, this W here is the same thing was a rational point. So there's no field extension. So this is actually nice. Okay, so now I'm done right so I had something here. And I supposed to figure out what this is, it's image here. So I need to figure out what it is here and here. And suppose that I know this map here, right. So I had something here, I know it here, I know it here so I know everything here. And also, I had told you that these new points there somehow, there's somehow easy. So also, I know everything here. So basically, in this big group here, I know everything except for this value here. And sometimes if I transfer it down all the way to here. Okay, but now, well, if I just pretended zero here for a bit transfer all of these things down and subtract it off from what it's supposed to be, and then invert this map here. I will find that I figure out what the last component is. And so then, what have we done. Well, it suffices to understand this map. And now we're done because by induction. Right, because I had now managed to reduce the D. And so now you can just go again and eventually you get to D equals zero and equal zero is trivial. So I'm sorry for running over time. But I hope I have given you some idea of how algebraic geometry proves this interesting result in material. Okay, thanks a lot for a wonderful talk Tom. Here. Let's see, are there any questions. Let's start with a question in the corollary the result with Maria Yarkersen. There was a, I think there was a number three showing up. Yes, possible to somehow explain why, why it's number three. Yes. So this number three has something to do with the following thing. Yeah, okay, so what about this read so I'm not sure if it's optimal. What is this right. So it has to do with the following thing you take F minus one has some transfers and F minus two has transfers. They all have transfers and some feeling is that the transfer this guy has transfers. Better transfers. Better better transfers. Right so just like how an E N right so if you take iterated loop spaces classically have like group structure be a group structure. And so the feeling is that eventually, so the F minus infinity should have frame transfers. Now actually, right so it actually turns out that already already F minus three does. And the F minus two, if characteristic is equal to zero. And the conjecture is that F minus one has frame transfers. But so one. Okay, so now that was a lot of waffling but the point is that somehow the contractions give you more and more structure and we believe that you somehow get all structure already after three steps, or after two steps or maybe after one steps and how you need this is the number this is where this number comes from. Right so this number three is because we can prove that after three steps here somehow at the at full structure. Okay, thanks. Then there's a question from Sean Tillson he asks, is there extra structure and that you have now from this scrollery, like more than the Abelian group structure. Yes, I mean. Yes, I mean so one thing which you do learn is that yes if you do pi zero of omega p one cubed sigma p one cubed F, right this is actually a homotopy module. So this has all the structure which you could possibly ask for. In fact, in the proof, we learned this. Right, so already if you're because I mean this is like if you take a three fold loop space and then you have good structure this is not like super exciting thing. The point of the proof is that this already right even without taking loops, you already have a lot of extra structure. So the answer is, yes, we definitely do get more structures on various things and this is basically the heart of the group. And there's a question is there an analog of maze recognition principle for multiple spaces. Well, in the world, does there exist one I hope so, can we prove it. I believe not right now. So we have right so there's there's some there's sort of S one loop space theory, which says can you deal with it in the S one direction and I would argue that this is probably reasonably well understood I'm not sure if it's like written down in this language but this sort of thing we can probably do. But the D D looping in the GM direction is very very hard so we do not have. Yeah, so we do not have recognition principle, right where cove mission principle for even something like P one loop spaces. And then GM loop spaces but the problem is that somehow because GM is not connected itself these things are sort of, it seems likely to me that you might have something like for P one or P one. That's the smash to the dub dub that for these you might have it, but we do not have any of these. We have we do have a recognition principle for motivate infinite loop spaces, which is right so like, like saying that infinity monoliths group like infinity are not group spaces so we have some like that group like spaces with frame transfers, but we definitely do not have this at finite stages. If we did then my result would be much easier or I mean as easy as proving this recognition principle. Yeah please work on that and prove it that would be great. I did not have any characteristic zero to prove your, your theorem, but it characteristics zero doesn't appear in a hypothesis of the fair. Yes, no I mean I did not. I use characteristic zero to make the already probably long and hard to follow argument, a little less annoying characteristic you do something which is roughly the similar story, but this reduction in the beginning. So this point here. This then becomes a few pages to argue that you can still do something like this in positive characteristic. But yeah so this is only I did it to simplify the exposition and only prove a special case. Another question. Can you make a comment on the zero and pi zero of your with your corollary with the accurate some. Well, what can I comment on, on what zero means, or sorry, I'm done quite well. Well, could you change it to pi one. No, I, well, I would like to believe that. What we do very much is confined to pi zero. So I think that, yeah, so I mean I would conjecture or whatever. You would guess that I can do something like pi I here, and then maybe and greater than I know something three I or three plus I don't know what it goes so that that that would be what you think, or what I would think, but I would expect that this probably requires different, different kind of attacks but I don't know so I definitely cannot I wish I could, but I can. Can you explain where the zero comes up and going from the transfer result to the corollary. Yes, I can, I can I think do that. So how does this work. The way this works is that I look at somehow. Right, so I look at the category of S one spectrum. And then I have the category of motivic spectrum. Right, so I can go to here, and I can go to here and basically I want to figure out what this composition is. So I can imagine this is happening somehow in spaces right so I can first smash it with GM and go to them somehow to this as such as one of K one, and then I go to two. And I keep going, and I can always factor it like this. Right, I mean I'm just saying that if you do an infinite iteration of GM loops and GM suspensions you can do one and one and one. And then eventually you always have to infinitely many. And what the what the proof eventually shows is that if you look at the hearts here. Right, so these categories in some sense right to the first one and the first one and so on so forth in a in a way which I find difficult to make precise they approximate this category, well I guess I can take some limit in the category in pure. But whatever. Right. And so what we prove is that shs one of K and heart right so it has this natural functor sh of K effective heart. And this is an equivalent for and greater than equal to three. And then eventually you'll get the corollary from that by some form of manipulations. And the way this works is that basically we know very explicitly what this guy is. And so we have to somehow describe these things very explicitly and why we just find our way through and eventually we see that what what are the objects in here they like these sheaves and they have some extra and then eventually we argue that this is all destruction. And so if you want to do it for like say pipe one right so then you don't have to look at the heart but you have to look at something like this, right, as one. Okay, so the concentrated degree zero and one. So I don't know some sort of motivate one types. And you could try to do the same analysis but it's going to get much more complicated right. And then you have to do it with three types and. Yeah, so that's my attempt at answering your question. Okay, great. Other other questions. Yeah, yeah, just just to understand more precisely. So you don't build pullback maps on the ROS complex. But instead you prove that's it. Instead you prove this independence results. Right. So that that yes, I well I haven't really thought about that. So, or another detail. Yes, so maybe I should say right well you have this thing you have to say it's a C zero of XF goes to C one of XF goes to C two of XF goes to answer and so forth right now you can do the same thing with why. Okay, and then the dream would be that you have some maps here, which make everything work. But the problem is that there is no transfer here and then you cannot write it down so that's the problem. Okay, so but instead. I mean, you can do the support and see thing. Okay, and then I mean this, this guy just goes away. I mean this just doesn't exist. And more don't exist. And then the point is this is formula which was written down how to do it. So there exists a map here and there is this one here and everything commutes right. And so the easy, I mean, the ideal thing would be to prove that this is the correct thing. And I think what my result shows, but I have to have to think about that is that sort of the lowest terms or if these are all zeros right then this map here it does the correct thing. So that there's this map which this fantasy map which you write down, but it will actually give you the thing which comes sort of implicitly out of what I'm doing with a transfer whatever. And I'm, my feeling is that you might be able to soup this up to learn that by some induction thing that it does actually the correct thing right so that that these maps here which you can write down that they all use the correct map and co homology. But I have not actually tried to do that. And I think it would be maybe annoying. But it's not, it's not out of the questions I'm not sure. But I think yeah I it says pullbacks with a rest with complex and I definitely don't do this in general. Okay, thanks. Thanks. Okay, anybody else. I think that concludes the question so thanks again Tom for a wonderful talk. Next up is Dylan. John also asked the negative stable homotopy sheaves and the answers. No. Okay, thank you.
 Well, I mean, thank you very much for the invitation to speak here and for organizing  a summer school in my living room.  It's super convenient.  Excellent, guys.  Okay, so the title is maybe a little bit cryptic, but I, right, so maybe I will convince  you hopefully that the result is not so much.  Okay, so let's try to get there.  So first of all, of course, well, that's why I prepared the first few things, which I want  to say because they are all obvious because you've been listening for more than five minutes.  So I will, what I want to talk about is some results in motivic homotopy theory.  So I'm interested in this category of motivic spaces.  What's a motivic space?  Well, it's a special kind of pre-sheaf valued in spaces.  And so a pre-sheaf on the category of smooth varieties and such a pre-sheaf, which happens  to be a sheaf.  So I will always be dealing with misnavage sheaves.  This is a typology.  Well, it's some sort of topology, which is somehow relevant.  It's the correct one for various reasons.  It doesn't matter hugely right now what it is.  And then of course, there's going to be some extra condition because it would be silly to  have like two names for the same thing.  And the extra condition is this so-called A1 invariance, right?  So you just, you look at all those sheaves F such that if you evaluate it on some smooth  variety X, or if you evaluate it on X times A1, you get the same thing.  And this is the so-called category of motivic spaces.  So there are of course, supposed to be lots of motivic spaces because supposedly studying  this category will tell us good things about the world by which I mean polynomial equations.  So for example, if I take any smooth scheme whatsoever, then I can try to, well, it will  not usually as a pre-sheaf, of course it will live here, but it may not be here.  But you can just sort of brutally move it into this category because this inclusion  has a right adjoint which is called the motivic localization.  And that's, that's, that's the obvious thing to do.  And usually I will not even write this L mode and I will not write this unada embedding  thing.  I will just say view X as a, well, this is not what I was going to say.  What I want to say is I want to view X as a motivic space.  And this just means that you should do this localization, which in general is of course  highly non-trivial.  But I want you to please imagine doing it.  And then there's, there's another class of examples, which is going to be very important  for my talk, which is let's say you take some sheaf of a B in groups, right?  So in this category here, you take some of the simplest possible objects.  Well, of those which are, I mean, you do write some not sheaves of spaces, you now make them  zero truncated, but you give them some extra structure.  This is a B in group structure.  So that seems like a pretty reasonable thing.  And then you could ask, is it the case that this, that this sheaf actually lives in the  category of motivic spaces?  And I mean, that's not always true.  There's a condition and I mean, it's just this condition here, but maybe for a sheaf  of a B in groups, this looks a little bit more familiar.  It just says that if you take F, you evaluate it on X or you evaluate it on X times A1.  And you should just get the same a B in group.  Okay.  So this maybe was not very exciting.  So what else can you do with a B in groups?  Well, you can look at their Eilberg-McLean spaces or in this case, I can look at the  Eilberg-McLean sheaf KNF, right?  So this is going to define some sheaf of spaces by definition.  And then I can ask again, does it live in this category of motivic spaces?  And well, you need this homotopy in various property.  And what does it mean to evaluate some, some Eilberg-McLean sheaf on a smooth variety?  You get some space.  The homotopy groups of which are the homology groups.  So in this case, the condition is that this homology, HI of X times A1 with coefficients  in F should be the same thing as the homology X with coefficients in F. And this should  hold for all I less than or equal to.  And so you see, in principle, as you make this N bigger, you get somehow a more interesting  object and you get, get a more stringent condition.  And then it's somehow natural to single out those pre-sheafs or those sheaves of a BN  group so I can do this for all N at the same time.  And those are called strictly A1 invariant.  Right.  So let's, let's just leave it at that.  Okay.  So now we come somehow to, to something which is close to the heart of the whole thing,  which of course, classically, you have this notion of connectivity of space.  I already implicitly used it by saying that like an Albert McLean space is somehow particularly  simple.  And what, what, what is, what this has to do with is that you some sort of sphere and  you build how you can attach spheres to your spaces, for example.  And more typically we somehow we have two spheres again.  I'm sure you know this, right?  So that's the usual sphere, usual circle if you want.  Sir.  Sir.  Sir.  And then there's somehow some algebraic circle, which is denoted GM.  And I mean, it's just you take the complement of zero in the affine line and let's say you  point at one and then these spheres, they define your point of motifs spaces, which,  well, they're, they're a bit like spheres somehow.  For example, I mean, the complex points, of course, of GM is just C minus the origin,  which has the homotopy type of S1.  So it looks like it's just the same thing.  That somehow this is a very topological observation and algebraically, they're different.  And then what this tells you is that you somehow get two notions of connectivity out of this  whole thing, right?  You can measure connectivity somehow in terms of S1 or you can measure it in terms of GM.  And I mean, there's no reason for this, for these to agree in general.  Now I just said there's no reason for them to agree.  So let me contradict myself.  So here's an important example.  If I take a smooth scheme and I take some closed up scheme, but it doesn't have to be  smooth.  And but I will assume that it has a co-dimension at least D everywhere.  And if you do this, well, I mean, whether you do this or not, you can look at sort of  the formal tubular neighborhood of C and X, right?  So I do this thing X mod X minus Z.  Right?  So X is a smooth scheme.  So it defines a motivic homotopy type and X minus C is a smooth scheme defines a homotopy  type.  So I can take the co-fiber of the inclusion, just define some motivic homotopy type.  And yeah, I mean, if Z was smooth, then you should think of this as some sort of algebraic  version of a tubular neighborhood.  And if Z is not smooth, then I know it's this algebraic version of a tubular neighborhood,  but maybe a bit harder to imagine.  And it turns out that this guy, I mean, if I view this as a pointed space, this is D  minus one connected.  D minus, that's not very good.  D.  And well, it's a D minus one connected in the S1 direction or in the GM direction.  It turns out in both directions.  Okay.  So how do you see this?  Well, first is let's say that the Z is smooth.  And then there's something called the homotopy purity theorem, which basically says that  this really does behave like a tubular neighborhood.  And it tells you that it looks basically like a tomespace of some, I mean, like the  tomespace of some vector bundle.  And so it means that locally somehow it looks like this sort of thing, a1 mod GM, which  is P1 and which is somehow, which is S1 smash GM.  So the point is locally it looks like it's connected, or I mean, if it's dimension D,  then it's going to be S1 to the, I mean, it's SD smash GM to the D.  And so locally it looks connected in both directions.  And then somehow the way the topology works, this is the name which topology works in the  TLCMU theory is that this also implies that it's globally connected in the sense.  And so what if Z is not smooth, it turns out that there's some sort of filtration argument  and you find that this result is still true.  Okay.  So this will somehow play a big role in what is to follow.  And now we can look again at our favorite strictly homotopy invariant sheaves, right?  So abstractly homotopy invariant.  Right.  So then my F defines a particularly simple type of motivic space.  And then what happens is that this space is discrete in the S1 direction.  Okay.  So it looks like the simplest possible thing which you could deal with in this direction,  but it doesn't have to be in the GM direction.  So somehow there's some, some, some structure going on, which is masquerading, but yeah,  so it's not immediately obvious, but it turns out that there's some more things going on.  Okay.  But not in general in the GM direction.  And I mean, I'm not sure that have anything.  That's a question for you, Tom.  Yes.  I saw the question, is there a nice reference for this filtration argument when Z is not  smooth?  Probably, but not off the top of my head.  So I mean, also when I make this claim, I'm saying that it's going to happen in the category  of S1 spectra.  If you know, like these sorts of details, I'm not claiming this in the sort of completely  unstable sense if I want to be very precise.  So I'm sure I can find you one.  If you email me, I can find something, but not right now.  If I clearly remember something is at the end of Röndig's erstware advances mass paper,  but at the end of the paper, this filtration argument.  Okay.  Thank you.  Yes.  Okay.  So I was, I was trying to explain why it's not always the case that these guys, right?  So if I take one of these guys, why is it not always somehow connected in the GM direction?  And I mean, I think just because there's no reason for it to happen, right?  There are these two directions and you picked something which by definition was sort of  discreet in the, in the one direction.  Why would it be in the other direction also?  So you can just work out some examples and you see that this happens.  So let me introduce some notation.  So in general, let's see sort of a spectrum.  For example, it would be discreet if and only if it's loops vanish, right?  So it seems important to study the GM loops of this guy.  So if this is always zero, then we would learn that these guys are always somehow discreet  in the GM direction, but it's not.  So let's give you the name.  This is called F minus one.  That's why we have this contraction, construction.  And the point is that this need not be zero.  And they, right?  So now let me just give you an example.  So what you can do is you can work out, this is not totally trivial.  The F minus, the discontent, this loop space is again just a, a sheaf.  And it's section over, let's say K is given by F of A1 minus zero mod F of K.  And I mean, you can put X instead of K in as an obvious modification of this formula.  And then you can just, that there are some strictly homotopy invariant sheaves, which  we know, for example, the witch heave.  So you take the pre-sheaf, which assigns to some, let's say, smooth variety.  It's a wit group or wit ring.  And then you take the associated sheave in the, in the, in the, in the same topology.  And this turns out to be strictly homotopy invariant, not obvious.  And what you can compute, what one can manage to compute is that if you do this contraction  business, you just get the wit guy again.  And this is not zero.  So, and also another very famous example is these KN millenowitz.  So the unremifined millenowitz K theory.  Also, if you contract it down, they're never zero and they form like this, this  internet GM loop sheave.  Okay.  So this is just something which happens and which may be surprising at the beginning.  But okay.  So we got to live with this.  Now I want to put on my, my topologist's hat for a little bit and think about, think  about loops based theory.  And so one slogan, which I have picked up is that somehow taking loops increases  structure, let me just roll this out here.  Increases structure.  So what do I mean by this?  Well, I mean, if X is a topological space, let's say pointed, right?  And then I take the usual loop space.  Now this has more structure, right?  Because loops can be composed.  So in other words, this is some sort of monite.  And if you want to be fancy, this is an E1 monite or  a infinity.  Okay.  So this is somehow the beginning of classical finite loops based theory.  So if you take higher loops, right?  So if I put an N here, then I make this an EN monite.  And somehow the point is that in fact, this is all the, this extra structure recovers  everything.  So that's somehow the crux of it.  Finite loops based theory.  The most basic result, I'm sure there are very difficult things which one can study.  So now what I want to claim is that if I do the same thing with my strictly  homotopy invariant, chief F, and I do this in the GM direction, I still get more structure.  So let's take F strictly homotopy invariant.  Right now, let me look at the first loop.  So I will denote it F minus one for reasons of tradition.  And now what's clear, well, what's almost clear is that this is going to be a module  over somehow stable maps from GM to GM.  And the point is this thing has been computed by Morrell to be the growth and  de-quittering of K.  Okay.  So what you learn is that this F, I mean, it was just some arbitrary, yeah,  the F, it was just some strictly homotopy invariant chief, which someone has given you.  You contract it down once.  And now suddenly everything becomes a module over the growth and de-quittering,  which I mean, can be a moderately complicated ring.  It's just some extra structure which somehow pumps out of the sky.  And maybe it will be useful.  And there's this, there's also more.  So there's also something called, well, I would call it monogeneic transfers.  And I don't want to go into too much detail exactly what this is.  So what it does is, if you have some fields, let's say,  finitely generate over K, and then you have some finite monogeneic extension.  Right, so this is finite extension.  And you've chosen a generator, then this yields a transfer map,  sort of tau x from F evaluated K of x goes to F evaluated K.  Right, so this is somehow some fancy way of adding things together in some special way.  And we will see later some other incarnation of this.  But all I'm saying is that, right, so also in our motivic setting,  it turns out that if you do take GM loops, you do find more structure.  And so what I was saying here is that, classically,  if you recover enough structure, then somehow this loops,  then you can reverse this loop operation, right?  So I do this omega n thing, and it goes, let's say, from n minus 1 connected spaces.  And then I go to en monoids.  Monoids, as I said.  This is somehow some way of encoding the fact that the first loop space has a multiplication.  The second loop space, the multiplication becomes commutative.  And then higher loop spaces becomes more and more commutative.  This is actually in equivalence.  So if you give me only the en monoid, which was the loops on your space,  then I can actually reconstruct this space.  And I mean, you just gave me everything.  OK.  And so what this tells you, for example, is that, whoops, why is this not OK?  Right, so if you have some, if x is n connected,  maybe it's n minus 1 connected, then, and y is arbitrary,  then if you do maps from x into y, well, first of all,  it is only depends on the n minus 1 connected cover of x.  Yeah, of y, only on x and, well, sort of the n minus 1 connected cover of y.  But then, of course, I can, I've already learned that n minus 1 connected cover  only depends on the loops.  So I can put an omega n here and I can forget about the covering, whatever.  Doesn't change anything as an en monoid.  OK, so this is maybe a bit of an esoteric observation.  But if you do make this observation, you could.  I mean, right, so now we have a motivic homotopy theory  and how does this work?  Well, usually something which works classically has an analog, hopefully,  emotively.  And then sometimes, if you're very lucky,  you can prove this analogous thing.  And then it gives some interesting result about algebraic varieties.  That's somehow secretly the plan of our motivic homotopy theory is supposed to be useful.  So now let's try to think about this analog while you might guess the following thing based on that.  So I put some motivic space here, which is highly connected.  What do I take?  Well, I take my x mod x minus z, right, from the first example.  Core dimension of z greater than or equal to z.  And now I'm going to map it into something else.  And let me put k and f here.  Right.  And so now remember, this guy here is d, so it should be a d.  So this guy here is d connected in both directions.  So if you believe that this analog somehow should told,  it tells me that this set of map, I mean, this homotopy classes of maps thing,  it should only depend on the default loop space of this guy in both directions  with its extra structure remembered.  OK, so it only depends on.  Well, if I take the default loops in the S1 direction here, I just get back f.  And I remember that it's the chief of a billion groups.  And then if I take the default loops in the GM direction, I get f minus d.  F minus d with its extra structure.  Oops.  What did I do?  OK, so we have to somehow figure out what we think is all the possible extra structure.  And what we think is all the structure, i.e. the transverse plus GW module structure.  I mean, of course, we don't know this.  Maybe there's more, which nobody has discovered.  But unless you provide me with more structure, which could possibly depend on,  I will just guess maybe that's all there is.  And it turns out that that's true.  So we do not have motivic finite loops based theory.  Certainly we do not have this.  But I mean, of course, this is an extremely special case.  And you might imagine texting this by other means.  And you can.  So this follows from a theorem of Morale.  So before I do this, let me observe that this set of homotopic classes here,  this actually has a more classically algebraic interpretation.  Because it's just the same thing as taking cohomology.  So mapping into this guy here is taking cohomology.  So this is some hd of something with coefficients in f.  And now I'm mapping out of x mod x minus c, which means I take cohomology of x  with support in c.  OK, so now this topological statement here, or this statement coming from topological intuition,  it says that this cohomology group somehow only depends on the contraction of it.  And when you get there and you know some things about motivic combotopic theory,  you will easily see that this is true.  So Morale, well, I mean, you will see this is true because of some difficult results.  So Morale says or proves that the so-called Rothschmidt complex,  and now we get to the title.  So it looks a bit like this.  This is some, you take your f and you view this as a sheet on the,  this name which side of x, and then you can resolve it.  So you see 0x at c1 of x.  And it keeps going.  And this computes cohomology.  OK, so this is not super happy yet.  I will try to answer your question in a second, Sean, because I mean,  I need to tell you something about what these cis are.  But so the point is, so this computes cohomology.  And so roughly what are these cis?  Roughly what I want to say is that the cd of xf,  this is going to be the sum over all the points of co-dimension d and x  of the default contraction evaluated at x.  So this is not literally true, but it is morally true.  And this is, right?  So, and then there's a differential.  And it uses transfers and stuff.  But the point is, you can check explicitly, right?  So now what does it mean to take cohomology with support in z?  It just means that I have to sum here not over things which are of co-dimension d,  but also inside z, right?  And the effect is just that you chop off the first d minus 1 term.  So this one goes away, this one goes away, and so on and so forth.  And so you find some complex.  And the terms, well, it starts with f minus d of something,  and then you take f minus d minus 1 of something, and so on and so forth.  And so you can easily convince yourself that also this differential,  which has some explicit form, it says, well, you do something,  you put it back to here, and you transfer, and you multiply, and it's a big mess.  But you never use anything which is not already encoded in f minus d.  And well, that's how you see that this topological guess here, it turns out to be true.  So now I have a chance question.  Can we view the extra structure?  So I don't think so.  So it's important that you do this with support.  So it comes from the fact that somehow you're supported in high co-dimension.  Or the extra structure on f minus d.  I still don't see how to get this out of the fact that hd with coefficients in f is a group.  I mean, it comes from the fact, no, I mean, it comes from the fact that GM somehow has special maps of GM,  give you a special structure on the loop space.  So I don't see how this is really related with them, with co-mology.  So I feel like this would more give you extra structure in the sort of S1 direction,  but we're dealing with sheaves of a billion groups, so it has already all the extra structure in the S1 direction.  Okay, so now we have this topological guess here, and I explained to you that maybe for some esoteric reason,  you want to convince yourself that this is true, and then Morale has proved that this is true.  But I mean, categorically minded people that we are, of course, we expect all of this stuff to be factorial.  Right, so if I change x to some other variety, then obviously this is more, I mean, then there's going to be some pullback map,  and clearly this also only has to depend on the contraction.  So that seems obvious enough.  So this is sort of this obvious question, but what about pullback?  Right, so let me amplify this, what do I mean?  Right, so we let, sometimes I don't know, let f from y to x be some morphism of smooth schemes,  and let z contained in x closed, closed, sorry, co-dimension,  greater than or equal to d, and then I have to assume, of course, that the pullback also has co-dimension greater than or equal to d,  this is not automatic.  And I let my f be one strictly homotopy invariant sheaf.  Okay, so then there's this pullback map.  Okay, I mean, that's because it's comodity with support, it's factorial by definition, whatever abstract definition you use when like injective resolution, so whatever.  And so this group here only depends on the contraction, right, on f minus.  Yeah, obviously this map only depends on f minus d with extra structure.  Okay, so this only depends on f minus d plus transfers, let me just write so this is just some shortcut for all this extra structure which we discovered.  And, right, so when I, when I, when I came to this, and this little exercise about the topological expectations and I noticed that Fabian has already proved that I was very happy, because,  well, this would allow me to solve some interesting problem, and I assumed that surely just souping it up a little bit whatever we've done so far, and it should give you should give you this.  And, so I will tell you in a little bit what I was hoping to do, but it turned out that I spent many hours days and nights trying to do this obvious thing.  And I could not do it, or it took me very, very, very long time so this turns out somehow, either I'm a bit dumb or this is much harder, or equally hard as proving this is not at all follow obviously.  So let me note this theorem.  And this is what this is the theorem which the title of the talk is about right, so the pullbacks for the rush mid complex exactly is supposed to say, Well, how do you want to write what you would want to think is that there's some some map, which you do on this rush  mid complex, and that should tell you how to do the pullback, and then you should easily deduce this result here.  And it's true.  At least this is let's say to then there's somehow a map which you can write down on this partial rush mid complex which you think should be the pullback and it would have the desired property.  But it's very difficult somehow to prove that this map which you do write down, indeed is the map which is supposed to write down.  And that's what the theorem is about.  Okay, so I hope the statement makes sense. So let me interject for a little bit.  So who the hell cares.  I feel like that that would be a very reasonable question. I mean,  the extent of course, we're just testing the waters of a motivic finite loop space theory I think by itself this is a reasonable thing but maybe also, it would be fair to say it's a little esoteric but so let me, let me give you one corollary which one kind of came from this.  So this is in joint work with Maria Yackelson.  And it says the following so.  So let's say that this is a perfect fute k.  I should have said this from the beginning so case always perfect file and attribute results.  Okay, so let me fix some motive space.  I'm pointing motive space.  Now what can I do what I can do is I can try to stabilize this guy right so spaces are hard.  It's simpler and how do you make it simpler basically by somehow is by smashing with p1 right by smashing with both the directions of connectivity with s one smash yeah.  So what I can do is I take my x.  And then I map it to omega p1 sigma p1 x.  In some sense, classically what happens is that this sort of Freudian tiles suspension theorem which tells you that somehow this map here, depending on maybe the connectivity of x it will induce nice morphism and some homotopy groups.  And this is this sort of stabilization phenomenon and once you observe this and in fact what you do is you do this a bunch of times.  Let's say this n plus one times here.  And basically you want to take the core limit of the whole system right and then this is going to be the right to this core limit here this is going to be the stabilization and this this space at the end somehow.  This is supposed to be the simpler version.  And so the point of the Freudian tiles suspension theorem is that if you just do this right you're starting with x and then you're doing a bit more and a bit more.  And these homotopy groups they will stabilize.  So there is this nicer stable answer at the end in the infinite infinitely far, but actually you already reach it at the finite stage.  So that's, that's I would say a very important result in classical topology, which unfortunately we do not have an equivalent of motivically.  And so, so this is a terrific result here, it can be used to prove that this map here well for entire says that you should get an isomorphism on the first couple of homotopy groups depending on how high up you go.  And so we can do this now on pi zero, which of course is much much weaker than the what you would hope for, but it says that this is an isomorphism.  If n is greater than equal to three.  Right, so you can imagine you have your space you have x and then you have loops sigma x and your loops squared sigma squared x and on to fourth.  And then you reach the stabilization here, sigma infinity x somehow.  And pi zero of Sigma infinity x is going to be the same thing as pi zero of Sigma three, Omega, no Omega three Sigma three X.  This stabilization this for entire thing, which is supposed of course to happen for all homotopy sheeps somehow, at least on pi zero it does happen and it happens at three step.  So I would I would like to believe that this is a, this is quite a nice result. And then hopefully it would justify expanding energy trying to prove this.  So now, the rest of my time I would like to spend trying to indicate to you how to prove this theorem.  I mean how to prove this theorem.  So I will not indicate how you get right I won't do this, because well that would be another half hour, but so I will try to do this.  And then at least for my taste. This is basically some sort of pretty, pretty hardcore algebraic geometry. But, okay, so let's let's try.  So how do we prove the theorem.  Well, I mean it's, it's, it's going to be, it's going to be a struggle. We have to fight. Now there, first of all, there's some easy cases. So there's an easy case, which is when F is smooth or more generally is flat, but somehow it doesn't.  And somehow does not help. This is, I don't believe this is known to fail. I mean, maybe it's, it's known not to like work in the most optimal bounds which you could guess but I don't know I don't think anyone knows these things to fail.  But we don't we we can prove it.  So I was a question if this was if this stabilization result which I proved for pi zero. Will it also hold is it known to be false for higher home to be sheeps and I believe it's not known to be false I'm sure everyone believes it to be true, but we don't know how.  The easy cases is if F is smooth and the reason is that in this case, there's a pullback map on the entire rostrum complex without support.  It's not compatible with the pullback in F because I mean basically you build it to do that. And then the universal properties sort of what the resolution means immediately tells you that everything works so smooth maps super easy.  There's another easy case, which I don't want to treat, which is if your F is what's called the homotopy module or an infinite loop sheet.  And the reason is that in this case, I told you there's this fantasy formula which you want to write down, but it uses transfers, and then you have a problem because you don't have transfers on C zero so you cannot make this map.  But if you have an infinite loop sheet and you have transfers on C zero. And again, you can then use this construction of Ross to write down the pullback map and you can check that this does everything that you want.  And basically the problem in general is that you cannot somehow write down this map on the Russian with complex because there's some somehow degree zero you don't know what to do.  But you can write it down in the higher degrees but then you have to somehow argue that it's the correct.  So here's a key observation.  And I'm just the fact about homotopy modules because somehow the whole point of this result is to prove it for us really homotopy and range of.  And they're definitely not all homotopy modules. So the key one key observation is as follows. Very simple.  So let's say I have my Z contained in X, what I mentioned greater than equal to D.  And then what I do is I look at the generic points of Z.  And I only look at those let's say of co dimension D on X.  I mean every point of Z has co dimension at least D on X. So I'm looking at those which have like the smallest possible co dimension, not every generic point needs to do this because I mean, it could be stupid things like Z is a union of two things and one is much smaller so it has higher co dimension.  Okay, so let's typically exit some smooth guy sees some close integral guy and then there's just one generic point and whatever.  And then what you do is you look at, you look at this map HD with Z X F.  And then I map it to this HD.  And what I do is I hand it to lies my X in this in this generic point so that's it need not be a close point of exit also this is maybe some sort of slightly if the algebraic thing to do but okay so that's the beauty of algebraic geometry we can do some slightly  things.  And okay so here the support Z should maybe Z intersected with a HCI but I mean it's going to be annoying to write.  The point is two fault one is that this map is an injection.  Right and the reason is just that you look at the most mid complex.  Here.  Right so what if I do come on to the support in Z. It means I chop off the first couple of terms, namely first D minus one terms. So it's going to inject into this thing here.  So I see why I see exactly those points of co dimension D which lie in Z, and I do the default contraction.  And so, since the hands realization does not change the rest of your field. This map is in an injection. And the other point is that that this map from the hands realization to access basically pro et al.  So, up to some pretension pretending it's it's it's all.  That's basically one of the smooth maps which we understand.  So pullbacks understood.  Okay, so what this means is that in general whenever I have to somehow pull, I write I have to pull back along some arbitrary map, I can at least sort of shrink this target in some et al neighborhood of certain points.  And this hopefully will allow me to make it a simpler.  So that's what I want to say, or how I want to summarize is that the some of this problem is local  in some specific sense.  Okay.  So now, now we come to the real thing.  So I will want to explain. Yeah, so I want to sketch the proof the following lemma.  And I will just show you the assesses follow so I assume that the field is characteristic zero.  And I let y and x essentially smooth.  So this is just some trickery which allows me to look at something like the hands realization it's not quite a smooth scheme but it's reasonably close to one.  So I give myself some map from y to x which is in fact a closed immersion of co dimension one.  And then of course I have to give myself a z content in x co dimension greater than equal to D. And I have to assume that if I intersected with y content in why still has co dimension greater than equal to D.  So I give myself some F script the homotopy invariant.  And then I have to prove and this is what I want to prove then this pullback map. I have a star HD was part in Z X F.  And see in the sec why I suppose why F, and I will call this thing here W. This only depends on F minus D.  Last transfer.  So this is this is the special case of the main theorem. It's a special case a because I'm assuming that the characteristic is zero and be because I'm assuming that this map F is them is the regular immersion of co dimension close the version of co dimension one.  So this, this is not a huge deal, because we've already dealt with smooth map so I can, I can reduce to write every map composers can be written as a composite of the smooth map and the regular immersion. So you have to deal with regular emotions, and the problem is also local.  So I can factor it locally as a composite of the co dimension one immersions. So you reduce to this co dimension one case easy.  So the characteristic zero assumption is somehow serious so I'm going to say some things and that they don't quite work in positive characteristic you have to argue more carefully but something along the same lines also works except you have to deal with annoying things like well regular not being the same as smooth  and blah blah blah.  This is, yeah, so I have only a finite amount of time and energy to explain things to you and I think focusing in character to zero somehow. It's already complicated enough to argument.  Okay, so we're 15 minutes I hope that we can, that I will be able to convey some ideas to you. So what do we do.  So the first step, we reduce to the case where access dimension D plus one, then of course why has dimension D and Z has dimension one.  And of course then W has dimension zero and I want to assume that it's just one point. And it's a rational point.  Okay, so how do you how do you do this. Somehow this is this is some sort of standard trickery. So what you do is, you replace X by the hands realization right so I mean what I do is, of course I pick.  Right so I pick my W and W.  Point of co dimension D and why, right because the problem was local somehow in why, right because of this business here.  The problem is local on points of co dimension D in why and W. So I picked this point and I just have to look somehow locally around this point, and I replace X by the hands realization in this point.  And of course I also replace why by the hands realization in this point and see by the hands realization in this point.  Okay, and then it's a, it's a sort of standard fact that this this inclusion here, then it admits in retraction.  Okay, so this is not immediately obvious but I mean, algebraic ways of seeing this a geometric ways of seeing this.  Please believe me that this is the case. And now of course because in character six zero disguise regular and disguise any field whatsoever so this map here is going to be smooth. Well, essentially smooth.  Okay.  So now what have I done.  Well, I have localized in this point so you will see that these dimensions things happen automatically. And I've, right and I replace.  By of course the residue field of w.  And so by this by this trickery, I have assumed that I can now assume that this w is indeed a rational point.  It's just I mean, the statement is so general and it's, it's, it's going to be hard enough so we be, we make our lives a bit more reasonable.  Okay, so now I still want some further reductions.  So I can assume that actually acts and wires move.  Right so in the assumptions and also because I did the centralization business here.  I only said it's essentially smooth so it's some co-founded limit of smooth schemes.  And so I mean, all of these things that are going to happen sort of at the finite stage in this co-founded limit. And so you can always, and then there's some continuity thing that the value on F also will be done the co limit.  And so you can always things happen at some finite stage so you can, you can assume that X and Y actually prop us I mean, not not proper but like real bonafide as smooth schemes.  And then I can assume this Z in principle could intersect.  Why in many points but I can as well I mean I can just throw out all the others so I can assume that C intersect Y is just W.  I guess it's already included here.  And I can assume that these move away from right.  Because I mean Z is a curve in character six zero it's only have gonna have finitely many singularities and I just throw out all of them except for W also then Z will be smooth.  No problem.  So now this Y into X is a regular margin of co dimension one so it's locally principles again by working locally around this point W. I can assume that why is the vanishing locus of a single function.  And finally I can assume that X is fine.  Again, all of this because it's somehow it's a local problem.  Okay, so I will admit that maybe it's not quite clear where we're going.  But let me let me try so I'm going to now make two claims which basically are the heart of the proof and which I'm not going to prove because, well, time attention span various reasons.  So the first claim is the following.  So that there exists a function you bar from Z to a one having a bunch of good properties.  And so first of all I wanted to not be zero at W.  And secondly, I want this product you bar F right there's also a function from Z to a one.  And this is going to be finite.  Okay, and also I want you bar not to have double roots.  Okay, so now how do you do this it's basically a Riemann Rock argument.  Plus using the fact right so you can use Riemann Rock somehow because it's mostly a smooth curve. And this allows you to basically get all of this and then how do you make sure it doesn't have double roots it uses the fact that it fields of characteristics zero are infinite right because I could always add some constant on it and then  no double roots.  Okay, so let's believe this for now.  And then what I do is I put five one equal to this product you go. Well, so I do sorry.  Right, so I use the fact that this this axis affine. So I pick some you from X to a one extending you bar.  And then I put by one equals this you have from X to a one.  Okay.  And now here's the second claim.  And the second claim is that there exists more functions that exist by two by three and so on up to five D plus one from X to a one.  And then the following holds.  So I write five for all of them together if I want to fight D plus one.  No, no.  And then I hold bunch of properties. So, fire w equals zero and fire set tall at all points of five one inverse of zero intersected with Z.  And the thing is there exists an open neighborhood you contained in a one over D such that if I base change you to there and I map it via five to you times a one.  This is going to be a closed immersion.  And this maybe if you know about these things it looks like Gabbers lemma or like part of Gabbers lemma and indeed that's it's very got it from as you can prove this because of an infinite field by general projection argument.  Okay, so now I'm sure you all remember all the 15 functions and schemes and everything that we had so far.  So just in case that you don't and also so that I feel like I'm, I'm telling you something let me try to give you an artistic impression of what's going on.  So I hope this is going to work and help.  But let's try. So what had we done right so we had my we had our scheme X or smooth scheme X.  We only have two dimensions so everything is only going to be two dimensional. And so X is just going to be some blob here. Okay, so that's X.  Now inside there. We had, we had the Y right this was from smooth scheme of co dimension one. So it's going to be some sort of curvy thing here. So this is my Y.  But that's it we have well, we had a Z right so we had some close up scheme also of whatever could I mention D but I mean I only the only choice I have is also going to be some one dimensional guy.  But it could have singularities and it turns out that the problem mainly arises if the singularities of Z of meet why. So this is sort of the kind of thing which is going to happen. So this is going to be Z.  And then we had the intersection with my which we call W.  And that the big deal is that our W is this point here, which is the singularity of Z. So this is what makes everything difficult.  Okay, so this is what we had started with.  Okay, so what else did we do. Well the first thing which we did was right we cooked up this you.  So let's see, let me draw this in here.  So the zoo that the you did with this will give me some some other guy right so this is going to be the vanishing locals of you.  And this white line is by the way the vanishing locals of.  Okay, and then we had like lots of maps the five one up to five D plus one so there's some map which I call by equals five one up to five D plus one.  And so this now goes to a n of course, a D plus one. So let me try to draw this.  Again, I only have two dimensions. So this is somehow an a one here on the coordinate x one.  And there's a one here on the coordinate x E plus one. And then there are some more coordinates in the middle of course but I cannot draw them.  And so what what happened here well what happened is that.  Right to the five one was the product of you and F. So this axis here is the pre image of that is going to be the union of the blue bits.  Right so it's going to be the union of y and some extra thing.  And so what else we had to see. Now the Z is going to do something.  I'm going to try to draw this maybe like this something like this. Okay so what's going on here well so the W the W still here and still this still this singular point we couldn't do anything.  And, but also what can happen is that this you write so this thing here this has some some could have some new intersections with Z right so for example here.  So this is some new intersection.  So this I cannot avoid. So this is some new intersection.  But I had asked that it should not have double roots so this is some new transverse intersection.  So at the beginning that the problem is basically when the Z has a singularity on the intersection. So basically what I'm telling you is there's a new intersection but it's, it's good we can understand it.  Right and then also what we have said is that on some open neighborhood.  This guy here.  The Z should be a closed immersion right so now my Z what was it it was the sort of curve here, and it turned into that sort of curve here but now it has a new double point.  So that's a problem here. So that's one of my bad points somehow.  It's problem, but that's fine. Right so what this is saying here is that it should be a closed immersion locally in this coordinate. So I just throw out this point right so then I might you.  I'm going to be a D, which in this case is a one, I suppose, and I'm removing this finitely many bad guys. No problem.  And then away from that it's supposed to be a closed immersion so maybe, maybe that's that's what I'm doing.  So, somehow morally what has what has happened here is that we managed to straighten out the right to the X and the Y. There were some arbitrary smooth schemes and now we've somehow managed to straighten them out into an  into a full a one. That's somehow what the what the gap a lemma always does it gives it gives you somehow enough room to get the full a one.  So it's the expense of having sort of this, this new guy here and these new guys here which we now have to deal with. But if you take nothing away from my talk and please somehow remember that that the point of Gabas lemma is to straighten out into an actual a one, whatever  whatever that means.  So now I have four minutes to try and finish the proof so what do I do. One thing which I do is I let you zero be the intersection of you with the thing where the first one is zero.  So in our case, that's just this point here but generally that could be more dimensions.  And then it turns out that there's some open subset here, which will be well chosen, and I will not tell you what it does the choice.  And then the following things going to happen. So what do I do.  Okay, so maybe. Yeah, so maybe I will abbreviate this. And what I will say is, so more steps.  I will tell you it suffices to understand the pullback somehow like this HD, the U a one over you have those two HD, the U zero a one over you zero with confidence in it.  So the whole exercise of this game was that I replace the situation on the left by the situation on the right.  So how is this any better.  So one thing which we can observe is that this Z.  It is finite over you.  So this is some proper morphism. So what I can do is zero was to you finite means that I can embed a one of course into P one, and then it will remain closed in P one.  And so I can instead, I can instead let's say, do this.  Right.  I can instead do this.  And it's also enough to understand this.  And now, what can I do.  Okay, so I have this HD, the U, P one over you with coefficients in F. Now I'm supposed to pull it back here to HD, C, zero, P one over you zero with coefficients in F.  However, right, so this zero zero, this is basically just finitely many points.  So this group here I can, I can work out using the wash me complex right. So that's just M minus D of W.  So that's this point.  And then maybe there's some stupid other points, which we have to deal with.  Plus, so a whole bunch of points some of I am minus the of why I so these annoying new points.  And now, now is where the transfer comes. So there is a transfer map here.  And I mean, yes, let me write it. So it goes to HD P of C you by which I mean just the image under the projection in this direction.  With coefficient, you with coefficients in F minus one.  Right. So if you ignore, if you ignore the supports here, or that's of course I did it wrong.  Right. So if you ignore the supports here, this is just saying what is HD P one over something coefficients in F.  But you just use the fact that P one is S ones, my GM. So you should remove one from the D and you should remove one from the F.  And that's how I got this. And you can check that you can play with the supports here. And this, it turns out is basically the nature of the transfer.  So this is what the transfer is. And I can do the same thing. Then I can put back here, HD minus one P of C U zero, U zero, coefficients in F.  And there's another transfer here.  And this diagram commutes since very elementary.  And I can also work out this thing here. So this is M minus D of W again because it's a rational point, plus maybe some other, some other things.  So I'm out of time, but also I'm at the end.  Right, because you had this. Right. So what does, what does the transfer do. I told you that this abstractly defined map, which has something to do with the transfer.  And well, you can see here in this simple case, right. So this is some sum of M minus these evaluated some points, and then they map down to some other points.  And so you what you need to do is that you need to find some kind of transfer. And I mean, that's this construction here gives you a transfer.  And you can see what this is. But now the point is we had designed this to be right. So this, this W here is the same thing was a rational point. So there's no field extension. So this is actually nice.  Okay, so now I'm done right so I had something here. And I supposed to figure out what this is, it's image here. So I need to figure out what it is here and here.  And suppose that I know this map here, right. So I had something here, I know it here, I know it here so I know everything here.  And also, I had told you that these new points there somehow, there's somehow easy. So also, I know everything here. So basically, in this big group here, I know everything except for this value here.  And sometimes if I transfer it down all the way to here. Okay, but now, well, if I just pretended zero here for a bit transfer all of these things down and subtract it off from what it's supposed to be, and then invert this map here.  I will find that I figure out what the last component is. And so then, what have we done. Well, it suffices to understand this map.  And now we're done because by induction.  Right, because I had now managed to reduce the D.  And so now you can just go again and eventually you get to D equals zero and equal zero is trivial.  So I'm sorry for running over time. But I hope I have given you some idea of how algebraic geometry proves this interesting result in material.  Okay, thanks a lot for a wonderful talk Tom.  Here.  Let's see, are there any questions.  Let's start with a question in the corollary the result with Maria Yarkersen.  There was a, I think there was a number three showing up. Yes, possible to somehow explain why, why it's number three.  Yes. So this number three has something to do with the following thing. Yeah, okay, so what about this read so I'm not sure if it's optimal.  What is this right. So it has to do with the following thing you take F minus one has some transfers and F minus two has transfers.  They all have transfers and some feeling is that the transfer this guy has transfers.  Better transfers.  Better better transfers.  Right so just like how an E N right so if you take iterated loop spaces classically have like group structure be a group structure.  And so the feeling is that eventually, so the F minus infinity should have frame transfers.  Now actually, right so it actually turns out that already already F minus three does.  And the F minus two, if characteristic is equal to zero.  And the conjecture is that F minus one has frame transfers.  But so one. Okay, so now that was a lot of waffling but the point is that somehow the contractions give you more and more structure and we believe that you somehow get all structure already after three steps, or after two steps or maybe after one steps and how  you need this is the number this is where this number comes from. Right so this number three is because we can prove that after three steps here somehow at the at full structure.  Okay, thanks.  Then there's a question from Sean Tillson he asks, is there extra structure and that you have now from this scrollery, like more than the Abelian group structure.  Yes, I mean.  Yes, I mean so one thing which you do learn is that yes if you do pi zero of omega p one cubed sigma p one cubed F, right this is actually a homotopy module.  So this has all the structure which you could possibly ask for. In fact, in the proof, we learned this. Right, so already if you're because I mean this is like if you take a three fold loop space and then you have good structure this is not like super exciting thing.  The point of the proof is that this already right even without taking loops, you already have a lot of extra structure. So the answer is, yes, we definitely do get more structures on various things and this is basically the heart of the group.  And there's a question is there an analog of maze recognition principle for multiple spaces. Well, in the world, does there exist one I hope so, can we prove it.  I believe not right now. So we have right so there's there's some there's sort of S one loop space theory, which says can you deal with it in the S one direction and I would argue that this is probably reasonably well understood I'm not sure if it's like written down in this language but  this sort of thing we can probably do. But the D D looping in the GM direction is very very hard so we do not have. Yeah, so we do not have recognition principle, right where cove mission principle for even something like P one loop spaces.  And then GM loop spaces but the problem is that somehow because GM is not connected itself these things are sort of, it seems likely to me that you might have something like for P one or P one.  That's the smash to the dub dub that for these you might have it, but we do not have any of these. We have we do have a recognition principle for motivate infinite loop spaces, which is right so like, like saying that infinity monoliths group like infinity  are not group spaces so we have some like that group like spaces with frame transfers, but we definitely do not have this at finite stages.  If we did then my result would be much easier or I mean as easy as proving this recognition principle.  Yeah please work on that and prove it that would be great.  I did not have any characteristic zero to prove your, your theorem, but it characteristics zero doesn't appear in a hypothesis of the fair. Yes, no I mean I did not. I use characteristic zero to make the already probably long and hard to follow argument, a little less annoying  characteristic you do something which is roughly the similar story, but this reduction in the beginning. So this point here. This then becomes a few pages to argue that you can still do something like this in positive characteristic.  But yeah so this is only I did it to simplify the exposition and only prove a special case.  Another question. Can you make a comment on the zero and pi zero of your with your corollary with the accurate some.  Well, what can I comment on, on what zero means, or sorry, I'm done quite well.  Well, could you change it to pi one.  No, I, well, I would like to believe that.  What we do very much is confined to pi zero. So I think that, yeah, so I mean I would conjecture or whatever.  You would guess that I can do something like pi I here, and then maybe and greater than I know something three I or three plus I don't know what it goes so that that that would be what you think, or what I would think, but  I would expect that this probably requires different, different kind of attacks but I don't know so I definitely cannot I wish I could, but I can.  Can you explain where the zero comes up and going from the transfer result to the corollary.  Yes, I can, I can I think do that.  So how does this work.  The way this works is that I look at somehow.  Right, so I look at the category of S one spectrum.  And then I have the category of motivic spectrum.  Right, so I can go to here, and I can go to here and basically I want to figure out what this composition is.  So I can imagine this is happening somehow in spaces right so I can first smash it with GM and go to them somehow to this as such as one of K one, and then I go to two.  And I keep going, and I can always factor it like this.  Right, I mean I'm just saying that if you do an infinite iteration of GM loops and GM suspensions you can do one and one and one. And then eventually you always have to infinitely many. And what the what the proof eventually shows is that if you look at the hearts here.  Right, so these categories in some sense right to the first one and the first one and so on so forth in a in a way which I find difficult to make precise they approximate this category, well I guess I can take some limit in the category in pure.  But whatever.  Right.  And so what we prove is that shs one of K and heart right so it has this natural functor sh of K effective heart.  And this is an equivalent for and greater than equal to three.  And then eventually you'll get the corollary from that by some form of manipulations.  And the way this works is that basically we know very explicitly what this guy is. And so we have to somehow describe these things very explicitly and why we just find our way through and eventually we see that what what are the objects in here they like these sheaves and they have some extra  and then eventually we argue that this is all destruction.  And so if you want to do it for like say pipe one right so then you don't have to look at the heart but you have to look at something like this, right, as one.  Okay, so the concentrated degree zero and one. So I don't know some sort of motivate one types.  And you could try to do the same analysis but it's going to get much more complicated right. And then you have to do it with three types and. Yeah, so that's my attempt at answering your question.  Okay, great.  Other other questions.  Yeah, yeah, just just to understand more precisely. So you don't build pullback maps on the ROS complex. But instead you prove that's it. Instead you prove this independence results.  Right.  So that that yes, I well I haven't really thought about that. So, or another detail. Yes, so maybe I should say right well you have this thing you have to say it's a C zero of XF goes to C one of XF goes to C two of XF goes to answer and so forth right now you can do the same thing with why.  Okay, and then the dream would be that you have some maps here, which make everything work.  But the problem is that there is no transfer here and then you cannot write it down so that's the problem. Okay, so but instead.  I mean, you can do the support and see thing.  Okay, and then I mean this, this guy just goes away. I mean this just doesn't exist.  And more don't exist. And then the point is this is formula which was written down how to do it. So there exists a map here and there is this one here and everything commutes right.  And so the easy, I mean, the ideal thing would be to prove that this is the correct thing. And I think what my result shows, but I have to have to think about that is that sort of the lowest terms or if these are all zeros right then this map here it does the correct thing.  So that there's this map which this fantasy map which you write down, but it will actually give you the thing which comes sort of implicitly out of what I'm doing with a transfer whatever.  And I'm, my feeling is that you might be able to soup this up to learn that by some induction thing that it does actually the correct thing right so that that these maps here which you can write down that they all use the correct map and co homology.  But I have not actually tried to do that. And I think it would be maybe annoying. But it's not, it's not out of the questions I'm not sure. But I think yeah I it says pullbacks with a rest with complex and I definitely don't do this in general.  Okay, thanks. Thanks.  Okay, anybody else.  I think that concludes the question so thanks again Tom for a wonderful talk.  Next up is Dylan.  John also asked the negative stable homotopy sheaves and the answers. No.  Okay, thank you.
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