 Let's go ahead and get started. So I'm happy to introduce Matthew Morrow, who's gonna tell us about aspects of motivic homology in two talks today and tomorrow. So Matthew did his PhD at Nottingham. He was a postdoc and a Simons Fellow at the University of Chicago. And then a postdoc at Bonn, and now he's Charger de Recherche at CNRS based in Jussu. So Matthew, oh, and I should say, he's known throughout the world for his contributions to algebraic k-theory and piata-coch theory. So Matthew, we'll take it away. Well, thanks very much for this generous introduction. I'd also like to express my thanks to all the other, all the other organizers for inviting me to take part in this, but also the PCMI staff for making it possible and seeing that they're doing a lot of work behind the scenes. And finally to my fellow speakers who have saved me some work because thanks to them, you have already seen some of the fun. It will be relevant to my talks. So already in the very first talk by Crashing, some motivic complexes appeared, albeit quickly. And Milno K-theories cropped up a couple of times. The relation that both these invariants have to a telco-homology has played a role. Again, particularly in Crashing's talk, that was block-carto-conjecture. And then in DeGlese's talk, you've seen some A1 homotopy theory and what else in some more categorical aspects, like categories appreciated transfer. And this has all mostly been for, maybe entirely, actually, unless someone wants to correct me otherwise for varieties typically smooth, maybe not all the time over some base field. So now what's left for me to do, we're gonna aim for the following. Firstly, I'm really gonna be interested in this motivic-cohomology just, well, from one point of view, just as an intrinsic invariant. So the co-homology that comes out of some of these categorical aspects that DeGlese is gonna be talking about, we'll just study the co-homology that it outputs, but particularly as a tool to analyze algebraic cases. And then the second topic, which I'll get onto pathway into tomorrow's talk will be some recent progress in mixed characteristics. So where we discard this hypothesis that we're working over base field, we try to prove some things in greater generality, not even with any smoothness or regularity hypothesis. So in light of this first goal that we want to say something about algebraic K theory, I'd like to spend a decent amount of time today just going over some classical calculations so that you can see some of the baby cases of where we'd like to get. So the first section will just be devoted to some, some of the theory of K zero and K one. And the phenomenon that I really want you to keep in mind, which if I remember, I'm gonna try to also make white and green is the following. So K theory is something very, very general. And what we're typically going to see is that it encodes other invariants and in a related phenomenon is that it breaks into simpler pieces, simpler is necessarily the right word. And these will turn out to be motivic in nature. So this story starts of course with this definition of K zero going back to Grotemdyk in the 50s. So let's start by recalling that. So we take some ring R, which could for the moment be any, I guess any associative unit or ring, but in fact, at some point I'm going to implicitly be assuming that all my rings are commutative because otherwise I'm almost certainly going to say something wrong. And then we define this K zero R, the so-called Grotemdyk group is the abelian group defined as follows. So we generate it by free, we generate it by finite projective modules over the ring, or perhaps more precisely by isomorphism classes of such. So here P is running over finite projective R modules. We mod out by the simple relation that we force direct sums or equivalently short exact sequences to become summation in the group. So modular the relations, the class of P is the class of P prime plus the class of P double prime whenever we have a short exact sequence relating zero goes to P prime goes to P double prime goes to zero. And let me mention parenthetically right away to just to save myself from doing it later that we can adopt the same definition for any scheme just by replacing these finite projective modules by vector bundles. Or we can do the same thing for any topological space by looking at real vector bundles or complex vector bundles. And that I suppose we'll do it some moment later. So the definition is pretty simple but it already outputs some interesting phenomena so we're going to explore this in a few examples. So let me start by recalling the case of a field then okay and I find out projective modules these are just vector spaces, excuse me. These are just vector spaces. And in this way I can produce an isomorphism given by the dimension function taking any such vector space and associating to it each dimension over the field. And one sees that this is an isomorphism because as soon as you know it, as soon as you know the dimension of a vector space you know the vector space after isomorphism and short exact sequences are compatible with dimension. So that's not so interesting but really place to start. So next case of interest with your up one dimension we're going to look at Dedekind domains. Let me take O to be a Dedekind domain. For example, you can take a wing of integers in a number field. So what's important is that then any ideal at least if it's non-zero, I guess even if it's zero even if it's zero, zero is a finite projective module. So each ideal is a finite projective module. I guess that characterizes regular wings that I mentioned at most one. And so we get its class inside k zero and what then turns out to be true is that we completely, we can break up this k zero into a short exact sequence as follows. So as before I've got some map to the integers given by taking my finite projective module and sending it to its rank. And we can completely understand the kernel of this in terms of the class group. So here I put the class group of O that is to say non-zero ideals under multiplication and so if I have a non-zero ideal in the class group what do I send it to in k zero? I don't send it to the class of I, I in fact I send it to the class of I minus the class of O itself, which is the identity element there which looks a little funny initially but this is what you need to do in order to produce a map that respects the additive structures on the two groups involved because the class group, the multiplicative law there is exactly given by multiplying ideals together whereas in k zero it's given by taking the direct sum. But that turns out to be compatible with group structures and produces a short exact sequence. And the moral to take away from that is that we break up k zero into the class group with some classical invariant one is interested in and a copy of z. So moving on to higher values and higher dimensions still let's look at the case now of some smooth algebraic variety over a field probably needed to be quasi-projective what I'm going to say or at least satisfy some other technical condition but I think we can ignore that. Let's take simplicity. So what do we do in this case? So suppose that we've got some cycle that is to say some irreducible closed sub-variety on x whose co-dimension we'll care about in a moment but not just yet. We can pick a resolution of the structure sheaf of z as follows. So I've got structure sheaf of z and I pick some resolution p zero through some pd where this resolution is given by some vector bundles and this is where we'll probably need some technical hypothesis like quasi-projectivity. And once we've got such a resolution we can then associate to this sub-variate z a class inside k zero as follows. I take the alternating sum of the classes of these vector bundles and one can check by identifying this is discussed a little bit more in this case in the exercises and the notes by identifying k zero with g zero defined in terms of all coherent sheaves on x. One can check that this class is really well-defined. So it's a well-defined element associated just to z not to this particular resolution that we've picked. And so from these classes we associate a filtration on k zero to find fill j of k zero x to be the subgroup of k zero generated by the classes of things having co-dimension at most j. So this is some irreducible closed co-dimension at most j. So I've got my orders right. We have k zero contains fill d where I suppose d is now the dimension contains and so on and so forth down to fill zero zero set of the filtration that contains zero. So it's a decreasing filtration. And I told you all this for the sake of the following the following theorem which is part of Lehman-Lach theory. So we have a map now I guess in fact, part of the theorem is that this map makes sense. I look at the Chao group of... I'm confused here. Oh, I made a mistake. I think you mean dimensionless and very quick. That's entirely possible. That's a minor mistake, but this is, but I'm gonna do a Chao upper j. You're right, I just put my indices somewhere else. Can I do fill upper j and then... You can, but it's still that the top one is... Okay, I'll just put a lower j. Okay. I'm looking bigger than or equal to j then. Right, right. No, no, sure. And then you've got to swap both. You can choose, but... I can choose, but I should choose in a consistent fashion. So I make that dimension and then I just need to do filled. Yeah. Now as long as I do Chao lower j, everything's okay. So I now take Chao dimension of my Chao group of j dimensional cycles, modulo rational equivalence. Thank you, Mo. I get a map as part of the theorem is that this map is well-defined to fill j modulo till j minus one by sending. Okay, I just take my cycle, I assume it's irreducible and I send it to... Well, we better move back. It's on one side to make it a little bit better. I take my cycle, I send it to this class that I've defined a moment ago. And the theorem say this is a well-defined subjective modus per definition and it's kernel is killed by j minus one factor. And here I'm okay, but I mentioned code, I mentioned not gonna... So one can say more than that, but I just would say simplicity, let me... How the j minus one is should be in the j. It should be a d minus j minus one, sorry. No, no, no. I do everything with code I mentioned you today, so. Go by d minus j plus one, greater mark on board two, there we go. I think minus one. No, it's minus, is it not minus j minus one? Or can you do better? No, it's just the code I mentioned. It's the code I mentioned minus one. Oh, it's the code I mentioned minus one, sorry. You're right, thank you. Anyway. Thank you. Thank you. In a moment, I'm gonna say small torsion, which has been sufficiently imprecise that I cover all the bases. So all the pieces of this filtration coming from this generally defined k zero identify with algebraic cycles. So let me just write down the mantra to remember up to small torsion, k zero x breaks into juggles. And what we saw for Dedek and domains in a moment goes in quite just a special case of that. So they have already with k zero, we see some phenomena that this, let's say that this generally defined k theoretic invariant breaks into interesting pieces in some sense. And so continuing this theme, let's go on to k one, which goes back to, I think also to the fifties defined by Bass. The idea of the definition being that we're going to classify isomorphisms between finite projective or finite free modules up to trivial changes of bases given by elementary matrices. I just see there's some activity in the chat. Let me check here. What are we doing J equals D is the automatic triples and it's going to be J minus one. Yeah, when J equals D, there won't be any kernel. Let Mark correct me if I'm K theory, but no. Matthew, you can ignore the chat. No, no, no, no, it just takes a second. I'm happy to. If people are too shy to ask out loud and the woman's going to have a critical. Okay. So let me move on to K one though. So we've got K one of my way. As I say, we want to somehow classify isomorphisms between free modules, but up to trivial isomorphisms. And then we do that as follows. So I take some, this infinite general linear group to find as follows. I take union over all N of GLNR, where, okay, GLN I include into GLN plus one by sending a matrix, just add a one at the bottom right. So in other words, it's infinite invertible matrices, which are eventually just ones along the diagonal and zeros and some outside some block. And I mod this out by elementary matrices. So this again is infinite union of all ENRs where ENR contained inside GLNR is a subgroup of what you get if you apply elementary row and column operations to the identity. I'll check for that. At least in passing to the limit over N gives me a normal subgroup. So as I say, some are precisely classifying isomorphisms up to obvious change of basis. So starting again with the case of, could start again with the case of a field, but more generally, let me make this make a comment that we always have the so-called determinant map from K one of R back to the unit of R, given by, this is represented by some matrix inside some GLN and I just spit out it's determinant in the usual sense and this is even gonna have a right inverse by taking a unit in the ring and just looking at the matrix starting with that unit and then being ones all down the rest of the diagonal. And so, okay, now we can do a calculation in the case of a field. So in that case, elementary matrices are just the whole special linear group. When it's supposed to be an F, that's Gaussian elimination, isn't it? Stating that if you take some determinant one matrix over a field, you can perform row and column operations to it until you get back to the identity. So that provides that equality of groups. And so we see the modding out by elementary matrices is the same as modding out by determinant one matrices. In other words, this determinant map provides us with just an isomorphism between K one of the field and the units of the field. So at the risk of presenting maybe a silly calculation, let me say that we can now already see some very elementary relation to a tau columnology in this case. I can look at K one now mod M. This is for M prime to the characteristic. And so this we've just identified with units mod M which we can then identify by a kuma theory with the first Galois co-immology with coefficients in mu M, which is of course very baby case of block cutter. So continuing upward dimension again, let me instead of treating arbitrary data can domains, let's let O be the ring of integers in some number field. Then it again turns out to be true that this determinant map identifies K one just with the units of O, but unlike the case of a field, this is now a rather deep theorem originally proved by Bass, Milneau, and Serre and there's a second proof to the cash done. But in both cases somehow you have to get your hands dirty with some representation theoretic properties of arithmetic groups. There's some real content in this. And in particular, I'd like to emphasize that it's not a nonsense statement about dedicated domains as we saw in the case of K zero. For example, if we take functions on a circle which is again a dedicated domain, then one can produce a non zero class in the kernel of the determinant map. So result three really, it fails for arbitrary dedicated domains. And what we see happening in this case and how does one check that this class is non zero and the K group one uses the fact that it corresponds to a loop going around the circle. And so let me say very roughly as another instance of this phenomena that what's happening in this case is that the K group is really it's decomposing into some data coming from units and some info about loops. And this information about loops turns out to vanish in the case of a ring of integers of an ownership. But again, implicitly hiding behind it there's some decomposition into other pieces. And so it's this phenomenon of decomposition that we want to explore in Motivic Co-Homology. But to do that, we need to first, these decompositions can concern what happens in higher algebraic K theory. So I want to start with before going on to Motivic Co-Homology, just say a few words about higher algebraic K theory. But I don't want to spend too long on it because that would be a lecture series in its own right. And I imagine that you already at least will be perhaps only briefly encountered it. So what happened? Very simplistic presentation of the history was that by the 60s, it was clear for a number of reasons that K zero and K one should really just be the beginning of some exceptional Co-Homology thing. So there should exist some higher homological invariance in so-called K groups for all n and bigger than or equal to zero. And the question of the time was, how can we define these? And Quillen came along and got his field's medals with the proposal of multiple definitions which I don't have time to present in all that gore detail, but I'd like to give some idea what happens anyway. So as I say, he gave multiple definitions of this K theory space, more correctly, a theory spectrum, infinite loop space, K of r. So here's one option which I quite like. As we've just seen, we've just seen K one and one of these constructions is in some sense, you derive this K one construction as follows. So the output's gonna be the whole K theory space as follows. So we again form this infinite general linear group then we take its classifying space. So that is to say we get some connected space whose pi one is just given by this monstrous group GLR and whose homotopic groups vanish. I'll say it's strictly bigger than one. And then we modify it, this is Quillen's insight, there's something called the plus construction which means that we, so we're gonna, it will change the space in such a way that forces the homotopic groups to become a billion. So at the moment, the first homotopic group of our space is GLR, which is highly non-habilian. And so what we do is we force all the homotopic groups to be a billion, but without touching the homology and Quillen's insight, that makes sense, produces some new space and remarkably, although you start with a space which only has anything interesting in the pi one, remarkably when you perform this process, some very mysterious information spreads out into all higher homotopic groups and these higher homotopic groups are precisely what we care about. And to be careful for pedants, I should throw in a copy of K zero R into this space just so that I get the right result of K zero, but for the higher homotopic groups, this is not relevant. So for the higher homotopic groups, the idea is really this, that you derive this K one construction, so forcing the homotopic groups of BGLR to be a billion and this produces all these wacky classes and higher degree, but you don't understand, but it seemed to be amazing. So, I mean, that's one of his, that was one of his constructions and he also had his Q construction, but I mean, this is, so this BGLR plus construction is great for doing hands-on calculations. It's what he used to do all the initial hard hands-on calculations in the theory, but for general theory, or at least from a modern point of view, it's not really what one uses. So, I just wanted to power and effectively make a few comments about that. That from a more modern point of view, it's some of the closer in spirit to K zero would be either the following. I can define, this can be a connective K theory spectrum by taking a set of the group completion, I take the infinity group completion of the infinity space given by looking at the category of projected modules and restricting to isomorphisms, that forms an infinity space, like infinity group completed and that will produce a mere connective spectrum, but if these words are meaningless to you, then don't worry about it. Another point of view is that K theory in a more general context, so not just of rings, is the universal, sometimes it's a universal way of linearizing. No, it's universal way of, again, it's sort of universal group completion approach. So it's universal invariant of stable infinity categories, taking exact sequences, now of stable infinity categories to co-fiber sequences of spectrum, plus some other conditions. And then from that point of view, you define the K-thever ring by looking at the K theory, perfect complexes, perfect complexes of armor, end of parenthetical comment, all that I ask you to take away from this is that in any case, output is these mysterious K groups, which more than one person in the room has been trying to calculate for a very long time. And the point of view that I really want to adopt in the remainder of these talks, and this point of view is not novel, that motivic homology is a useful way to do it, and variance of motivic homology, motivic invariance is a very general sense that I'll try to get onto in more detail next time. But just the remaining for a moment on the subject of these mysterious invariance, let's see just how mysterious they are. So Quillen in 73, he did one of the very, very few, wait, maybe it's the only, maybe it's still the only complete calculation of K groups that we have, which is the case of a finite field. So in degree zero, I just get a copy of zero, excuse me, in degree zero, I get a copy of Z. In even degree, I don't get anything. And in odd degree, I get a cyclic group of order Q to the degree plus one is a degree plus one over two minus one. But then everything goes downhill. We don't even know the K groups of the integers. In fact, we know that it's hard to know the K groups of the integers because it's a theorem that the vanishing of the mod four K groups, the K groups of degree divisible by four is equivalent by a theorem of Kuhri-Hara at the date from 92 to the van der Veer conjecture. So the van der Veer conjecture is a statement about cyclotomic number fields, stating that if you look at the maximal real subfield of a cyclotomic number field obtained by joining a P group of unity, then the class number is not divisible by P. That's widely open, which I guess we understand most of the rest of the K groups of Z. So there's calculations that we don't understand, but then there's also general properties that we don't understand. Suppose you take some regular ring, which is finitely generated over Z, then the K group should be finitely generated. That's for all N. And that's, I mean, that's totally, okay, it's known if the dimension is at most one, again, by some like classical calculations and everything else, everything else, we don't know what to do. As another example, okay, this is this calculation where there's gonna be a positive result. Let's take F to be a field of characteristic prime to some N, then, okay, let me stick this in green because it's supposed to be an instance of the phenomenon that we're keeping an eye out for. This is again, special case of blockato, K2, not too many twos, K2 mod M will identify second Galois coromology coefficients in mu M tensor two. M tensor two, that's in the query of Swaziland theorem, which is now a special case of blockato, which we'll come to a little bit later. So then again, we have a description of this abstractly defined K group in some terms of some previously, some previously interesting invariant. Depending on your point of view, you can either view it as a calculation of the K group, which is nice because we want to calculate K groups, or you can view it as some new description of what we see on the etal coromology set. Both points of view have their role to play. That's 150, must be between the 70s. So then finally, in this list of properties and expectations, what I mentioned at least, it again goes back to Cullen's original work in the 70s. So partly motivated by zeta functions. So there were conjectural formulae, particularly explored by Lixon-Bau, giving special values of zeta functions in terms of K groups, but then you can make similar predictions of such descriptions in terms of etal coromology, and then they lead you to expect some relations between the K groups and the etal coromology groups. And what this finally led to, let me just say it in the case of a number field to keep things easily. So for a number field, f and any m bigger than or equal to one, they conjectured, then just say relation formula, say formulae for the K groups of f with mod m coefficients in terms of etal coromology. And we're gonna see the precise formula a little bit later, so let me not bother writing it down. Indeed, in fact, here I've cheated a little bit and I've reversed engineered the result because when you look at the original conjectures, they usually stated for the ring of integers or even the ring of S integers with respect to some fixed set of primes S, but then you can pass the limit over S and get some results for the number field itself, which will just be, it'll be easier to state in terms of the way I'm gonna present things. So instead of etal coromology, I could have said gaol coromology, but let me do my usual game of coloring this in. This is supposed to be an instance of the sort of phenomenon we're gonna keep an eye out for. And it's describing K theory in terms of some other known invariance. Did I want to say anything else? I wanted to make a quick remark on K theory with coefficients because it's going to start appearing. So this is K theory with coefficients. So in the usual way, I mean, once you've got the K theory spatial spectrum, you can construct a mod M version of it whose homotopy groups will then fit into short exact sequences as follows. Where this final term is the M torsion in the group one down. And so this will appear. This will appear quite a lot. And I mean, it should maybe be remarked that even if one understands some K group of coefficients, it doesn't mean that one understands these individual terms. It's very often not very clear how the information gets wrapped together to produce the information in the middle term. So even if you understand all K groups of coefficients, you very often can't reverse engineer the information to understand the K group itself, which is a shame, but that's what I like to do. So with these preliminaries on K theory out of the way, now we can really jump in, oops, and jump in to the arrival of a modific format. And so the labored this point a little bit, but here's the sort of question that I really want to explore. So in what generality do the above phenomenon of, when can we describe K theory in terms of other known invariance? When does it break up into nice pieces? That was quite, I guess I better into topological example, that it's always true in topology. So in the following sense, so if you've got any topological space X, you can associate to it, it's topological K zero defined using complex vector bundles. So again, you do the same as before with a bit complex vector bundles up to isomorphism classes of complex vector bundles and force short exact sequences to give you sums. And then you can do a little bit more work using suspension and bot periodicity tricks and bot periodicity to produce a whole slew of K groups. Yes, I've got to go over all the integers. So it gives you the analog of the algebra K theory that we're studying, but with the topological spaces, much older than the story of algebra K theory. But one then has this theorem of a theorem Hertzberg that these four ways can be described in terms of prior existing invariant, namely singular co homology. So there's a spectral sequence coming from various copies of the singular co homology of the space and converging to these topological K groups of space. And it degenerates rationally. So if the only capital is going on up to small torsion, we can even decompose all of our topological K groups just in the singular co homology. So right there down topological K theory breaks into singular co homology. So motivated by all these phenomena that we, again, I'm probably maybe I'm misrepresenting the history a little bit, motivated by all these phenomena that we've seen above of special phenomenon K7, K1 and what Lixin-Barmel and Quillen were conjecturing should happen to the K groups of rings of integers of number fields. And what we see here coming from the topological picture, the following framework of conjectures was proposed, excuse me, by Balanson and Lixin-Barmel in the 80s. They proposed the existence of the typical conjecture, which says what? I want to adopt the axiomatic point of view that we expect existence, certain co homologies with certain properties as they did. So let me restrict for the moment to smooth varieties over fields. So for any such gadget that should exist a co homology theory. So that should exist some complexes, which I'm going to call Z-motivic weight J of X for all J bigger than or equal to zero. So as I say is weight J-motivic co homology of X with a whole bunch of properties. Not so interesting otherwise. So let's start looking. What do we expect? So not the most fundamental one that I really care about is a relation to K-thread. Namely, we have an analog of the two hertz of spectral sequence. I mean, that for me is really the, next time we'll discuss variations on this theme. We want to describe other K-theoretic invariance in terms of other motivic invariance. And the test that you've got the right things is that you have such a spectrum sequence. And so in this case, we had a spectral sequence coming with the same index and I haven't messed things up from the motivic co homology of X. But now we've got to take account of all the possible weights and again, converging to the K groups of X where just to be clear about notation. So by the motivic co homology groups, I just mean I take the co homology groups of this complex. I want to make certain statements directly in terms of the complex rather than passing to the co homology groups. And it should degenerate rationally. And so then rationally the K groups exactly decompose just into some sort of motivic co homology groups. But in general, we can still say that modulo, the degeneration issue of the spectral sequence, we see that the K groups are splitting up and decomposing in some sort of these descriptions in terms of motivic co homology. In low weights, one wants a good description of what's going on. So the weight zero motivic co homology, this is not very interesting. It's just going to tell us something about the number of connected components. So it's just going to be sitting in degree zero and there it's going to give us the number of connected components. A little bit better is weight one that will be given by just taking the whiskey co homology of units of a wax star and then moving it a little bit to the right. So let me make some silly comments about where these complexes live. So in the case of degree zero, that's of course just supported in degree zero. And in the case of degree one, well, I'm really now, if I go to the right-hand side, the co homology of units on a normal scheme, and it's only sitting in degrees zero and one. We have a nice resolution for the structure sheath on a normal scheme. So that's going to be supported in degrees. Oh, since I've shifted it a little bit to the right, the final thing is supported in degrees one and two. And what in general is supposed to be just want to tell you what is true. The vanishing to the left, as we'll see in a moment, is that the motivic complex in weight J should be supported in degrees zero through two J. Even, I mean, this will turn out to be true, even in degrees less than J if X is the spectrum of a local room. Next one has, next, we can begin to discuss relations to other invariants. Otherwise, somehow it's not interesting. We don't want, we don't just want to understand algebraic K theory. We want to understand algebraic K theory in terms of existing invariants. And so now we just say a few words about how the motivic homology is supposed to be related to other invariants. So we start with the relation to algebraic cycles. So we'll say a word in a moment about higher child groups, but let me just stick with usual child groups for the moment. Then the degree two J, weight J, motivic homology is child up J, no, Mark, worries me these days, but I think it sounds okay. Well, the J is upstairs, so it should be all right. The general rule. So we detect algebraic cycles in terms of motivic homology, and then really I can finish on the most important part, which is relation to tau homology, which is really the core and what is very often just goes under the name, Baylor's and Nixon's Banconjecture, which states that these motivic homology groups, if I work with finite coefficients, so as previously that means I take my complex, I take it mod M, and then I take the homology that mod M complex, these will just on the nose be a tau homologies with coefficients in mu M and the J take twist of mu M. If M is prime two characteristic of the base field, this is the important part, I think I should have put it first, I is at most J is in the range where you're computing the motivic homologies up to the weight to understand it on the nose in terms of a tau homology. And so let's now, let's do a calculation with that to see what that really gives us in practice. So let me take, I want to make this, I mentioned a moment ago, conjecture of Lixon-Baumann-Quillen about k-fairy of number fields. So let's make that precise now, let's compute k groups of number fields, number fields F as predicted by Lixon-Baumann-Quillen. At least what are we gonna do where we only gonna be able to compute them, modulo M for M odd. So how do we do this? We look at the mod M version of the Atea-Herzburg spectral sequence. So here's where I put my H0 Galois F coefficients in I guess just as you mod M. And then as I go down the vertical axis, I get Galois-Cormorative F coefficients in mu M, then I got an H2 Galois coefficients in mu M tensor two. And then I would get an H3 with coefficients in mu M tensor three, except I don't, because for odd, because M is odd, the homological dimension of F being a number field is at most two. So I don't get any H3. And so now I can continue computing and continue running down my spectral sequence. Here I'll pick up an H0, again, coefficients in mu M. Here I'll have an H1, coefficients in mu M tensor two. Over here I'll have an H0, again, coefficients in mu M second twist. Now I'll have an H2, coefficients in mu M tensor three. And I'll be able to continue. My spectral sequence will continue down like this and everywhere else, I get lots and lots of zeroes. Here I try to, I don't need to worry about an H2 here. Maybe I'll put that in H2, Galois. I don't really care about what happens to K0 anyway, but I don't think it's there, so. I don't think it's there, but I don't care for this. I don't actually really care what happens with it. And so what do we conclude? We see that in fact, there's no interesting arrows in the spectral sequence. The arrows should go like that. So all the arrows are zero. And so we can just read straight off that. I've read the answer earlier. I'm gonna put it in green because it's a very nice example of the phenomenon that I'm supposed to be promoting. We compute that the odd K groups with Z mod M coefficients, these are just given on the nose by certain Galois homologies and that the even index K groups will fit into short exact sequences involving an H0 on the right. Is that right? Yes, that seems okay. It'll be weight J and the edge map coming in. I've got to put together this term and this term. Edge map coming in will be an H2 of a mu M tensor J plus one. I haven't messed up the index, I think it's okay. And so we see that we can really extract or the indices legible, see that we can really extract some awesome on the nose description of the K groups in this case. So that's the power of the machine, if you like. We have on the one hand this abstract relation given by the Ateo-Herzburg spectral sequence just breaking up K during demotivate cohomology. But then we have these other relations to a tau cohomology coming through Balencien-Lichtenbaum and together we make lots of progress on analyzing K groups. Okay, so I stated all that all that above as a conjectural framework of Balencien-Lichtenbaum but the theorem is that it's almost all known to be true and thanks to can include to many people Locke, Levine, Friedlander, I've tried to put this in alphabetical order, Rost, Susslin, Weyvotsky, the above conjectures are true except, and it's a big except that it's a pain, except for the part that was that's known as Balencien-Sule vanishing, namely that these gadgets are supposed to be supported in positive degree. So as it stands, one might get negative cohomology which is not very satisfactory, but we don't know better. So having stated this, I thought it would be good to present at least a definition of these motoric complexes which I think has not yet appeared in any of the talks because it is surprising how quickly you can write down a definition. I mean, then what's extremely hard is checking the properties which don't actually go through the definition represent but the fact is that in five minutes, I can write down the definition of these complexes that turn out apostioid to do the trick. So let's do that. The original definition is blocks, higher cycle complexes. I mean, one of the great things about the definition actually is that it continues to work in greater degree of generality. So this I'll come on to in the next talk. Not only does it produce something interesting in the singular case, but it continues to work in greater degree of generality than just working over fields. Still seems to produce the expected motivic invariant for reasonable schemes over dedicated domains. For example, over the integers in some sense, the only case you need to treat largely thanks to work over the V. Probably have time to say a little bit about this about this next time. For the moment, let me stick to fields, to varieties over fields. The right might as well drop the smoothness hypothesis. So for variety X over a field, we look at the following sort of free abelian group on cycles. I'm gonna take the three, what sort of cycles am I gonna take? I take the free abelian group on pretty sure this time I'm right about code dimension. On code dimension, J irreducible closed sub schemes of what not of X itself, but of X times this algebraic simplex. So let's, okay, let's finish the sentence and then we'll define some terms intersecting all faces properly. Okay, so what do we need to define? We need to define this gadget there. This is what I say is this algebraic, the n simplex, take spec of K joint T zero through Tn, or the n dimensional one and mod it out. This is really all just some game in normalizations by the sum of the Ti's minus one. So that's isomorphic to just affine n space. Oh, sorry, I guess my K is, there's no name for my field. I guess my base field can be, let me take it to the f and let me put it in the fixed system. So that's isomorphic to affine n space, but for the understanding the simplificial interactions between the different dimensions one works with this normalization, because now sitting inside of this, I have various faces given by delta m's where m is smaller than n. I look at vanishing loci of various Ti's, they cut out from the various copies of delta m's, where m is at most n. And so then when I talk about faces of x times delta n, I mean these various x times delta m's, m less than n have various copies of these x times delta m's sitting as faces inside, and when I say that the cycle that I've got to intersect these faces properly, I mean in the usual way in the expected co-dimension. So there's no strange intersections happening. And then complex fire cycles is defined by sticking all these together with the boundary maps given by alternating sums of all these intersections. As I did find the boundary map, I take some cycle, and I send it to alternating sum from zero to n, I'm just posing the line, in little zj xn, and I send it to the alternating sum of the various intersections, say the ifh face x times delta n minus m. n, I hope I'm not gonna mistake the indexing of the, I think the indexing is okay. I've got n plus one copies of this face sitting inside, intersect z with all of them, I take the alternating sum of all those intersections. I mean, it's really written down as a complex, but in fact, it's really a simplicity of being a group. So that's where this alternating sum comes in. Great. And then the blocks higher child groups, the co-homology, homology, this hn, I think I should do h minus h minus m. Well, h lower n. Right, I was gonna do h minus n instead because I'm imagining it, but I can still just go homologically. It's a homology kind of guy. Right, right. Because the problem is some of the spectral sequences. Right. So block side child groups and then just given by the homology of his complex. And I told you all this because part of the previous theorem that motivical homology exists and has the expected properties is really a comparison that I can define my, we can define motivical homology by taking blocks, complex of cycles and just when we just think about the index, so I move it to the right, I move it to the right, homologically, maybe then it's a way about difference with homology and homology, but I think I'm okay now. So the fact that this has the expected properties. But as I commented to actually, to actually prove the properties of the previous theorem, you, things like Bennings and Lixon found you, you can't do it by passing directly through this definition. When block introduced, this high child groups, you could prove some bits and pieces of the previous theorem, but by no means everything. And one router has to pass through the type of machinery that Degleese is presenting in his talks, along with an independent verification that the output of the machinery he's presenting actually produces the same as these high child groups, which is a highly non-chival comparison and of a slightly different flavor. Nevertheless, I wanted to present that because it shows in principle that one can write down and in the end it took closer to 10 minutes, a definition of these more different complexes that it interests us. And so I'd like to finish by returning to some more concrete aspects of the story, namely Milner-Kay theory and Block Carter, because these in principle have nothing to do with higher algebraic-Kay theory, more motivic-cohermology, but that prove passes through all this machinery. So theme is, you know, how are these fancy invariants related to Block Carter, in fact. Since I've only got a few minutes, I can luckily fold back Crashing's first talk that let me, I guess I just kept up the case of fields, so let me do that. If we've got some field F, we can cook up the Milner-Kay groups, which are these very explicit invariants. I don't even need to be a field, given by tensoring together J copies of the unit and modding out by the so-called Steindrug relation. And Tate then showed that for an integer prime to the characteristic, we have the so-called Tate or Galois or co-homological symbol from the Milner-Kay group, that might as well go mod M, into one of the Milner range of the Galois co-homology, namely into the Hj, with coefficients in the J-Tate twist. And the Block Carter conjecture, says that this is supposed to be an isomorphism. And it's of course no longer a conjecture, it's now theorem of Rostin-Verivodsky, states that Tate symbol is an isomorphism. So a priori, it's just related to Milner-Kay theory and Galois co-homology, what's it got to do with algebraic K-theory and notific co-homology? That's thanks to the following remarkable fact that Milner-Kay theory shows up in notific co-homology. So this is a theorem originally due to Nestorenko and Sluslin, and there's an alternative proof by Tataro, and there's even alternative proof, I think just of the surjectivity by Kurtz and Muller-Schnapp, stating that for any field F, we can identify a piece of notific co-homology in terms of this Milner-Kay theory. We should say there are natural isomorphism, for the natural isomorphism, okay, I can say for each J. Between the Milner-Kay theory and what we call the Milner range of the notific co-homology, it is the top degree of the notific co-homology. So I look at the Hj of weight J of my field. I was gonna tell you how to define this map. I don't quite have time to do that. Maybe I could include that in the exercises. Because I can now instead just finish quickly and explain how you get from this plus the sort of conjectures we talked about above to Block Carter. So if we now go mod M, what do we see? We see that Milner-Kay theory mod M is going to identify with notific co-homology with mod M coefficients and weight J. So here, if you're paying attention, you'll see that in principle, there could be a little obstruction term coming from an Hj plus one, but I included in the list of conjectures that when you're local, there's no Hj plus one of weight J. There's nothing beyond the degree J of weight J. And so in fact, I can, there's no obstruction coming and going mod J. I can just pull the mod J inside but then Valence and Lichtenbaum, since I'm now computing a degree which is at most the weight, Valence and Lichtenbaum tells us that this is just the same as the etal co-homology. And so that produces overall for us the Block Carter isomorphism. But in fact, Milner is true. Let me even just say, let me write down what we've really done. So thus Valence and Lichtenbaum implies Block Carter but the converse is basically also true. And if you want to see more about that, I strongly recommend the book of Hessemeier and Weibull. The second chapter goes into this in detail and really shows how Block Carter conjecture, the Valence and Lichtenbaum conjecture and the Hilbert-90 style conjecture are intertwined and shows you how you can reverse this sort of argument. You throw in some roots of unity and apply some standard tricks to go backwards. So in this way, we see that this, one of these main conjectures that we now know are true in this framework of motivic homology at the end of the day, some of they were used to these very concrete, to these very concrete statements that compute K-groups for us. And so I'll finish there today and tomorrow I'll begin to get into what happens when we discard this hypothesis that M is prime to the characteristics. We'll start looking at piatic phenomena in characteristic P. We'll look at variants of this type of story. Here I've really been looking at standard K theory on smooth things. We'll explore some variants of that and then finally get into the most recent progress of focusing on what goes on in tau locally in a piatic context. Thank you. Great. Let's thank Matthew for a great talk. No, thanks for coming. There's been a lively discussion in the chat. Yeah, I see. I've been scared by these numbers going up. But I think so far everything has been answered. So hopefully there are some more questions for Matthew. I was just wondering, you did computations for K-groups for number of fields. And I'm just wondering, do we do the same thing for piatics? And if you can, if you know it for piatics for different primes, will you be able to gather the data for the number of fields just in the spirit of local principle? Yes. So we can again calculate K-groups of local fields and rings of integers in local fields. Provided that we, as in the calculation that I did, provided that we work with finite coefficients. That's not going to let you completely, not anywhere I see immediately reverse engineer to get the calculations for a number of fields. You get some formal description of some of the whole K theory spectrum in that way. But if you're trying to get your hands on the individual K-groups, that won't quite give it. That will not give it? Don't think so. No, it doesn't seem. I mean, you'll get some formal, let's say you should get some formal description of the K case. No, not even, not even because no, no, no, no, I know that I think of all the arithmetic square looks like this would let you extract if you knew everything that you wanted to know about the number field and the local fields and the rings of integers inside the local fields, then you would get some information about the ring of integers of the number field. That's the best you could do. But even that wouldn't give you a precise calculation. But no, if you just know what's going on, say for all the QPs, you can't reverse engineer the information about Qs. I see, okay. No, that's a good question. I mean, it's a natural approach that you take, but maybe I could propose very briefly. Weibull has a great survey paper on this. It does, fantastic. In the handbook of K theory, and he presents both the local fields and the number, and you can sort of compare and contrast there. Okay, thank you. Other questions? I have one. What of the things we have seen depend now on these missing part of the theorem, this balance and soul benefit? Oh, nothing. I mean, nothing I've said depends on that. Okay. No, no, the results I presented are all unconditional. Okay. So that's a good question. I mean, for the moment, no, but nothing I said requires balance and soul benefit. So that's missing for what then? For which part? It's a good question. I should ask myself that while I was preparing the notes. I guess you need, maybe Mark can chime in. I mean, do you still need that to construct the category of motives at some place? It's the first, I mean, it's the first obstruction to having, so we have these triangulated categories of motive. But if you wanted to have an abelian category of like motivic sheaves, the first obstruction is the balance and soul a vanishing. That's what you need to have sort of a good theory of motivic tape sheaves. Sort of saying the X groups of untaped motives vanishes in negative degrees. And if you had an abelian category of tape motives that whose derived category was the image in the triangulated category of motives, then that would be the case. So it's a question of whether you have a nice abelian category sitting inside these triangulated categories. It's the first obstruction. There are many more. There are more. There are more. That's the first one. Maybe it's worth mentioning too that it is now reduced to rational. Right. Question by what you've taught us. Right. Like I showed you that the motivic cohomology mod M just looks like a tau cohomology and there's no negative at tau cohomology. And we also understand it mod P even when P is the characteristic. We'll see that next time. And again, one sees there's no negative cohomology groups. So it is just a statement about the rational motivic cohomology. And therefore you can even convert it into a statement just about the K groups. You take your K groups rationally. These decompose into Adam's summands which are the same as the motivic cohomology. And these are supposed to vanish beyond a certain point. So if you had other conjectures like finite generation then that would do it. But the problem is you don't have these conjectures. Right. You don't have the problem of infinite divisible subject. So for the moment you might get weird things. I guess you get a copy of Q appearing in negative degree. There's no way to discount that. No, that's right. You can't. But the thing is there's no way to detect it either. No reasonable biology theory has negative cohomology. So there can't be anything. Great, more questions. Actually about that, is there an analog of Valence and Suley vanishing for the HZ twiddle of degrees and Fizel and Comes and Bachman, Ostveer? I think it's the same. Yeah, I guess so. So that would be this. Okay. Well, there's no vanishing in negative degree for vitkomogy because it's periodic, data periodic. So I know that. I know that there's vanishing along the, you know like that thing that was zero is now non-zero. But this question, this question mark kind of above like in the upper two quadrants, like here. Like here we got it. You mean Valence and Suley or Valence and Suley? Yeah, I associate Valence and Suley with like a question mark up here and the extra vit stuff down here in quadrant three. Yeah, but it's still in negative comogical degree. But about that, the mean of vitkomogy just is a direct factor of the multi-ecomogy rationale and vis-vit part that we can compute. Great. So there's no, the mystery part is the key theory part. That's great. Thanks. Still, yeah. Any other questions for Matthew? Yeah, I have another one, which is that the Valence and Licton bound identifying this etal comology with the motivate comology. I was wondering about relating that to representing etal comology as a spectrum in SH. We have the spectrum, the motivate comology say mod L and we inverse the special element tau. It's a relation between that specific calculation of representing etal comology and Valence and Licton bound. I don't know. I'm not sure I really understand. The question is. The means to represent. I think it's a weaker statement. Which one? The one that you get etal comology by motivate comology with coefficients and then inverting tau. It certainly follows from Valence and Licton. The implication doesn't go the other way. You could have things that die by multiplying with a high power of tau going from motivate comology to etal comology. One more time, a little more slowly. It's saying if you invert tau and the etal comology on the motivate comology, you get etal comology. So you can gain a lot and lose a lot by localizing. Yes. Right. So it doesn't say that in a fixed degree, the H, the Z mod M of Q in the degrees where P is less than or equal to Q that you get the etal thing that it just says if you multiply by a high enough power of tau you get a high enough power of tau multiple of the class you were trying to get. And there could also be a kernel that would get killed by a power of tau. So I think this kind of statement that the localization after tau is, this I think was for K theory was proven by Thomason. Something like etal K theory is algebraic K theory and sufficiently high degree, some statement like that. I'm not absolutely sure, but I think he did prove something on the app before. He had, you know, quill and elixin bomb, which was a K theory analog of valence and elixin bomb. I think it's a weaker statement. Just great. I get it. I think that's super helpful. Okay. Questions for Mark. Yeah. Any other questions at all? I can pass. Sorry, I do have one more, but I can also, I'll be quiet as clear. Okay. I propose, Kirsten, you ask your question and then we can go over to Sokoko for 15 minutes before the next talk. You're super kind and thanks for your patience. So we had this way of getting chow from K-naught and we have this relationship between K and all the motivic homologies, but we have this cycle class map, which sends something in chow to something in lots of homology theories, like in particular motivic homology. So do we get anything cool if we do both? So we first, we take K-naught and we go into chow and we do cycle class map to motivic at someplace else in K theory. I mean, at worst you get multiplication by J minus one factorial up to a dimension co-dimension issue. But don't, aren't you in a different K group when you? Hopefully not. Okay, thanks. Okay. Let's thank Matthew for his great talking conversation.