1/3 Motives from the Non-commutative Point of View
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Transcript: English(auto-generated)
00:17
Right, so the title is on the blackboard, it's going to be three lectures, and I'm very
00:22
grateful to the organizers for setting this up in the first place and then for persevering and doing this in this format, better than nothing. I mean, there are multiple issues now which don't appear in normal conferences. So I have to apologize, today I have to run to finish five minutes earlier and
00:41
run away to another seminar in Moscow, like from Paris to Moscow in five minutes. So today I will not be able to take questions right after the talk, but I will also give you a lecture on Thursday and then on Monday next week, and so you are more than welcome to ask questions and answer them later. Yeah.
01:01
All right, so what I want to give you is some kind of overview of really, I mean, there will not be that much proofs and maybe not that many theorems, it's rather a viewpoint on motifs from something which eventually will be non-commutative, but it will come along kind of naturally.
01:23
And so I have to again apologize, so it's a long story which developed over the years in kind of strange way. My own background is from algebraic geometry, this is where I come from. And this is where the story comes from originally, is some of it. But a large part of it was developed by a algebraic topologist, actually.
01:42
And so we saw some of it in Kino's great talk yesterday, and we'll see more of it today in the third lecture. And since I'm not an algebraic topologist, I mean, I might not be very good with say references of who proved what.
02:01
So I will give you some statements which I know are correct, but maybe I will miss some attributions, so I apologize in advance for that. It's very good that we actually have also a course by Tino on these books that Sian, Matz, and so on to work, which forms a very important part of the story.
02:21
So I feel kind of covered on that part. So anyway, but let me start from algebraic geometry and even not algebraic, but kind of usual geometry. So if we have some kind of X, which is a sine, sine, sine manifold, and we want to
02:43
consider this cohomology. And as we know, there are several ways to define it, which will give you in the end the same answer.
03:00
I mean, there are many ways, but kind of the more standard ones are as follows. So first of all, you can represent, you can think about homology and you can represent it by cycles, some actual geometric objects in X, or if you want,
03:20
you can do singular homology and simplices in your X. This gives you homology and cohomology, integral coefficients or any other coefficients which you want. This is a nice theory, which also works in topological spaces, one approach.
03:40
Another approach is instead of this, you can think more cohomological, you can think in terms of sheaves on X and sheave cohomology. But more classically, maybe you might consider check. So instead of considering cycles, you consider open covers, nice enough for open
04:04
covers, you write the check complex, all that kind of stuff. Again, this gives you cohomology with Z and a pasteluri and any other coefficients. And then there is a third thing which is specific for manifolds now, and this is
04:23
Dirac homology. And it has no coefficients, meaning that coefficients are reals, coefficients are basically represented by forms and it gives you the same result.
04:45
So all, I mean, it gives you the same result when you compare it. So Dirac homology is the same as usual, singular homology, coefficients in R. So all give same result and they share some properties which are
05:05
kind of natural when you want to think about something as a cohomology theory. So, first of all, the thing is homotopy invariant, if you have your X, you multiply
05:22
it by an interval, say, you want the round maybe an open interval, so it is not manifold, then you get the same thing as you had before. And then a version of it, which in the C-infinity setting is basically the same
05:44
statement. But let me state it separately, nevertheless. So if you have something which you can think of as a manifold, for example,
06:02
submersion and you consider the cohomology of the fibers, then they form a local system downstairs, so they are locally constant, so fiber of some point, now this
06:20
is locally constant. Again, this has several guys, you can consider relative cohomology, which will give you a locally constant shift downstairs, when the round cohomology, you can have some vector bundles downstairs of that connection. The point is that if you move along your base of your family,
06:42
the cohomology doesn't change. OK, now let's go to algebraic setting. And the point is that in algebraic setting, kind of all three approaches survive to some extent, but they give you different things now. Let me stick to smooth for now algebraic.
07:05
All right, over simplicity, let's say over a field k. The first approach, geometric approach based on cycles, eventually you have to think long and hard and invoke first part theorems, gives you what is known as
07:22
a child groups, child groups, or algebraic k-theory. So the two things are intimately related. Maybe k-theory is more important to me because it's somehow more fundamental,
07:41
but the point is that this is a very good theory in the sense that this is really intrinsic. You don't impose anything on your variety, you just work with whatever is there already, either cycles for child groups or vector bundles, so you can choose for k-theory, but so it's intrinsic, very
08:01
interesting, but hard to compute. Notoriously hard to get. Okay, so this is the story, this is kind of the best
08:20
replacement for singular homology in the geometric sense. Now the story about check coverings. Well, the reason it worked in the usual topology is that you can always cover your synchrony manifolds by something, by some affine sets which are small enough to be contractible.
08:41
Algebraic is not possible anymore because if you take an algebraic variety and you take, I mean the only topology you have a priori is risk topology, and when you take kind of the smallest possible open cover, it will not be contractible at all. So if you're over C, say you have a curve over C and you remove points,
09:01
and that's the only thing you can do, you remove points. You end up with an open curve without many points, and actually it's not contractible at all. The more points you remove, the more contractible it becomes. However, it is always of topological type KP1, so it's defined by its fundamental group. And then there was this great idea of Grothendieck that you should generalize your notion of cover.
09:23
Instead of just open subsets, you should consider coverings of open subsets. And if you develop that idea far enough, and if you're Grothendieck, you end up with a tall chromology.
09:47
Now this still has some limitations, because in algebraic geometry we can only consider finite covers, we don't have infinite covers. So instead of the whole fundamental group of something, you only get
10:01
profane completion. And as a result, so you get some theory which is nice, which behaves nicely, but the coefficients have to be, well it can take finite coefficients, but for practical applications people prefer l-adic homology, so coefficients are QL where this is some l prime,
10:24
and in characteristic zero it's okay, but if you're in positive characteristic, then this should be different from the characteristic of your base field. And the story I'm going to present is mostly interesting, the situation when you're in positive characteristic. So this will be some
10:43
kind of P, and this has to be different from P. And now there's the third story, the rough story, and this again works, homology. So it again works nicely by theory
11:06
and you can just do the very naive thing, you can define differential forms as exterior powers of the contention bundle, and you think about it as an algebraic thing. And then if you're over C, a non-trivial but true statement is that this computes the same homology.
11:25
So any homology class can be represented by differential forms with polynomial coefficients. So this works, this also works in characteristic P, but of course
11:41
here we have a problem which is that the coefficients of our homology theory are by definition when you're working the same as the base field. And originally lots of this was developed in the course of proving the make-injections, and there people wanted to have something positive characteristic, but homology
12:02
theory with coefficient characteristic zero. So there are some kind of extension of this, which of course came up, which is called crystalline homology. So this has coefficients k, but then there's crystalline homology, which is sort of better.
12:25
So first of all, it can now be defined for a singular variety just as well. So x can be singular, doesn't have to be smooth anymore.
12:43
And moreover, it can have coefficients, so if you start with something over k, there is a version of homology which is defined of coefficients in the ring of width vectors. So for example, if your k is just fp
13:01
of the prime field, then this will be a p-adic number, p-adic integers. I will speak more about this later. And not only they give you different things, it's not even clear how to compare them. So for example, if you want to compare Dirac homology and Atalic homology,
13:23
then you see right away that if you're in positive characteristic, you cannot do it because the coefficients are different. One is a logic where l is not p, whereas the other one, the de Rameau crystalline story, has coefficients either in fp or maybe in zp or qp, it's still the same p.
13:41
So there is a very interesting involved and long kind of long story about what comparison you can do, which is called p-adic posh theory. And I mean it's due to many people from ten findings, and then in the end balance on, and then in the end I think it was finally completely resolved not very long ago by Bhagavad Gokhad, Peter Scholz and Matthew Morrow,
14:07
but even there the statement is not obvious because it's not hard to find, it's not that obvious, you don't even find the context where I mean comparison theorem can be made. And as for k-theory, normally there is a
14:21
map from k-theory to either of the other two, but it's very far from being an isomorphism, we don't expect it to be an isomorphism. And what we expect to be able to do in the best of possible worlds is to be able to correct our kamochi theory somehow, that in the end they compute something which is closer to k-theory.
14:41
So that's the general problem. But let me now discuss the behavior of all these three theories in respect to those standard operations which I mentioned in the beginning, so sort of homotopy invariance properties. So here the story is again slightly
15:14
different because we don't have an interval anymore. An interval is not an algebraic thing. You can either take the whole line, which is a fine line, or you can think really in
15:23
phinitesimally, you can take, I mean we can't do it in the usual topology, but we can do it in algebraic, so we can consider in phinitesimal neighborhood of a point, very small neighborhood of a point. And here the telekomology behaves exactly as you expect
15:43
a kamochi theory to behave. So first of all, it's what is known. So I think the terminology is due to Vyvotsky. It was not originally,
16:01
I mean originally it was not thought it was it would be very useful, so it's it's kind of a great discovery. Vyvotsky leads to very interesting theories. So the property is called a-y homotopy invariance, and this means that the telekomology of some x, it means what you think it means.
16:24
If you multiply x by the fine line, you get the same thing. And this kind of a global statement. And another important thing here is that and where the banner is that k-theory behaves similarly
16:49
the same way. If you localize it at some l or maybe complete
17:03
some l with the same progress as before, so l is not equal to characteristic. By the way, if there are any questions along the way, I don't think I can see it very well, so if some
17:22
I don't know if Grigorik could. All right, sorry to have interrupted you. There is a question for crystalline k or dvf in the question chart. For crystalline,
17:44
let me go back to that slide. Where? W of k, f or dvf?
18:06
So k will be the, I guess, k will be the residue field of k, and coefficient is wk or fraction field of w. Either, I mean, there are both. There's wk, you can take the
18:21
fraction field out. Yeah, I was about to answer, but okay, thanks. All right, and then for k-theory, this is actually a highly non-trivial statement. In the end, this is, I mean, just what I wrote is
18:41
actually easier, but some generalizations are not. So it is basically a discovery of Susten, which is called Susten rigidity. And then another thing which I
19:01
mentioned, you can work with the algebraic geometry, is infinitesimal rigidity. And this is again just true. So a telekomology of some x which is maybe non-reduced, maybe some kind of you know infinitesimal neighborhood of x and some ambient variety is exactly the same as telekomology with reduction. It doesn't show any important examples. So this is the
19:27
story for it all. Now, k-theory. So as I said, if you consider k-theory, say find k-efficient or complete it at some l, then it's fine.
19:43
But generally, it's important to realize that k-theory is not a one homotopy in y. Well, it is if x is smooth. But as I said, all the theories actually
20:04
make sense for singular x, and it's useful to consider them in full generality. So for smooth x, it's still true,
20:20
but if you just multiply by a fine line, the k-theory doesn't change. It's one of the basic theorems of Willem when he can develop k-theory. But in general, for singular x, it's just not true. In a very bad way, I mean very, very, very much. And it's not infinitesimal like
20:48
what I put like this. So it's not true anymore that k-theory of x is the same as k-theory of its reduction. So if we consider some kind of infinitesimal neighborhood, then k-theory would change very much. In fact, we already saw this yesterday
21:03
in this lecture where you consider, for example, well, basic example, z mod p and z mod p squared. And k-theory of z mod p, we know k-theory of z mod p squared, we don't even know, so that, I mean, not completely different, but they change drastically. There's a whole story about how k-theory changes when you're doing
21:22
infinitesimal deformations. So in this sense, k-theory is not does not behave in the way you would expect homogeneous. So here it's maybe it's useful to mention that there is this complete, especially if you're in positive
21:41
characteristic, there is this complete dichotomy. So you can sort of take your k-theory and localize it at some prime different from the characteristic, or at the prime equal to the characteristic. And the answers are totally different. Already for the point, so if you just consider the prime field, both have been computed by Quillen, but the
22:00
answers are different. So if you computed l coefficients, l not equal to p, then you get some kind of both periodicist, you get some polynomial algebra in one generator with degree two, well, if you do fp bar maybe. So it's kind of similar to topological k-theory. And if you do it at p, then k-theory will point only, there is only k-zero, there is nothing else.
22:23
So already here the behavior is different, but when you start taking some kind of infinitesimal neighborhood of points, so for example lifting it to z mod p to the power n, or maybe considering truncated polynomials, then there is this relative k-theory
22:41
which we saw, and that relative k-theory is entirely p-local. So if you localize at l, you get rigidity doesn't change. And if you look at p, then it changes very much. So the two kind of stories behave in a very different way. And the kind of periodic p-local story is not
23:02
at first similar to homology. And now the crystalline homology. Dmitry, there is a question, does there any kind of A1 invariant k-theory? You can force k-theory B to be A1 invariant,
23:24
but for my purposes, this is exactly what you don't want to do, because it completely kills off what you want to study. This is some kind of dichotomy. So there is like the whole theory of motifs of way, what is kind of based on the ideas that
23:40
things should be A1 and y. And it really leads you very far when you do a logic stuff. But for periodic stuff, I don't think it's, I think it basically kills off the interesting parts. So I would prefer not to do that. But even before you go to k-theory,
24:01
which is very difficult, let's discuss what happens with crystalline homology. And here it's like this. So it's kind of an inpatient invariant, I mean not completely invariant, but so at least if you have a family,
24:26
it's kind of locally constant. So you have a family and you look at the fibers, then downstairs you get a bundle with flat connection. Of course the problem is that, say if you don't even do crystalline, but
24:43
say do just the Ramka homology in a positive characteristic, it's true that you get a flat connection. It's the same story over any base field. The problem is of course that in characteristic p-flat connections give you less than you expect. So it doesn't give you like a trivialization to all orders. But at least it gives you a
25:03
trivialization of some kind of what I call Frobenius neighborhood. So up to the power of p you're okay. And in general, I think the full statement is that crystalline homology doesn't change if you do some kind of invisible neighborhood which has an
25:22
additional structure called divided powers. So there is some story you can do there. Kind of locally constant. But certainly not A1 homotopy invariant. Not A1 homotopy invariant.
25:44
And actually there is a very easy geometric way to see why this happens. So now I have to, yeah, it's a geometric reason.
26:04
Now I don't give you the definition of crystalline homology nor of K-theory. Well because there is no time of course, but also because it's not that important. What's important is the properties. And the basic property of crystalline homology which you need to know is the following. So if you have some x over this k, sorry,
26:38
for now let me fix k to be a perfect field of characteristic p.
26:51
Say finite field of characteristic p. And it can happen that this thing can be actually lifted to something defined over this with vectors.
27:04
W of k. And there is some kind of lift. Then it always happens. If it happens then the crystalline homology of x just computes to the drunk homology of this lift. And then if you have a lifting,
27:34
crystalline homology of x is the same as the drunk homology of this lift. And if you look at something like a projective line, for example,
27:45
then it has of course a lifting. And so crystalline homology of p1 is what you expect it to be. So there is this base which is
28:02
W of q is one dimensional. There is a special point in the base and the fiber of special point is p1. And then you want to lift. Okay, now if you want to consider a1, you just take some point the special fiber and remove it. So there is no point.
28:21
Does this have a lifting? Well, it does have a lifting but it's not what you expect. So here we have a1, 1 minus infinity. But this lifting is not this minus any kind of lifting with infinity.
28:48
Formally by definition this twiddle has to be basically complete. So you have to complete this. This means that you not only remove one point, removing one point would correspond to removing a single section of your family. But basically you remove all the
29:04
sections which pass through this point infinity at zero. So you remove this, this, this, all this has to be removed. And in the end you end up with lots of holes in the general fiber of your story, lots of holes in the general fiber of your lifting. So it's not just p1 minus infinity.
29:23
It's basically p1 minus anything which reduces to infinity in model P. And this has lots of commodities as you expect because you take a curve and remove lots of points. Each point creates your commodity class. So h1 crystalline on a fine line is huge. And so some people think
29:43
it's a bug and try to correct for this. And there are ways to correct for this using rigid analytic spaces and maybe with log structure. I mean there are lots of interesting stories about the correctness and make crystalline cohomology behave more like you would expect a homogeneous theory to behave. But on the other hand you can
30:02
think about it as not a bug but a feature, as something which is intrinsic to the nature of crystalline cohomology and work with it. And this is the viewpoint I want to adopt. Admitri, sorry to have interrupted you. Willie Kartinas asks, when a lifting exists, that crystalline cohomology
30:22
with p-adic integer coefficients lift to the RAM with same coefficients or you need to pass to p-adic rationals? No, no, no, it's through integrally. So a lifting has to be smooth in large break sense but then it's through w of k
30:40
coefficients. Of course you can after that invert p but you don't have to. But what you have to do, you have to complete your lifting periodically. So it really has to be some kind of formal scheme because the story is really about infinitesimal, a series of infinitesimal lifting rather than any geometric lifting with a junior geometric time.
31:03
So it's through integrally but it has to be adequately complete. So this for a1 what you get will be actually torsion but there will be lots of it. So it will be annihilated by p but
31:21
there will be lots. Okay, now my general goal in these lectures would be the following, roughly speaking. So I
31:46
want to leave completely aside the tall story which I'm going to present but I want to think about k-theory and crystalline cohomology and I want to show that the two phenomena are actually related.
32:00
So that the two phenomena mean that neither k-theory nor crystalline cohomology really behave in a way we expect from what are related and actually if you start thinking about k-theory it
32:27
just inevitably leads to crystalline cohomology. If you don't know about crystalline cohomology at all but you think about k-theory long enough you discover crystalline cohomology automatically
32:41
and that's kind of the message which I want to explain. And then as an additional bonus it will turn out that this whole story actually just is naturally defined in a much bigger generality, namely you don't start with a ring which is commutative, you don't have to start with
33:01
algebraic variety, you can actually do it for any associative unit or not necessarily commutative ring or even digital algebra. So this whole story turns out to be non-commutative which is kind of an added bonus which I think was originally expected but now it's acceptable. But I mean let's not rush,
33:21
you'll just come up by itself. Okay, so let me start on this story. I told you what the story should be and now let me actually start doing it. So I need to tell you at least something about crystalline
33:47
cohomology, right? I told you one thing is that when you have a lifting it gives you a little thing. But kind of one other thing I want to know about
34:03
crystalline cohomology is the following. Assume that x is smooth now, again our k which is perfectly characteristic. Then although it was not the original definition, there is a way to compute
34:25
h-crease of x in a way which is similar to the Dirac cohomology. Then it's canonically as a morphic, so cohomology of x. So there is a topology of coefficients and certain replacement of the Dirac complex, w omega x. So this w omega x is
34:49
something which behaves as a Dirac complex. So it's canonical with factorial respect to at least say local isomorphisms and gluings and so on. It's called the Ram-Witt complex.
35:14
I think it was the original idea of Bloch and it was later developed in
35:20
the story of fully done by Illusi but with lots of input by Deligny apparently. I mean the paper is by Illusi but if you read it he says very often that this and this and this was done by Deligny. And it's the story from like mid to late 70s.
35:42
So I don't really want to know, don't need to tell you what the whole complex is, but its first term and what replaces functions. It's something which has been around for quite some time already then. So w omega 0 of x.
36:07
Omega 0 of x is just functions and then here we have something which is this thing called ring of with vectors. Now this already appeared of course but there I did it for a field and if you didn't know what this was
36:23
you could just think that this is, you know, like zp. But now all x is certainly not a field, it's a polynomial ring of many variables, some relations. So I need to explain you really what with vectors are. I saw recollection kind of this was recollection of h degrees and now kind of sub-recollections.
36:44
What is this? So at the very least it's a functor from commutative rings to commutative rings. But actually at this point the fact that it's a ring will not be that
37:02
important to me. Let me first describe it as an abelian group. So it's a functor from rings to abelian groups. And there are many ways to present the story but one is the following. You first define something bigger.
37:26
So as an abelian group consider something which is denoted blackboard. So you have start some a now which is commutative ring.
37:46
Let's know the requirement. And then you find some abelian group which is called big universal with vectors and denoted by blackboard w of a. And by definition it's the following. So big vectors. It's basically the most
38:22
naive thing you can do. So you have something which is possibly in characteristic p and the goal is to have something which is nothing nothing nothing related by it. You have a commutative ring, you have an additive group and that's of course the same characteristic as a but you also have the multiplicative group. So let's look at this. Of course multiplicative group of a may be
38:44
very difficult but you can add one formal variable of the denoted by t. So you can see the form of power series in a one coefficient in the one formal variable t and consider invertible power series.
39:02
So there is the map which so power series is of course invertible if its leading term is invertible if and only if. So there is a map which sends something to its leading term and this is split because you can always take any invertible element and a you can take it as a constant power series.
39:23
So this splits as so this is a star plus something else and this something else. Sometimes you can always write as you can see in literature it's written as 1 plus t a. So these are form of power series
39:45
of leading term 1 and the group operation is multiplicative. Now there is a way to look at it which is slightly which looks a little bit too complicated. So instead of considering the
40:06
multiplicative group I can do the form. So instead of a star we can actually consider algebraic k-theory
40:22
but not like higher k-theory but just k1. Take k1. Now the definition of k1 we actually saw yesterday fortunately. It's actually very close to a lower upper star so this is
40:43
a upper star occasionally plus something else but there's something else it can be non-trivial but it will be the same for a and for power series so when you do kind of the difference it will not change. I mean formal definition is you
41:07
take gl infinity and you take its first homology or abelization and you can take determinants so this gives you a map to a star. It's not always an isomorphism but it's not that far and
41:21
it certainly will be kind of relative isomorphism if you look at power series. So in fact you also it's also true that k1 of a is k1 of a
41:40
canonically split into this plus w. Now why is it useful? So I give you I mean we saw the definition of k1 in terms of Jelen but there is also a more invariant way to define it and the point of more invariant definition is that
42:04
it's actually not an invariant of the ring but an invariant of the category of projective finely generated models over k.
42:27
So actually k1 and in fact all k-theory depends on the
42:42
let me denote this by p of a which for me will be the category projective finely generated a modules and particularly its functorial aspect. So you have some functors in this case
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and there is one obvious functor if you have some vectors
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let me now stick with situation a is k algebra for simplicity or some okay then if you have some k-dimensional vector space
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flip to the endomorphism then you consider a functor which just you know just as a product of and it's a functor from a of t from projective models for to itself. So you take your module your module tensor it with a and you
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twist the action of t by this small a here and so this is a functor so it uses a map on on k1
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so for any g a we get an endomorphism of
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example take some integer consider the cyclic group take the corresponding well group algebra
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and dimension vector space and a let a just act by the generator so it's just you know shift just the free action of g mod n all itself kind of permutation of the cycle of other n permutation so
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this is a vector space in the dome so it gives you a well defined map so get
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epsilon let me denote by epsilon n but then if you take a product of v with itself you see that as a vector spatial endomorphism is just a sum of n copies of it right this means that
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this epsilon n is actually almost an idempotent it squares to n times itself and this can be used so if
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so we're now in situation where so it's a over k so assuming that now that k is characteristic p is a that i always assume then it's
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easy very easy to prove and easy to see actually that as a group as an abelian group this w of a is actually p local so everything not divisible by p is invertible
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this means that if n is not divisible by p you have a well-defined endomorphism of your w of a you can divide divide this epsilon n by n this is well defined and this is an
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idempotent so it squares to itself and in fact you can show that the whole w of a actually splits into a copy of so you so you have a family of commuting idempotent endomorphisms
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numbered by all positive integers not divisible by p and in fact what happens is that the whole w of a just splits into a product of copies of a single thing which is
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denoted now by just w without blackboard and this is
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called the group of so this is just if you want the common kernel of all those and so this is known as a group of p-typical with vectors and this is what i want where am i p-typical vectors
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and there's additional theorem that this has a product so it's actually a ring and this is my with vector but for me it's not important that it's ringed now but what is interesting is that it turns out that exactly the same construction now gives you the whole the rhombic context and this was not realized at first but so there is one correction so i was
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working with k1 but we have the whole algebraic theory so we have a n and then of course sorry of course formal power series by definition i'm just projective limits
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of truncated things right to take a of t modulo t and plus one maybe i'm taking inversely with respect to m you can ask whether it commutes with k-theory and it does commute with k1 but not with high k groups so what i want to do i want to find
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uh say completed a of t
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as this inverse limit well it has to be a multiple limit maybe there's some type of things there of k groups of truncated polynomials again there is always the
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augmentation map to the leading term which is split so you can say that this thing splits as k-theory of a plus something
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but you know that x for now exactly the same so again all k-theory
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depends on the category of model so you have the same idempotence kind of relative so this x this second term and quote
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you can also think about it as relative k-theory of this this is p-local so you can always
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also look at the those idempotence i have look at those kernels and the theorem is that this gives you exactly the terms of the the rhombic complex so a is now smooth commutative maybe maybe finitely generated so
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just algebraic smooth algebraic all right i find it for time to enumerate the test is complex and then in degree i this thing is naturally identified to this common kernel seven i on relative k-theory so let me rewrite
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this i take this k complete relative degree i plus one and i take the common kernel
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of all those idempotence epsilon and acting on this guy and this gives you the term of the rhombic complex so the point is i mean the gist of this is that if you just replace k1 with the
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higher k-theory you automatically without thinking are led to crystalline commode the rhombic complex in crystalline commode now this theorem has a long history so actually it was originally conceived by bloch because the paper where he actually introduced the ideas for the rhombic complex it was called on p-typical k-theory but it was maybe even before or at least right
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after quillen's k-theory appeared so he worked with the previous version which was millner k-theory and he could only do it in like small degrees and so on but then the story developed and i think eventually the theorem is new probably to large scale i think people in the world sorry a quick question
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uh is it obvious that the chain complex structure on these relative uh k-hats it's a priori spectrum well when i denote ki i write uh i mean i mean the homotopy group a posteriori it is actually uh but it's not obvious though
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so it is an alberg-machman spectrum it is a chain complex and then so this is one question and then another question is so i get this w individual terms another question is what is the where is the differential right the drum differential and uh this is kind of the next question which i'm going to address
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on thursday and the kind of one line uh kind of punchline is that it comes from circle action on this relative case area but this will really have to do i mean there's no time for that today i will explain this on thursday okay um yeah but the result itself is
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due to lars hasselhoff from i don't know like so he has a paper whose title is very similar to blocks it's called on p-typical curves and quillen's case but even then it uses lots of technology so i'm not really sure what is the good reference for this it's also direct to proof and i'm not going to prove it in these
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lectures but what i'm going to present in these lectures is the way to see that this thing here is actually very computable it's related to some pure and combinatorial invariant which you can compute and then how to identify that to the with complex as a separate story which is not that interesting actually in
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mathematics but kind of the the main thrust will be this and i will explain first the first thing i explain on thursday is how to get the the round differential on this thing so it's really in the terms in the drum will be complex but the whole complex okay now i think i have to run now
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more or less okay let's thank the speaker for today's lecture and so yeah please prepare your questions and comments for thursday and monday lectures please yeah so see you very much thank you thanks