1/3 Algebraic K-theory and Trace Methods
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Transcript: English(auto-generated)
00:17
Okay so I think everybody is back so now so again do not hesitate to ask
00:24
questions just I wanted to say that if you use the Q&A Chan then you can put anonymous questions for those who are shy but you can also use a public Chan if you want. So now it's it's time for Tina Gaard from the University of Michigan and she will speak about
00:45
Algebraic K-theory and trace methods. Thank you. Thank you all for joining me today. So I'm going to be giving three lectures over the next few days and my goal for this lecture series is to introduce algebraic K-theory and the
01:01
so-called trace method approach to studying K-theory and I'm going to talk about some exciting recent developments in these areas. These talks are really aimed at early career researchers like graduate students and postdocs so I'm not going to assume any familiarity with algebraic K-theory and I really want to start right at the beginning of the story. So what we're going to
01:22
start with in the lecture today is to address the question first what is algebraic K-theory? So I'm going to talk about the algebraic K-theory of rings and I'm going to let A be a ring. So I'd like to take a historical perspective for a while and talk about the development of this field throughout
01:44
time so we're going to start really at the beginning of the story of algebraic K-theory. So we're interested in studying rings and when you want to understand a ring one thing you could consider about it is modules over that ring. So I'm going to let P of A denote the monoid of isomorphism
02:05
classes of finitely generated projective A modules. So recall that a projective module is a direct sum end of a free module and I claim that this
02:24
is, excuse me, I claim that this is a monoid which means we have some sum operation so if you take the direct sum of two projective modules you get another projective module. But this doesn't have inverses in it so it's not a monoid but not a group. So what is the zero... Just a question, do you assume that your
02:44
rings are associative, commutative, etc? Associative yes, commutative no. So what's the zeroth algebraic K-theory of our ring? Well it goes back to work of Grothebeek and the definition is as follows. The zeroth algebraic K-theory
03:03
of a ring A is the group completion of this monoid. So we have this monoid and group completing essentially means well I didn't have inverses in it but I can just formally add those in to get a group. So let me illustrate that
03:22
perhaps best with an example. So let's say our ring was actually a field so for a field F what happens? Well first I want to consider this monoid of isomorphism classes of finitely generated projective modules over a field. Well modules over a field are vector spaces and up to isomorphism
03:43
they're classified by their rank. So P of F is just isomorphic to the natural numbers. And so what is the zeroth algebraic K-theory of a field? Well I'm supposed to take the natural numbers which doesn't have additive inverses and formally add them in and what do you get then? Well
04:01
you get the integers. So the zeroth algebraic K-theory group of any field is the integers. So the definition of so that's zeroth algebraic K-theory and the next sort of group to come along in history is the first algebraic K-theory group. So this comes out of work of Whitehead and Bass in the 1950s. And
04:25
here's what the definition of K1 is. So the first algebraic K-theory group of a ring is what you get when you take GLA, the infinite general linear group, and abelianize it. Now it turns out that the abelianization of GLA can
04:41
also be written as GLA modulo a subgroup EA generated by certain elementary matrices. Okay so this is the first algebraic K-theory group of a ring. So in our historical overview we're sort of in the 50s and in the 50s and into the 1960s people started to see that these algebraic K groups, although
05:02
they're defined purely algebraically, started to have interesting applications to topology. So one illustration of that is through the famous S-cabortism theorem. So the S-cabortism theorem was proven independently by Maser, Stallings, and Barden in the 1960s. So what question is
05:28
the S-cabortism theorem aiming to answer? Well it's the following. Let's say we're in dimension at least five, so for n greater than or equal to five, if W is a, let's say, a compact n plus one dimensional cabortism between two and
05:51
manifolds, sorry let me try to spell, between two and manifolds, which I'll
06:00
call X and Y, such that the following is true. When you look at the inclusion of X into the cabortism and the inclusion of Y into the cabortism, that those are both homotopy equivalences. So that's set up where you have a cabortism with that condition that those inclusions are homotopy equivalences. That's called an H-cabortism. And what the S-cabortism
06:23
theorem is addressing is the following question. When is the cabortism trivial? So what do I mean by trivial here? Well the question is asking when is the cabortism really just a cylinder on X? Okay so that's the question of the
06:45
S-cabortism theorem. And it turns out that it's not always the case. There's an obstruction to that being the case. And the obstruction lies in a quotient of algebraic K-theory. In particular it lies in a quotient of the
07:02
first algebraic K-theory group of the group ring on the fundamental group of the manifold X. So this is one of the first instances we see of this purely algebraic construction of algebraic K-theory having really interesting applications to sort of geometric topology. So there was a lot of
07:22
interest in these groups in part because of these applications. And around that same time in the 60s Milner gave an algebraic definition of the second algebraic K-group K2 of A in 1967. In the interest of time I'm not
07:44
gonna go into what Milner's definition of K2 is. But the thing that you should know about it is that it's again a purely algebraic definition in this case in terms of generators and relations for the second algebraic K-group. So one question you might have in your mind looking at this is well I've told you
08:02
that there are these groups K0, K1, and K2. The naming certainly suggests that they're related to one another. But why? Why are we calling all of these things algebraic K-theory? So one answer to that is that at the time we're sort of in the 60s in history at the time there were known relationships between these groups. And there were a lot of them but I want to mention
08:24
one of them. So there were a lot of known relationships between these lower K-groups and here's an example. Let's say we have A a ring and I an ideal of A. You can algebraically define what are called relative groups. So there's a
08:47
way to algebraically define groups K0 of A rel I, K1 of A rel I, and K2 of A rel I. And then it was known that there's a long exact sequence of the
09:05
following form. So there's a long exact sequence that relates this relative algebraic K-group K2 to the K-theory of the ring A to the second K-theory of the quotient A mod I. And then it continues on to the first algebraic
09:24
K-theory, the relative group, the first algebraic K-group of A, and so on. Okay so this is the kind of thing that suggests that these groups are all instances of some common object which we're calling algebraic K-theory. So in the 1960s and 1970s of course the big question became well we have K0, K1,
09:44
and K2 and it turns out they've been really useful for applications to things like topology. And so people were asking well can we define all algebraic K-groups? So the question was can we define KN of A for all N
10:00
greater than or equal to 0? Now what would we want out of this? Well we'd to agree with the known definitions. So we want this to be agreeing with the known definitions of the lower K-groups K0, K1, and K2. But more than
10:21
that I needed to also extend all these relationships that we know between the K-groups. So and extending the relationships between them. Okay so that was the question. Now looking at these properties for instance the one I've
10:40
written down I haven't told you some of these other relationships between them but looking at these properties they start to feel reminiscent of the kind of thing you see in topology. I mean for instance this long exact sequence the kind of thing we're interested where we typically see in a topological setting. So that's not a coincidence. So the first definition of algebraic K-theory is due to Quillen and Quillen did this work in the early
11:04
70s. And here is Quillen's definition of higher algebraic K-theory. Quillen says the nth algebraic K-theory group of a ring A should be the nth homotopy group of BGLA plus when n is greater than 0. So let me unpack
11:22
that a little bit. What is this? Well GLA is the infinite general linear group again we saw that earlier. BGLA is its classifying space so we have this method in topology of associating a space to a group and that's the classifying space. The superscript plus is something that we now call
11:41
Quillen's plus construction. And what that is is it's something that you can do to a space that doesn't change any of its homology but does something very specific to the fundamental group. So it kills a maximal perfect subgroup of pi 1. Okay so this is the first definition of higher algebraic K-theory. We often call this now the plus construction definition of K-theory. Now
12:04
if you go into the literature though if you're interested in algebraic K-theory and you look into the literature you'll see that these days there are lots of definitions of algebraic K-theory, lots of different perspectives on this. And that's in part because now we can take the algebraic K-theory of a wide variety of things. So I'm talking right now about
12:20
the algebraic K-theory of rings but these days there's a notion of algebraic K-theory for spaces, for ring spectra, for varieties and schemes, for exact categories, really for any category with a notion of co-fibration and weak equivalence, etc. So we can take algebraic K-theory of a
12:41
lot of different kinds of objects now and partly for that reason there are lots of different ways of defining algebraic K-theory. But for any of those definitions if you restrict your attention to rings it still agrees with Quillen's original plus construction definition. So this is a perfectly fine way to think about the algebraic K-theory of rings. So Quillen made
13:02
this definition of algebraic K-theory and at the same time he did a really beautiful calculation which is the following. Quillen computed the algebraic K-theory for all finite fields. So he computed that the algebraic K-theory for the finite field with q elements is, well we've already seen that it's the integers in degree 0 but he computed that it's z mod q to
13:25
the i minus 1 for n equal to i minus 1 and it's 0 when n is greater than 0 and even. And I should note that Quillen later won the Fields Medal for his work in algebraic K-theory. Okay so now we're in the early mid 70s in
13:43
our historical overview and you know we're feeling really good about things because now at last there's this definition of higher algebraic K-theory and frankly it looks simple enough, right? It's supposed to take home Joby groups of some space. And so then of course people tried to start computing it. We have Quillen's beautiful calculation for finite fields
14:01
and we'd like to compute the algebraic K-theory of other rings. So what else might we ask for the K-theory of? Well Quillen tells us the K-theory of maybe we should look then at the K-theory of z mod p to the k. There's a funny story about this that one of the senior members of my field told me which
14:21
is that he was advising PhD students in the mid 70s and he gave this to one of his PhD students as a thesis problem. You know compute extend Quillen's work compute the algebraic K-theory of z mod p to the k. The PhD students struggled, had a really difficult time with it and ultimately just really got stuck. And you know this is not a sad story. The student moved on to a
14:44
different project and eventually graduated. This happens of course and I think at the time the advisor also realized that maybe he didn't know how to solve this this problem with algebraic K-theory and it turns out that now in retrospect almost 50 years later we know that this was a
15:00
really bad thesis problem for a PhD student in the 1970s because still today nobody knows how to do this. So this is still unknown. Okay so maybe that was too ambitious. You know maybe we should just start with the K-theory of z mod p squared. You know some things are known about that. It's known in low degrees and there's some stuff known about it but in some broad sense this
15:23
is also not known. Now what's another brain that we might really be interested in the K-theory of? Well we might be interested in the K-theory of an enormous body of work that has been you know going on for 50 years and a
15:41
lot is known about this now but this is still not completely known and I will come back and say more about that in a few minutes. So if this were not my field and if I was in some other field and I was looking at this I mean something that I might think to myself is you know so why are we trying to compute this? I mean just because we can define a ring invariant doesn't
16:04
necessarily mean that it's a good idea to study it and I've just argued for you that algebraic K-theory is really difficult to understand in even these like really pretty simple situations and so maybe this just isn't a useful invariant of rings. Well so why do people study algebraic K-theory? I mean
16:22
one answer is of course that the mathematics around it is beautiful but maybe even larger than that algebraic K-theory is one of these sort of magical objects in mathematics that appears across mathematical fields. So K-theory although you know we've seen it in a variant of rings it has
16:42
applications to many areas of mathematics including homotopy theory, number theory, algebraic geometry, geometric topology and so part of the interest in K-theory and K-theory calculations is because of those connections. So there's a question maybe can you say what about the tools
17:02
that Quillen used to compute the K-theory of finite fields? Yeah so right it's very it's a very interesting story Quillen sort of well I don't know I wasn't even born then so maybe I shouldn't claim what he was thinking but his work on the K-theory of finite fields came up in work that he was doing on another conjecture having to do with the image of what's called the J
17:24
homomorphism and it turns out that he was able to identify this BGL FQ plus as a kernel of some Adams operations on related to topological K-theory. So part of the reason that it's difficult to sort of extend Quillen's
17:43
work to other situations is because that because it was sort of specific to the situation of these finite fields. So it's beautiful work and by what he does is essentially he's able to identify this space BGL FQ plus as the kernel of something having to do with Adams operations and then that's much easier
18:02
calculationally to get your hands on. So that's sort of the idea but also that doesn't extend broadly to other K-theory calculations. Right so what I was saying is I was claiming that algebraic K-theory has these connections to other fields of mathematics and I'd like to tell you a couple stories about what I mean when I say that it has connections to
18:22
other fields. So let me give you an example which comes from number theory. So this example is a conjecture and it's called Vandiver's conjecture. So it's called Vandiver's conjecture but actually it's due to Coomer in 1849
18:42
and he wrote this conjecture in a letter to Kronecker and here's the conjecture. It says for prime P let's let K be the maximal real subfield of
19:00
the rationals adjoined a primitive P through to unity. Then the conjecture is that P does not divide the class number of K. So I'm not going to dwell on what any of that means. It's sort of not necessarily important for what I'm saying
19:20
today but what I want to say about this conjecture is that this is a very important conjecture in algebraic number theory. So this is the kind of conjecture that people you know prove theorems based on you know assuming Vandiver's conjecture is true then blah blah blah blah blah. So this would be a huge step to be able to establish Vandiver's conjecture. It is thought
19:41
that if there are counterexamples they are indeed rare so it has been verified computationally for all primes less than two billion one hundred and forty seven million four hundred eighty three thousand six hundred and forty eight which is not just some random number that's two to the thirty first. So if there are any counterexamples that have to be the
20:02
primes would have to be bigger than that. There is I should know that there there was a couple months ago a claimed proof of Vandiver's conjecture on the archive. I haven't had a chance to look at that too much myself and since it's not published let's consider Vandiver's conjecture to still be open. So this has been a long-standing conjecture in algebraic
20:25
number theory. So why do I mention this in the context of what I'm talking about today? Well Kurahara proved in 1992 that Vandiver's conjecture is equivalent to another conjecture. So Kurahara proved that Vandiver's
20:40
conjecture is equivalent to another conjecture and that conjecture is that the K for M the algebraic K theory groups of Z of the form K for M are all 0 for M greater than 0. This to me is fascinating. I mean if you think
21:02
about that algebraic K theory of the integers it was defined as some homotopy groups of some space and when you look at Vandiver's conjecture here you know you don't see any homotopy theory. I mean you don't even see the integers right? So this is sort of a fascinating connection between algebraic K theory and a deep conjecture in algebraic number theory.
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And it's telling us that the K theory of the integers is somehow capturing some very deep arithmetic information. Okay so that's one kind of connection that I mean when I say algebraic K theory appears in unexpected places. Let me also mention connections between algebraic K theory and algebraic geometry.
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So you know one of the subjects of this summer school is motivic homotopy theory and the connection between algebraic K theory how it appears in algebraic geometry is related to motivic homotopy theory. So for those who aren't motivic homotopy theorists in the audience you know what
22:01
is broadly speaking the idea of motivic homotopy theory? Well you know it was an effort originally due to Morel and Voivotsky to bring tools from topology into the study of objects in algebraic geometry. So I'm going to make a very rough dictionary of what that might look like. So I'm gonna make a
22:21
little dictionary here of topology and algebraic geometry. And what am I gonna put in my dictionary? So let's think about our first course in algebraic topology. What do we study? Well we're interested in studying invariants of spaces. And what's usually the first invariant of spaces
22:40
that you learn? Well it's usually singular cohomology. Now when you go on in algebraic topology you learn well they're actually generalized cohomology theories and so maybe we also learn about the invariant topological K theory which has to do with vector bundles on our space. Now if
23:01
we want to translate these kinds of tools into algebraic geometry what are the analogs going to be? Well maybe in topology I said my fundamental objects were spaces but in algebraic geometry we might want to consider varieties. And whereas we had singular cohomology in algebraic geometry we have a cohomology theory that we can use to study varieties called motivic
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cohomology. And then the claim that I want to make is that the analog of topological K theory in topology in algebraic geometry is actually algebraic K theory. So what is the content of that statement? Well one way
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to think about what the content of that statement is is well one one tool we have in topology that relates singular cohomology and topological K theory is the Atiyah-Hertzberg spectral sequence. Atiyah-Hertzberg spectral sequence and what is that? Well
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roughly speaking it's a spectral sequence that has e2 term in singular cohomology and converges to topological K theory. So there's a large body of work I'm trying to establish the algebraic
24:23
geometry analog of this kind of Atiyah-Hertzberg spectral sequence and it turns out there is such a spectral sequence which is often called the motivic spectral sequence which has e2 term in motivic cohomology and converges to algebraic K theory. So in other words the fields of
24:41
algebraic K theory and motivic cohomology really inform one another through these kinds of tools. Okay so those are just a couple of stories of how algebraic K theory is appearing in these different branches of mathematics. We also saw one through the S-cobortism theorem of how it appears in geometric topology. So hopefully I've convinced you that these algebraic K
25:02
theory groups although they're difficult to compute and challenging to understand are important and interesting to study. So what I want to move on to then is a different question which is well how are these groups computed? So how are algebraic K groups computed? With that I'm also
25:26
going to take a bit of a historical perspective and sort of go through history and talk about the development of this method of computing that I want to focus on in this series and I'm going to drag an example through history with me just so you can see really concretely how you
25:40
know at different points in time the computational powers changed. So the example that I'm going to drag with me is the algebraic K theory of Z join X mod X to the M. So this is truncated polynomials over the integers. Okay so in the 1960s and 1970s if you look at the K theory literature from that time
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what you see is you see a lot of low dimensional calculations. So around that time most of the literature is low dimensional calculations using algebraic tools. So what you see is a lot of calculations of k0, k1, k2 even after
26:21
Quillen's definition of higher algebraic K theory because it wasn't really understood how to compute with that. So for example in this example that I'm going to carry with us there's a theorem due to Geller and Roberts from 1979 where they compute the second algebraic K theory group of Z
26:43
join X mod X to the M, rel X and they compute that that's Z mod 2. So let me just say a brief thing about this relative group. This K theory of Z mod X to join X to the M has a copy of the K theory of the integers sitting in
27:01
it and the relative K theory is basically saying what do you have beyond that. So that's what they're computing is what's what do we have besides a copy of the K theory of the integers. Okay so you know that's a calculation of the second algebraic K group and they do that using the algebraic definition of k2 but we'd really like to understand all these
27:22
higher algebraic K groups. So really our goal is you know for a ring A we want to be able to compute the algebraic K theory of A for all Q greater than or equal to 0. That's what we'd like to be able to do. And I claim that there's a tool that's going to help us do so. So I claim that a
27:43
tool that can help us and has helped over the years in these calculations. I'll try to in someone raise hand. Sorry maybe can you ask your question to Q&A and then go on. I claim that a tool that's gonna help us do this is
28:05
equivariant stable homotopy theory which is also one of the themes of the summer school. So what is equivariant stable homotopy theory for those that are not familiar? Well it's an effort to study spaces or spectra that have a group action on them. Now if you think about that too hard you may be a bit
28:23
surprised because I've defined K theory already. I've talked about the question that we're trying to address and nowhere were there any group actions right? This wasn't an equivariant question and that's a fascinating thing that we're seeing a lot recently is that sometimes equivariant tools can be used to address questions which on the surface are not
28:44
equivariant questions. So algebra K theory it turns out is going to be a great illustration of that but another one that you might be familiar with is Hill Hopkins and Ravenel's solution to the curverin variant 1 problem. So that's a classical problem in differential topology that really is
29:01
not an equivariant question but it turned out that they use z mod 8 equivariant stable homotopy theory to solve that. So that's sort of a surprising thing. The specific approach I want to tell you about is something called trace methods and the trace methods approach originally comes out of ideas of Goodwillie and it's also due to Bocksted, Chang and Madsen. Okay so
29:31
the idea of the trace method approach is simple enough the idea is the following. Algebraic K theory is really hard to compute directly from the definition so maybe we can approximate it by something that's more computable.
29:43
Maybe there's a question for in the new preceding example is the relative k1 trivial? Oh no I just picked k2 as an example of the of a low
30:04
dimensional calculation there were other low dimensional calculations k0 and k1 were also known at that time although I couldn't off the top of my head give you the citation for who did those calculations back in the 60s but it was known. Okay so what's the idea of trace methods? Well the idea is that we
30:23
want to approximate K theory by something more computable and we'll see that what we're gonna do is we're gonna make successive approximations to move closer and closer and closer to K theory with the idea that hopefully eventually we get close enough that we can actually recover something about
30:42
the K theory itself. So what is our first approximation going to be? Well our first approximation is going to be via another familiar ring invariant which is something called Hochschild homology. So let me tell you what the definition of Hochschild homology is. So I'm going to let Bsic of A denote
31:06
what's called the cyclic bar construction on my ring A. So what is that? Well it's a simplicial abelian group and it's defined as follows. So
31:23
for a simplicial object I have to tell you what are the simplices. So at Rth level this thing is R plus 1 tensor copies of A. And then a simplicial object I have to tell you what it is at each level and also the face and degeneracy maps relating them. So what are those maps? Well the face maps,
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the ith face map, takes a tensor A0 through AR and the way to think about these face maps is most of them just take two adjacent elements and multiply them together. So the ith face map takes Ai and Ai plus 1 and multiplies them. And that makes sense as long as i is less than R. What
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does that last face map do? Well that last face map is going to bring the last element around to the front and then multiply. So it's going to be AR A0 tensor A1 through AR minus 1. Okay so those are the face maps and we
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also have degeneracy maps. So the degeneracy maps just insert the unit after the i coordinate. So Ai tensor 1 tensor Ai plus 1 and so on. So inserting the unit and that makes sense for all i. Okay so we have these face
32:42
and degeneracy maps and it's not helpful maybe to have a picture in your mind of what this looks like. So what I'm saying is that my simplicial object looks like this. I've got A and then A tensor A and then A tensor A tensor A and so on. And I have these face maps in this picture. They're going downwards, downwards direction. I have a bunch of face maps and then the
33:02
degeneracies go in the opposite direction. Okay and that picture continues. Now I wanted to notice, note that this that this thing also has a cyclic operator on it. So it has a map TR so on the rth level it has a map that's
33:22
going to take my tensor and just rotate the last element around to the front. So this goes to AR tensor A0 through AR minus 1. Okay this is the cyclic bar construction. Remember my goal was to tell you what is hock shield homology. So what is the hock shield homology? Well from this cyclic bar construction I
33:42
can get a chain complex. I'm going to call it C of A. And what is it? Well it's this cyclic bar construction. But right now I have too many maps. It's not a chain complex the way I've presented it so far. But I claim that if you take the alternating sum of these face maps that that squares to
34:03
zero and that gives you a chain complex. So you can check the combinatorics of that yourself if you want. And so what is hock shield homology of a ring? Well hock shield homology of a ring is just the homology of that chain complex. Now the Dold-Kahn correspondence tells us
34:21
that we have another way of characterizing this which can also be useful for us. The Dold-Kahn correspondence tells us that the homology of that chain complex is also the homotopy groups of the geometric realization of that simplicial object. Okay so that's what hock shield homology is.
34:40
Let me make a note about this which is the following. So I said that this was a simplicial abelian group and it is but it's actually more than that. So this cyclic operator that I talked about here that's not part of the structure of a simplicial object. That cyclic operator actually makes it into what's called a cyclic object. So the cyclic bar construction, hence the
35:04
name, is a cyclic object. And why is that important for us? Well it turns out that by the theory of cyclic sets due to Kahn that when you geometrically realize a cyclic object it has an S1 action.
35:24
And it turns out that S1 action is going to be important for us. So this is the first time we're seeing a group action. Maybe the equivariance is starting to come in but it will be later we'll see why that's important. But I wanted to mention it here that we have that. Okay so the goal remember was that I was trying to approximate algebraic K
35:40
theory. So the next question is well in what sense is this an approximation? So there's a map from algebraic K theory to hock shield homology and that map is called the Dennis trace. Maybe I won't go into total detail
36:01
about that but let me give you some idea of why you'd have some map like that. So the rough idea is that this Dennis trace map is induced is induced by a simplicial map that looks like the following. So remember algebraic K theory was something about BGL
36:22
A and I have a simplicial map from the nerve on GLNA to the cyclic bar construction on n by n matrices with entries in A and then there's a map I claim from there to the cyclic bar construction on A. So what are these maps? Well what is
36:44
something what does an element of this nerve look like? It looks like an r tuple g1 through gr in simplicial degree r. Now that thing needs to map to an r plus one tuple in the cyclic bar construction and what it maps to is it maps to you take the product g1 g2 through gr
37:03
the product of all of them and take the inverse and then tensor g1 tensor through gr and so that gives you a map from that gives you this left hand map and then what is the right hand map? Well it's some kind of multi-trace and that's where the trace comes into
37:21
the trace methods is from this kind of trace construction. So I'm asking the question maybe a bit late but someone asked why do you use trace to call this map? Yeah right so here it is. So this map is really a common notion of a multi-trace so this map from n by n matrices
37:42
sorry this map on the cyclic bar construction is induced by a map from these tensors of n by n matrices to tensors of elements in A that is sort of a classical multi-trace where you sum over a bunch of indices and take some trace here. I'll get the formula wrong if I try to write it off the top of my head but it's a classical sort of multi-trace and
38:02
that's where the trace comes from the sort of trace methods. Now of course this is a map from gln and we really wanted a map from the infinite general linear group so you need to take some co-limit as n goes to infinity and then also you need to pre-compose with a harevich map because we wanted a map from homotopy groups so there's a bit more to it than what I've written but this is what induces that dennis
38:22
trace. So the plus minus on the dennis trace is the following um so uh on the one hand we were trying to compute algebraic k-theory by something that's more computable and hochschild homology is much more computable than algebraic k-theory so that's good um the downside is that it's just not a very good
38:43
approximation and maybe that's really not so surprising because algebraic k-theory was this invariant defined using homotopy groups um you know really it was a topological invariant and hochschild homology is as we just saw something purely in homological algebra so we wouldn't necessarily expect hochschild homology to capture all the
39:04
deep information in algebraic k-theory but it turns out there is something that we can um we can learn uh about algebraic k-theory from even this algebraic perspective. So Goodwillie in the 1980s um showed the following so Goodwillie
39:25
proved that the dennis trace lifts through what's called negative cyclic homology um so in other words my I had this map from the algebraic k-theory to the hochschild homology and Goodwillie
39:42
proves that actually factors through another invariant which is called negative cyclic homology and he further proves that rationally uh negative cyclic homology is in some situations a good approximation to algebraic k-theory. Okay so let me unpack that a little bit.
40:04
I have a quick question here so can you interpret this multi-trace map as uh using dualizable object and things like that or monoidal category? Yeah I mean certainly people have
40:27
thought about like sort of the categorical versions of these kinds of traces there's even um been recent work on that by uh like Kerry Malkovich and Kate Ponto and that group of people um I don't I'm I'm not going to be able
40:44
to make a concrete statement on the top of my head but yes there are definitely people thinking about sort of categorical versions of this in my lectures tomorrow and Thursday so uh excellent thank you um so uh
41:02
right so let's unpack Goodwillie's theorem a little bit so negative cyclic homology I'm not going to write out the full um definition of what negative cyclic homology is for you but uh the thing to know about negative cyclic homology is that it's another classical invariant from homological algebra it's purely algebraically defined you actually define it using the hochschild complex
41:22
but you make it into a bicomplex and um do some other things to it but it's purely in algebra now when I say rationally it's a good approximation to algebraic k-theory what do we mean well I could make it a little more precise so in particular he proves that if we have um i and a a nil potent ideal
41:43
that when we look at the relative algebraic k-theory of a rel i once you tensor that with the rationals then you get an isomorphism to uh this negative cyclic homology so in other words when you tensor with the rationals what happens well you
42:02
retain information about the rank of your algebraic k-theory group but you lose all information about the torsion so what was happening in the 1980s is that various authors were developing tools to compute algebraic k-theory by computing this negative cyclic homology but that what that's going to tell you is it's going to tell you about the rank
42:22
of the k-theory groups so if you look at the 1980s you'll see in the literature a lot of work on the rank of k groups using this kind of tool using good willy's theorem so for example if we look at our our sort of sample calculation that we're taking with us throughout this historical story there's a theorem from of soulez from 1981
42:44
where soulez computes the rank of the algebraic k-theory of z adjoint x mod x squared again the relative groups and he proves that this is rank one if q is odd and zero if q is even that was generalized in uh 1985 by
43:03
stash f and stash f looked at uh the k-theory of z adjoint x mod x to the m so not just x squared and proved that the rank of that is m minus one if q is odd and zero if q is even so in the interest of transparency
43:21
soulez soulez proof actually doesn't use good willy's theorem uses some other methods to get at this stash f does use good willy's theorem in order to generalize this to see a joint x mod x to the m so that's where we were in the 1980s which was that there are a lot of calculations that were able to be done at that time of the ranks of k groups but the big question then
43:40
became how can we understand the torsion so in 1990 good willy gave an address at the icm um based and based on results of himself and wildhausen and others he conjectured that there should be what he called a brave new version of this story so what do people mean when they talk about
44:02
brave new algebra well brave new algebra is the idea that you know we have all these classical constructions that we know and love in algebra and perhaps we can define topological analogs of those constructions to translate those objects from algebra to topology
44:21
and those topological um analogs may have some deep uh information in them that were lost in the algebraic context so what good willy conjectured is that there should be topological analogs of hawkshield homology and negative cyclic homology and that those ideally would capture information about the torsion of
44:41
algebraic k-theory as well so that vision became a reality not so long after and the first one of these topological analogs is called topological hawkshield homology so what's the idea behind topological hawkshield homology well it's we want to do this brave new algebra
45:02
perspective we want to we want to take something from algebra and ask for topological analog of that in order to do that i need to think about what do they really need in algebra to define hawkshield homology well hawkshield homology was an invariant of rings and to form that cyclic bar construction i took tensor products of my rings i didn't write it in the notation
45:22
but implicitly i was always tensoring over the integers and once i had that i was able to define the cyclic bar construction on a ring maybe i'll put a tensor in the notation to note that i use tensor product and from the cyclic bar construction we were able to recover hawkshield homology so that's what happened in algebra now i'd like to bring this into
45:43
topology and how am i going to do that well i'd like to replace rings with a topological analog and that topological analog is what we call ring spectra so these are topological analogs that have sort of a multiplication in the sense of a ring and uh tensor product then is going
46:01
to become smash product in this topological setting and so one way of thinking about what's happening is that i'm changing my ground ring instead of working over the integers i'm changing my ground ring to work over the sphere spectrum once you've done that if that all makes sense then the cyclic bar construction you can just define in the same way that we
46:21
did before so we can define the cyclic bar construction by now smashing together ring spectra but it's the same definition of that type of simplicial thing and from that cyclic bar construction then i should get some topological version of hawkshield homology which is called thh now the first person to sort of execute that idea was box dead um box did this quite a while ago so
46:45
he didn't have a lot of nice things that we have today um like uh nice categories of ring spectra with associative smash products so he had to do a bit more work but he was certainly trying to execute um this idea of translating this algebraic construction into topology so
47:03
there's a topological hawkshield homology is supposed to be yet another approximation to algebraic k-theory so there is a map relating them which is often referred to as the topological dennis trace so that's a map from algebraic k-theory to topological hawkshield homology and if you're following really
47:23
carefully you might be a little bit confused about what i just wrote because i set up here the topological hawkshield homology is something that we do to ring spectra and then right here i wrote topological hawkshield homology of an actual ring so what does that mean well let me just note that i'm going to continue to write that when we write topological hawkshield homology of an actual ring
47:44
what we mean this is just notation for topological hawkshield homology of the eilenberg mclain spectrum of that ring so to a ring we can associate one of these ring spectra called the eilenberg mclain spectrum but for i don't know ease of notation we just write that as thh of a okay so topological hawkshield
48:05
homology it is uh it is an approximation to algebraic k-theory it's better than hawkshield homology as an approximation so we're moving in the right direction but if you think about good willy's theorem there was that that negative cyclic homology that was quite close to k-theory rationally we'd like to
48:20
have a topological analog of that as well so in order to develop that we need to notice a few things about thh so topological hawkshield homology of a has an s1 action for the same reason hawkshield homology did we're geometrically realizing something cyclic and really you can define this as
48:42
what's called a genuine s1 spectrum so we're going to get more into that tomorrow of what that means and thinking about this as a spectrum with an s1 action versus a genuine s1 spectrum but for now let's just think it's a topological thing with an s1 action and we'll unpack what that really means tomorrow so we're seeing again that equivariance
49:03
now it turns out the topological hawkshield homology also has what's called a cyclotomic structure so for today i'm going to black box this uh cyclotomic structure bit um not because it's not important but because i'm going
49:21
to really talk a lot about that tomorrow um we're going to see today where it comes in and tomorrow i'll tell you really what it means but it turns out that that's going to be something that's really crucial to this story so we'll talk about that in depth tomorrow so how does that help us the topological hawkshield homology is this kind of equivariant object well we want to define a version of cyclic homology in this topological
49:45
setting and that will be called topological cyclic homology so i'm going to tell you now the classical perspective on this which is due to bokstad shang and madsen but let me mention that there's some amazing recent work of thomas
50:03
nichlaus and peter schulze that sort of reframes this definition of topological cyclic homology and we're going to talk about that tomorrow about the nichlaus schulze work and how it relates to this classical definition but for now let me tell you how topological cyclic homology was defined classically so what did bokstad shang madsen do well they said let's look at a cyclic
50:25
subgroup of s1 so topological hawkshield homology has a group action and one thing you can ask about when you have a group action is you could ask about fixed points so i could look at the cp fixed points of topological hawkshield homology now i might ask well how is that
50:44
related to the thing that i had before i fixed anything and one answer is well i could include the fixed points so that gives me a map f which is inclusion of fixed points okay so i'm going to draw a picture so we can keep track of what's happening here so i had algebraic k
51:02
theory and i had a map the topological dennis trace to thh and now i'm saying well okay let's look at the cp fixed points of thh and i said well there's a map this f map that is relates the fix cp fixed points to the original spectrum but actually i could keep going like that right i mean i have a map i
51:24
could include the cp squared fixed points into the cp fixed points and so on now it turns out that there's also a second map which i'll call r which relates the cp to the n fixed points
51:41
to the cp to the n minus one fixed points this map which we will discuss um at length and tomorrow is called the restriction and i'm not going to define it for you today there's not a nice one sentence thing way to say what it is the same way there is for this f map but the thing to know about the restriction for today is that it uses
52:03
the cyclotomic structure so in order to define this map you need to know that you have that thus far mysterious cyclotomic structure that i referred to so we have an r map like this and box at shang and madsen proved that this
52:20
topological dennis trace this map that i've got down here actually lifts through all of these fixed points and then they so i give you this diagram on route to defining topological cyclic homology so here's the definition of topological cyclic homology so box dead shang and madsen defined
52:42
topological cyclic homology of a at a prime p is the following i'm going to look at all of these fixed points of thh and then i'm going to take a limit in some homotopy theoretic sense over all of these maps so in other words in my diagram topological cyclic homology lives somewhere up here
53:02
and they proved that this topological dennis trace lifts all the way to a trace to tc so this is box dead shang and madsen defined what we now call the cyclotomic trace from the algebraic k-theory to the topple try that again to the topological cyclic homology
53:25
of a okay and then the thing that makes makes this trace method so powerful is that in nice situations tc is a good approximation to algebraic k-theory so there are lots of what we call
53:44
comparison theorems that make that more precise like what do i mean when i say it's a good approximation that's quite vague so there are lots of different comparison theorems depending on the exact calculations you're interested in but let me tell you one of them that's very powerful which you know one way of thinking about this story is that we wanted an integral analog of that good willy
54:04
theorem to tell us about torsion and there is such an analog due to dundas good willy and mccarthy and what does their theorem say it says well when you have a nil potent ideal then when you consider the relative algebraic k-theory
54:25
and compare it to you can define relative topological cyclic homology that that's actually an isomorphism so that's the best kind of thing we could ask for the topological cyclic homology actually completely captures the algebraic k-theory
54:41
so that may seem kind of abstract you know we went through um equivariant homotopy theory we need to understand all these fixed points but i claim that this can yield really concrete calculations in algebraic k-theory so let's first look at that uh computation we've been dragging through history and see what uh history tells us now so where we left it in the 80s through
55:01
sulei's theorem we were able to understand the rank of those k-groups but not anything about the torsion and so excuse me i have a question so the question is to go from tcap to tc of a did they use originally a fracture square definition so i'm not sure yeah right okay so i've been
55:21
i've been uh careless maybe so um if you look at my diagram i defined topological cyclic homology at a prime and then in terms of the cyclotomic trace i gave you a statement that sort of for all primes at one time you can make a similar diagram where you do it for all primes at one time instead of just one prime but then you have these restriction and f maps anytime you have division
55:43
um and it's just messier so i wrote it one i wrote it for one prime at a time um but you can reassemble the like the t c of a at p essentially captured is like what's happening at the prime p and yes you can reassemble the sort of integral information um from that so uh often we work one prime at a time
56:03
in this area just because it makes the calculations simpler um but you can write the same kind of diagram and the same story to define it um all at once um okay so what i was gonna do is i was gonna revisit that calculation that we
56:21
sort of brought with us through the talk and uh through work of big like angle bite myself and lars hessleholtz um we looked at this calculation again using sort of modern trace methods and equivariant stable homotopy theory and we proved that the odd algebraic k theory groups z to join x mod x to the m rel the ideal generated by x
56:42
so sule told us that those groups had rank m minus one or actually stash up told us those groups had rank m minus one but they may or may not have torsion and we proved that indeed there's no torsion in those groups and then the even groups for this uh same ring sule's theorem tells us that it's rank zero
57:02
but it may or may not have torsion and we proved that there is torsion and that the order of that torsion is mi factorial times i factorial to the m minus two so this trace method approach can give you really concrete calculations in algebraic k theory so that's the particular calculation that we sort of brought with us through
57:21
the talk to see the development but i want to mention some very other very important calculations that were done via this method so shortly after the cyclotomic trace was defined bokstad and madsen um computed the algebraic k theory well for primes greater than two they computed the algebraic k theory of the p-addicts
57:42
let me just note that i don't know how to note this but in their original work there was a conjecture for part of it and the conjecture was due to seletus so let me say that plus seletus for a conjectural piece of it um rognus revisited that for the prime two so he did the analogous calculation at the prime two um a number of years ago now has the whole madsen used this kind of
58:03
trace method approach to study uh the k theory of local fields some local fields um and they proved the quill and lippenbaum conjecture in those cases so that was a while back um and many others i mean i could go
58:27
i could go on there are lots and lots of important algebraic k-theory calculations that have been done with this trace method approach let me just note uh and i've been focusing on rings the k-theory of actual rings but these tools this topological hochschild
58:43
homology topological cyclic homology are naturally defined for ring spectra so you could also ask about um about more generally about ring spectra not just rings so for instance there's a body of work by osani and rognus um where they look at the algebraic k
59:02
theory of topological k-theory so a little k u here is the connective topological k-theory spectrum okay so there have been lots and lots of algebraic k-theory calculations using these trace methods and it's been very very fruitful so let me just close by mentioning what i want to do tomorrow in my next lecture so what i've said so far is that well you know the trace
59:27
methods allows you to reduce your k-theory calculation in good situations to computing topological cyclic homology and uh and that's good but the claim is that topological cyclic homology is supposed to be more computable right that's only helpful if we
59:43
can compute tc and so tomorrow i'm going to tell we're going to talk about well how do you actually compute tc so how does one compute topological cyclic homology what tools do you use and we're going to see that that leads us to really needing to understand this idea
01:00:00
of what it means for a spectrum to be cyclotomic. So we're gonna talk about cyclotomic spectra. And what we're gonna see when we do that is that these above calculations that I've mentioned, all these calculations that I've mentioned here, use some really serious equivariant stable homotopy theory to do those calculations. But there's beautiful new work
01:00:21
of Thomas Nicholas and Peter Scholza, which gives us a new perspective in this area that allows us to actually move away from some of that equivariance. And so I'm also tomorrow gonna talk about the Nicholas Scholza approach to cyclotomic spectra and topological cyclic homology. And we'll see some specific calculations where that can be advantageous
01:00:41
to think about it in that way. So that's the plan for tomorrow. But I think that was everything that I wanted to say for today. So I will stop there. Hey, thanks a lot, Tina. So now we have a question. So I'm sorry, I could not ask all the questions directly
01:01:03
during the talk because it was too much. But if you have matured your question, you can now ask it to Tina and I will relay it. So there's one question about the comparison of, can you use about Quillen's computation? So can you use topology, to TC, for example,
01:01:23
to recover Quillen's computation of the K-theory finite fields? Yeah, so when you, yes. So you can recompute the algebra. I mean, you can recompute for instance,
01:01:42
the K-theory of FP completed at P easily using, easily is the wrong word. It has been done using the trace method approach for sure. And that's written up nicely in some old notes of Eve Madsen. So he, what is it called?
01:02:02
What are his notes called? Something like algebra K-theory and traces in that calculation is written out explicitly. So yes, you can recover the K-theory of FP completed at P using this kind of trace method approach. Okay, so another question linked, do we have cases where TC and K-theory coincide?
01:02:24
Yeah, so the Dundas-Goodwillie-McCarthy theorem is giving us cases like that. So for instance, one of the calculations that I, right, so I mean, so the situation it's capturing
01:02:41
is sort of things like you see in this theorem here, where you have relative algebra K-theory, rel some nil potent ideal. And in those cases, the TC and the K-theory do coincide exactly. There are other theorems, I'm telling you one than this Goodwillie-McCarthy theorem. And there are other theorems due to people like
01:03:01
Hasselholtz and Thomas Geiser. And I'm gonna forget if I try to list everybody, but there are lots of comparison theorems in specific cases. And what usually happens is that depending on what kind of calculation you're doing, you may look for maybe a slightly different comparison theorem. But the hope is that you have a comparison theorem that either tells you that they coincide directly
01:03:22
or is gonna give you some information about the relationship between the two. But you probably have to be complete, right? K-theory, because if you look at the prime, it's a different story. Yeah, I mean, so done this Goodwillie-McCarthy, so the original Dundas-McCarthy theorem,
01:03:41
McCarthy theorem, involved a peak completion. And then there's, they've sort of, their most recent version of this theorem that I mentioned up here is an integral theorem. But maybe to get to your point of what you were mentioning about the peak completion is that, you know, lest I've left you believing
01:04:00
that trace methods is gonna solve all of our problems, let me make a comment, which is that, you know, you look at this, I claim that many years ago now, Bakshin and Madsen compute the algebraic K-theory of the P-addicts. So you might look at that and feel very optimistic then about the algebraic K-theory of the integers, right?
01:04:20
Like surely that should be closely related to the algebraic K-theory of the P-addicts. But unfortunately, algebraic K-theory doesn't behave nicely with respect to peak completion, the way topological cyclic homology actually does. So when they compute this, they look at the cyclotomic trace to topological cyclic homology, and that cyclotomic trace is an isomorphism.
01:04:43
Now it turns out that topological cyclic homology, that for TC, this kind of like peak completion inside and outside behaves nicely. And this is the same as the topological cyclic homology of the integers P-completed. So one might hope, well, I have a map from the K-theory of the integers to TC.
01:05:02
So I have a map like that, which is the cyclotomic trace. And so you might look at that and think, well, then wouldn't it be nice if I could say something, you know, nice about how these two are related to each other. But K-theory doesn't share this property that TC does, that this peak completion inside and out
01:05:22
gives you this isomorphism. So you have to be a bit careful. So for instance, Box-Sachang-Matson's result is not gonna tell us about the K-theory of the integers and the K-theory of the integers is in part, I'm not gonna say intractable, but part of why it's difficult is it's not very approachable via this method because in that case, I've highlighted too many things,
01:05:44
in that case, this cyclotomic trace map over here is not an equivalence or something that we can really understand very well. Okay, so next question is, is there an extension of TC-THH2 schemes?
01:06:03
Or nice schemes? Yeah, yes. So you can study topological cyclochromology and THH for schemes. I don't know that I have a lot of good things to say, tell you about that, but I will give you a reference, which is that work of Lars Hessel and Thomas Geyser,
01:06:21
for instance, looks at how to study these things in the algebraic geometry setting as well. So next question is in the theorem on Galois and Hessel-Holt and it is that Geyser or no, or good really? Oh, it's me.
01:06:41
It's a, it's a, sorry. Is there, do you know the group structure on the torsion rather than the order? Right, okay, so when we look at this theorem that I mentioned, the Wieglich angle by myself and Lars Hessel, we compute the order
01:07:02
of these groups and even dimensions, but I haven't told you what the group structure is. No, we don't know the group structure and using the sort of methods that we use, that question is very intractable. And I'm trying to see if I can explain why using what I've said so far. So it turns out that these calculations
01:07:21
that I've done here, and we'll, maybe it'll be more clear tomorrow why this is the case, but these calculations are gonna reduce to some equivariant homotopy calculations in ROS1 graded equivariant homotopy theory. And those calculations that go into this K theory calculation, with current technology, there's no expectation
01:07:42
that you'd be able to recover the group structures. And so we're able to like do some tricks to get the order of the groups, but we don't know the group structures except in low dimensions, like I couldn't do it off the top of my head, but for some small values of I here, you can explicitly say what these groups are, but in some general sense, no,
01:08:01
we don't know and it's not so tractable. Okay, next question. So we still have four questions. So is it, is something, is it, do we know something more about the structure of K, Q, Z, X, mod X, X to VM, comma X for Q even?
01:08:24
Yeah, I think that's the same question. I mean, I think that what that question is getting at is about whether we know the group structure here. And I know, except in low degrees. Okay, so next question is about the action of S1
01:08:44
on BC, CI, A, by, can you define this by using the trace map? So the way to see the action on the cyclic bar construction, let me back up a little bit,
01:09:01
see if we have a picture of the cyclic bar construction. Okay, here it is. We have this picture of the cyclic bar construction. I said sort of mysteriously, well, it's a cyclic object and therefore it has an S1 action, it's like magic. So the way to think about that is what a cyclic object gives you essentially is that like in simplicial degree,
01:09:22
this is, there's like an off by one thing that's a little weird. This is simplicial degree zero and one and two. In simplicial degree one, you have a Z mod two action, which is like permutation of these tensors. In simplicial degree two, you have a Z mod three action. In simplicial degree three, you have a Z mod four action, et cetera. And what's happening with this S1 action
01:09:42
on the geometric realization is essentially saying, well, all of those group actions assemble in some nice way into an S1 action on the geometric realization. So that's due to Alan Kahn and sort of the theory of cyclic sets. So that's how to think about what that S1 action is, coming from in the cyclic bar construction.
01:10:04
Okay, so that question, but I still ask it. Do we have a over interesting application of TC, THH, other than two algebraic K theory? Yeah, that's a great question. So I would say, you know, up until recently, if you talked about TC and THH,
01:10:22
people would usually say, well, it's mostly as a means to compute algebraic K theory. That's like one way of thinking about what those are. And that's certainly still true. But arguably right now, you know, people are starting to see a lot of interesting applications, particularly of THH on its own. So there's this recent work of
01:10:42
Bhargav Bhatt, Matthew Morrow, and Peter Schulze, where they look at connections between topological Hochschild homology and pietic Hodge theory. And I maybe won't say so much about that, but maybe to say one thing about it is that I mentioned briefly that there's a way to put a filtration on algebraic K theory
01:11:00
so that you get a spectral sequence relating motivic cohomology and algebraic K theory. And Bob Morrow and Schulze look at, well, what happens if you try to put such a filtration on topological cyclic homology and topological Hochschild homology, and they're able to get relationships between those objects with other theories such as crystalline cohomology. So yes, in recent years, there's a lot of interest in,
01:11:23
particularly THH as its sort of own invariant to study, but historically, they were certainly like means to understand algebraic K theory, but we've seen a lot of new developments in the last few years about that. Okay, so still three questions
01:11:41
because new questions arrive. So about computation, so I have one. Is it easier to compute TC, for example, TC of a ring, for example, A and C mode P and things like that. Easier than K theory, for example.
01:12:02
Oh, I may interpret the question, but yeah. It just depends. Sometimes yes, and sometimes no. I mean, like, so for the integers, yes, like topological cyclic homology of the integers is definitely more accessible than the K theory of the integers
01:12:21
because as we can see, sort of, oh, I don't know where I was saying it. Maybe it's, oh, here we go. Topological cyclic homology of the integers was actually computed by Bakshad and Madsen many years ago but the K theory of the integers is still unknown. So yes, in some cases, TC is definitely more accessible
01:12:41
than algebraic K theory. And so there are certain calculations where the issue is not having a good comparison or not understanding the comparison between K theory and TC. That's definitely something that happens. But another thing that happens is that, you know, you just can't compute TC in the first place. So depending on a specific calculation, there are sort of different roadblocks
01:13:02
that come into play. Okay, so same kind of question. So is it still hard to compute K theory of Z mode P to the N? So even using the DGM theorem, so I guess it's doondas, good willy, and I don't want to get the last name wrong.
01:13:21
McCarthy. Yeah. Only potent ID or techniques like that. Yeah, so the K theory of Z mode P to the N is still open. Certainly people have tried it. There's work of Morton Bruhn and well, he is more working on THH of Z mode P to the N. And be like Engelbeit has thought about this
01:13:41
and other people I think are thinking about it now using sort of new technology. So, you know, people have tried it with trace methods and ideas have included like trying to filter it somehow and access it that way. But it's not, it's certainly, it's open. It's not easy using this approach. And I think a lot of people have given thought
01:14:02
to whether it's possible, how to make it possible. So there's a question about computing K theory of Z XY, Z bracket XY mode X via good Y, X comma Y. Yeah, so the question's about the K theory
01:14:21
of the coordinate axes. So Z join X and Y mod XY. And the answer to that of whether this is computed is yes. The K theory of this was computed by Viglek Engelbeit and myself. And it is similar to this calculation up here
01:14:42
in the sense that, let me get this right. In the odd degrees, we compute the groups exactly. And in the even degrees, we compute the orders of the torsion groups for sort of similar reasons that I mentioned in that other calculation. But yes, these are understood. And also if you work over like a field,
01:15:02
perfect field of characteristic P that was understood prior by work of Hesselholtz. And tomorrow we'll touch more on like, why a little bit on why we consider these kinds of rings and how that would be approachable at all using this method.
01:15:23
So last question this time, but I'm very happy that we have so many questions. So first of all, is there, so can we understand TCA comma P in relation with THH as K theory with P completed coefficients relates to K theory
01:15:43
with integral coefficient of ZP, let's say. Is it the analogy that? I'm sorry, I'm reading the question. I'm trying to understand what it's asking.
01:16:03
So maybe we can just answer the second question. It should be easier. So we can, there is a construction that can, that of TC of A as THH of HA of the island, I'm not going to speak to HA.
01:16:20
Can we apply the same construction to KA? No, so the construction where you take topological Hochschild homology of A and build from it topological cyclic homology of A relied on the fact that on topological Hochschild homology of A, you have an S1 action.
01:16:40
We use that in a very important way, further that it was a cyclotomic spectrum. K theory doesn't have an S1 action. So it doesn't make sense to talk about like taking those fixed points and building that tower in the same way that you do for THH. So it's just the, right? You just can't do that construction at the level of K theory.
01:17:01
Okay, so maybe the other question we will continue in the comments after the talk. Another question I did, so let me, so we talked about negative cyclic homology. Is there notion of non-negative cyclic homology?
01:17:25
Yeah. That is negative topological cyclic homology. Right, okay, so yes, yes on all fronts. So, okay, there's Hochschild homology, which I defined for you. There's negative cyclic homology, which I didn't define but mentioned.
01:17:42
There is cyclic homology and then there's another one called periodic homology. So the way to think about this is that Hochschild homology, I defined for you the chain complex that you use. You can extend that to a bi-complex and cyclic homology is what you get
01:18:01
when you only consider the positive part. Negative cyclic homology is what you get when you only consider the negative part and periodic homology is what you get when you consider the whole thing and take the sort of total complex of these things and take its homology. So we have algebraic theories of all of those. And then your question about topological versions of that is an excellent one. So for Hochschild homology,
01:18:23
we have topological Hochschild homology. For periodic homology, we do now just recently, people have started to consider something called topological periodic homology, which I will define tomorrow. And then for negative cyclic homology, it gets a little confusing because I said,
01:18:42
well, topological cyclic homology is like our analog of negative cyclic homology. And that's true, but if you look at the actual definition of negative cyclic homology, there's a more direct topological analog, which until recently, people didn't really consider because it doesn't behave well.
01:19:01
It doesn't do the thing that you want it to do. Turns out that it plays an important role in this work of Nicolaus Schulze. So that's what we call topological negative cyclic homology or some people negative topological cyclic homology. I don't know, there's no agreement there yet. And again, that I will define for you tomorrow.
01:19:21
So yes, there are more algebraic theories here and they have topological analogs, which are interesting and will come up when we talk about how to really get our hands on TC. Okay, so I think we have finished the question now. So thanks a lot, Tina. Again, we will hear you in the next talk afterwards.
01:19:45
So let me look. We, so tomorrow you'll have your second talk and then on Thursday.