So thank you very much for this invitation

to all the organizers. I can imagine that organizing this conference in this COVID chaos, it was not easy. So thank you very much for all the work. So I'm gonna talk about these motives of singularity categories.

And well, oh, sorry, I have a discord. Wait, I'm sorry, I lost my, how can I see again, my share screen? Sorry, I did something happen here.

I don't see my screen. Oh. What's? But at the moment it's shared. Okay, you can see. I'll type, yeah, yeah. Alt tab. Try hitting the zoom thing again on your laptop. Well, it doesn't share.

Yeah, I don't see the screen. So the screen is sharing now. It's sharing fine. You just have to make the correct window appear again. Or you can stop the sharing and start it again, maybe. Maybe it's gonna be easier, sorry.

Sorry, I'm sorry for this. I don't know what is going on. So let me go again on this. Yes, you can see now. Okay, I'm sorry, I'm sorry.

I won't touch this again. So yes, so as I was saying, thank you for organizing this call. So I will be talking about this story of motives of singularity categories. So I will lead a bit of preparation to talk about this.

So let's say more or less the contents of the talk. So first, if you have any questions, please stop me. Second, the results of this talk were obtained together with Antony Blanc, Bertrand Tomain, and Gabriele Vizzozzi some time ago.

And more recently, there are some progresses in this topic. And I want to briefly, by at the end, to mention two progresses concerning this topic. So the first one is concerning the PhD thesis of Massimo Pippi, who just defended. And another is this project, this program

of Tomain Vizzozzi about the block conducted formula. So I will very, very briefly mention these two applications of the story I'll try to tell. Okay, so here's the plan for the talk. So the first part of the talk, I will spend some time explaining the story of singularity categories

and the relation to matrix factorizations. I will tell a bit of where these things come from, and I'll tell the motivation for what we're going to do next. So in the second part of the talk, I will make a bit of a digression and talk about motives of singularity categories,

but more general, generally motives of digi categories. So this comes in the continuation of the Gonzalez-Dabuada previous lecture. So there is an overlap between the two talks. I'll try to explain this. Then I will try to explain the connection

between these singularity categories and its vanishing cycles. So that's the main theorem of this talk. And as I said, in the end, if I have some time, I just want to briefly survey some of these results and the new processes. Okay, so let's start with this review

of the singularity categories and matrix factorizations. Okay, so let's start with how this, one way to enter this story is to start with this theorem of Serre. That allows us to understand, to detect the singularities of a scheme,

just by looking at its derived category. So essentially what this theorem of Serre says is that a scheme is regular if and only if perfect complexes, the derived category of perfect complexes and derived category of bounded coherent sheaves agree. So I want to say more concretely what this means.

So this means the following thing. So let's first make a remark. The first remark is that there's always an inclusion going in this way, meaning perfect complexes inside the derived category of bounded coherent sheaves. It's not exactly always, well, you need some hypothesis on X,

eventually co-connective, but in the cases of this talk, it's always. Then the complicated part is the other inclusion. And this is not true in all generality. The key lemma that we have to look at is this lemma of Serre that allows us to,

gives us a criterion to test if a scheme is regular by looking at the shape of resolutions of coherent modules. So the first theorem is this one. It says that a scheme is regular if and only if any module, coherent sheave on M, let's say in the heart of the derived category,

admits a finite resolution by vector bundles, which is finite. So meaning I start resolving my module by projective models, and at some point this stops and this stops at the dimension. So, and this is exactly what allows me to say that the derived category of this derived category,

all objects here actually have a resolution that live in here. So the two categories agree. So this is the starting point for this story. And what we are not interested in this case, we are interested in something a bit more general. So we want to understand what happens when X is not regular.

So in this case, we want to understand the excess of this inclusion. So this is always living inside. So perfect complexes live inside the derived category of gradient sheaves. So we are interested in understanding this excess, because the theorem of Serre tells us exactly that this excess vanishes,

if and only if the scheme is regular. So most of the time in this talk, I will be working with the scheme over C, or over a field, so I will use regular and smooth, to say the same thing. So the first thing we can try to do

when this is definition due to Orlov, is to isolate this excess. So this excess, called the singularity category, is exactly the quotient of the coherent sheaves on X by the perfect complexes on X. So the only thing that remains after this quotient are exactly the things that give obstruction to smoothness.

So we take the verde quotient because here, everywhere here, I mean, digi categories. Okay, throughout the talk, I mean, digi categories. Or actually, in this case, in this particular case, you can actually mean a triangulated category. So the first thing we want to understand in this talk

is how to control this excess, this piece of information. So Eisenbad gave us a formula to control this excess in a very particular case. And this particular case is the following. We suppose that our X, our scheme,

is actually given by the zero log of some function on an ambient space U. So I have X is just the zeros of some polynomial function on some U, and U is smooth. And then in this particular setting, I can say something about the singularities of X.

So what can I say? Well, I can say the following thing. So let's look at this formula. I'm sorry I wrote it already, but let's try to isolate piece by piece what I'm saying here. So let's take again here a coherent module on X.

So X is my upper surface, it's zeros of this polynomial. Let's take M. So if X was smooth, then all I would have would be this piece of information, a finite resolution by vector bundles, only this. And this actually leaves, it's a perfect complex. So the theorem of Eisenberg tells us that,

well, in the case of an hyper surface with singularities, what's going to happen is the following. The resolution does not stop. It continues all the way to infinity. But there is a strange phenomenon happening is that it's the fact that after the dimension, this resolution becomes periodic, too periodic.

So the term here is a term here, the term here is term there. So although it is infinite, we can control it by these two periodicities. So how do you prove this? Well, if you go for the computations, it amounts to just using the Auslander-Buxman formula in commutative algebra.

But this is the idea. So essentially, in the case of an hyper surface, any M coherent on X, I can resolve it by two kinds of data. One is this finite resolution by vector bundles.

And another one is this infinite length with two periodic resolution. So let's try to axiomatize a bit the situation here. And this is exactly what Eisenberg did. It introduced a category, which he called a category of matrix factorizations.

So U is our ambient scheme, F is our function, X is our hyper surface inside U. So we define the category of such strange objects consisting of a pair of vector bundles, Q and P, two maps of vector bundles, okay, vector bundles on U. This is a point of attention.

So a pair of vector bundles on U, two maps of vector bundles, such that when I compose them independently of the direction, the map I get is the multiplication by F. So you see that in particular, this is something that lives over U.

But if I go to X, meaning I kill F, when I kill F, this thing here becomes a two periodic complex because multiplication by F, just by definition is killed. So it just gets zero. So in a way, these objects of this category become presentations for such kind of complexes,

infinite, but two periodic complexes. So the starting point of this construction I will mention today is this following theorem of Voronoi, that says the following. What perfect complexes can only be seen inside the derived category of Boolean sheets. This is always for the purpose of this talk.

And then there is this excess, meaning the quotient of these by these. And the theorem of Rolovov says that I can identify this quotient exactly with this category I just described, matrix factorizations. And how does the procedure goes? Eoristically, it goes exactly like this. If I have a module M,

the finite piece of resolution by vector bundles lives in this part. And from the other side here, I only get the two periodic piece of information. So in a way I have an exact sequence of categories where I store these two kinds of information

on the extremities. So this is the starting point of this talk. Of course, by the theorem of Serg that I mentioned before, if this excess piece of information vanishes, then X is smooth. And this is the new finale. So this category controls all the singularities

of the hypersurface. So this is the starting point of this talk. And now I want to briefly mention two computations one can do just to get a feeling of what is matrix factorization.

The kind of information that we can have stored in these categories. So the first one is called as neuro-periodicity. So, yes. Yes, there's a question. So is there something similar for regular embeddings of higher co-dimensions? I will come back to this by the end of the talk.

So this is exactly what I said. So this is the concerns, the progresses in the thesis of massive PPP. So I'll come back to that at the end. Okay, any more questions? No. Okay. So let me just, as an example,

briefly mention two different computations one can do that can give us an idea of what these things are. So the first example, this one, is a computation of matrix factorizations where the ambient scheme is just a two. And the function is X squared plus Y squared.

I'm working over C here. Sorry, there is another question just to add. Does the category of matrix factorization have a structure of a triangulated category in a natural way? Yes, yes, in a natural way. Yes, yes, yes. In a way, if you want, either you see it through this equivalence, this is a Verde quotient and the Verde quotient of two triangulated categories

as a canonical structure of a triangulated category. And this equivalency is compatible with this unit equivalence. So yeah, so back to this first computation,

we can show that if I take the ambient space A2 with this function, then I have an equivalence of two periodic DG categories. So this is something I didn't mention here, but as you can guess already, by the structure I put here, this is two periodic resolutions.

This category is going to be two periodic. I didn't say anything about these two periodic. It is not obvious that this category here also has a two periodic structure, okay? At least from the description I gave here. But in fact, it is true,

it has a two periodic structure and these two things are compatible. Which takes a bit more time to find out the two periodic structure on this category. So I was saying, first of all, we can try to compute this simple example and what we get is called non-periodicity. That says that MF on A2 with this function,

X squared plus Y squared, is just MF on the point with a function zero. And if you see what MF on the point when the function zero means, well, we just get exactly two periodic complexes. So complexes with maps going in different directions with compositions is zero.

So this is an exercise we can do. And as another feature of these categories is this Tom Sebastiani property that says that MF of a product, A2 is just a product of A1 and A1,

is actually the tensor product of MFs of each copy of A1 with the corresponding functions. So this is called the Tom Sebastiani Theorem. And in the generality that I'm going to use here, it has been proved by Pregel for matrix factorizations in Pregel's thesis. So just two examples to give you an idea

of what this kind of information systems contain, although it doesn't say much for now. Okay, so now I introduce you these categories of matrix factorizations. Now you know that they somehow encode the existence of singularities.

So I want to give you an idea of where we're going. And to start to give you an idea of where we're going, I have to mention you this first result by Tobias Dikaroff and at the same time by Anatoly Pregel that says the following thing.

Let's suppose I take a regular local ring R with an element f, okay, a non-zero divisor. Then I can define this category MF. So this category MF is suppose, I expected to give the singularities of the zero locus of f, meaning spec,

if I write it here carefully, I want to expect the singularities of spec R mod f. These are exactly the zero locus. So this sits inside spec R. And the result of Dikaroff is,

so in this case, there are two results I will mention. So the first result is that this category here, we can show it has a compact generator, an explicit one. We can get very good control of this compact generator, meaning every object is generated by under shifts

and called limits by these objects. And the second thing we can know, once you compute the explicit in this compact generator is to compute the oxford homology of this category, because it becomes just the oxford homology of the endomorphism of this compact generator. And the computation of Dikaroff shows that the oxford homology of this category

is Jacobian ring. So maybe I should write here, Jacobian ring, where you quotient the ring by the derivatives of it, by the Jacobian of f. Let me write it this way. And so this is, but the degree of this

lives in degree the dimension of R. So this is an important point. Okay? So if you want, this is the first sign that the story of how we are taking this story is the first hint. And the second hint explains, yes? There's a question here.

Do you assume R is essentially a finite type over a field? I don't think so. I don't remember. I have to look at it. I don't think so. There's also another question. H H star is graded and the Jacobian ring

is in general not graded. No, I'm saying that it's the concentrated in degree the dimension. So this is not graded. I'm saying that the only piece is going to be this. This is the graded part of this in dimension D. So the question about essentially a finite type

is because how do you define the partial derivatives? Ah, probably need some condition of finite type.

I don't have it in mind, I would have to look it up, but probably you're right, probably you're right. If it poses a problem for the derivatives, then you probably need it. And there's another question. So what happens to the degree if you apply the product structure of R? The product structure of R,

I'm not sure I understand the question. Same. Okay, let's see. I think what this means is gonna become more easy to understand in the next slide, I hope. So the way to go to understand this,

it's a sequence of results and computations that I'll write this way, that compute the periodic cyclic homology of this category. So we already saw in the previous talk,

this periodic cyclic homology appearing. So there is a computation due to several people that connects this periodic cyclic homology of matrix factorizations to a homology of vanishing cycles. And this is gonna be the starting point for this talk. I will come back in a few minutes

to explain the vanishing cycles part of the story. I just want to use this result as a motivation for what we're gonna do. I hope that what we're gonna do makes this picture also a bit clear. So if you have questions, I suggest you ask in the end, and then maybe what I'm gonna say next can answer your question.

So the first part of this result identifies the periodic cyclic homology with the twisted the Ram homology, meaning I take the Ram complex, but I twist the differential by this wedge with the differential of F. So this computation is due to Efimov and Dikirov.

And then there is a second half of this computation that due to Saba and Koncevich, that identifies this twisted Ram homology precisely with a homology of vanishing cycles. So what we're gonna try to, what we're gonna do in this talk is to explain you why these two things, this one and this one agree,

but we will explain you from the motivic point of view. So here's the program for the talk. So here's the idea. We're gonna try to conceive an object which you're gonna call the motive of the category MF. We're gonna look at the theory of vanishing cycles from a motivic point of view,

and we'll try to establish a comparison between the two. So this motivic vanishing cycle side of story, this was developed in Joseph Hayub's thesis. And this one, I will explain in a second what this means. But the whole idea, that's what I will explain,

is to go, explain how to go from this side to this side and we will avoid this. Okay, we wanna establish a direct comparison. Yes, there's a question. So there's a question. HP is taken over Laurent power series here? It's taken over Laurent power series with a variable in degree two. There is a two periodic phenomena

that we have to take in consideration here. So this form of writing is a very naive, it's very informal, let's say. There are some subtleties in how I take this HP, and I was trying to put them under the carpet for the purpose of the talk, especially because I'm gonna concentrate on this part,

and this part I hope is gonna be clear. So this is what I said. Let's hope this will clarify something. Okay, and there's a further question by Birkan Muze. Is there a six-unctor formalism in the non-commutative setting? A six-unctor formalism in the non-commutative setting, you mean for non-commutative motives?

Yeah, I guess. Well, the answer is, I don't know. And I don't know means that I thought for a long time about that and I could not prove it. So I tried to do it a long time ago. I could not prove this six-unctor formalism.

Yeah, so the answer is, I think it's, I don't know. Maybe someone has thought more about that. Okay, there is a four-unctor formalism for sure. Okay, the question is the shrieks.

So, okay, so let's try to explain what this means. So I will start from this side to try to explain you what is the motive of the category of singularities of MF.

And this is where there will be some overlap with what Gonzalo Tawada explained in the previous lecture. So this is first part. I'm going to give you a general regression on non-commutative motives, but I'll try to go very quickly. So this story of non-commutative motives

started with some ideas on Kontsevich. Your suggestion came from Kontsevich. And later on, Gonzalo developed a formalism with, and later with Gonzalo and Tawada and Sisinski. They developed a formalism for these non-commutative motives. But this formalism is mostly a comological.

Actually Gonzalo in this lecture, he kind of explained this comological because in order to relate it to stable motivate theory of schemes, he had to use a duality. So in this talk, we're going to take a more homological approach. And by this, I mean that something closer in the spirit

to the original construction of Morel-Vevotsky and the motivica motivate theory. So here's the idea of what we'll try to do. So the motivica motivate theory of Morel-Vevotsky, it's a construction that takes a smooth scheme and assigns an object in this category we called SH.

Well, this SH is a very formal construction. And actually I will try to explore a bit what is the universal property of this construction. And what we'll try to do is to mimic exactly the same steps of this construction, but starting from DG categories.

Instead of starting from smooth schemes. So the first thing here, the first subtlety I'm going to do here is I will define non-commutative spaces, not at DG categories, but at DG categories with a knock. And the only reason I'm going to do this is to have the same from reality. And then I will introduce this construction

like a non-commutative version of the Morel-Vevotsky construction. And no dualization is going to be needed to compare these two things. So how are we going to get this? Well, the first step is just to tell you how this is constructed through an universal property. So what is the universal property of this Morel-Vevotsky construction?

So I'm going to write here the following theorem. The following theorem is just, I mean, this is more like a, it's a characterization of the, of functors out of this category. And it says the following. It says that every time you have

a symmetric monoidal functor, going from smooth schemes to any category that is stable and presentable, and such that F sends these navigation coverings to push out squares, F is A1 invariant, meaning precisely what is written here,

and such that F sends the motif of P1 pointed at infinity to some tensor invertible object, then there is a factorization. So I forgot to write this. So I wrote the theorem in such a way that this functor is the initial functor

that refines all these lists of properties. So any other functor, SM to some D, F, sorry, to some D, that verifies the same list of properties

it factors canonically through SH. You have a question, I think. Yeah, first, so I have, I think you mean D tensor is a monoidal infinity categories, right? Yes, this is a symmetric monoidal, yes.

Infinity, okay. And there's also a question by Hana Zuri. Can you explain in what sense that what has approached is homological and yours homological? I had a remark about this in the next. So let me just write this, yes. Symmetric monoidal infinity category, yes.

Then the result, well, the first thing it has to do with this op here. Okay, so it corrects, I correct the duality in the beginning

instead of correcting in the end. So homology theories are usually something that go from smooth schemes op to some abelian category, okay? And in this case, we're not taking homology theory, we are taking a homology theory that goes to SH over S.

So it has to do just with the variance or contra-variance of the theory. And of course, one is related to the other and this is what I want to say in the next slide, if I can skip it, is this piece of information here. It says that the theory we're going to get,

I'll come back to next slide, the theory we're going to bet is actually dual in PRL stable to Tabuada's construction. So meaning precisely that SH of this talk is just functors to spectra from Tabuada's talk to spectra.

So it is exactly dual, okay? So the only reason why I'm mentioning this one is because we want to have a comparison that does not, where the duality does not play a game. So I'll come back to this in a second. But first, let me go back to what I wanted to say.

So this gives you a, so this result gives you a characterization of this construction. So what we'll try to do is to define a theory of non-commutative motives that in a way is the universal thing that satisfies the analogs of these properties

for DG categories. So what would this be? So the first thing is we define non-commutative spaces to be DG categories of finite type. So finite type just means that they're obtained by attaching finitely many cells up to homotopy.

So I have to tell you what is an isnevige square of DG categories of finite type. So I have to, I mean, I didn't say what was an isnevige square of schemes. So in this case, I will just have to assume that you know what is an isnevige square of schemes. So, but if you know, essentially,

I define this isnevige square of DG categories to be pullback squares, such as this one, such that both these maps have the properties you would expect from an open immersion, meaning a localization. And the kernels of these open immersions have to be isomorphic, which is exactly what we expect from the et al. property

from the isnevige square. So the interesting, the only thing you have to know from this isnevige squares is that geometrical isnevige squares through the functor perf, so perf of an isnevige square isnevige in this sense.

So that's the only thing relevant for this story. Yes? There's no condition on the functor from U prime to U. There is, it's a localization. It has to be a localization.

All the maps in the square are localization. Only, sorry, only these ones. Yeah, and what about the map from U prime to U then? There is no condition. Exactly like you have an isnevige square, down you have open immersions, and then the map is et al, but there is no condition.

Yeah, so I'm not asking for any condition because it's gonna be automatic because of this. Okay. So that's the trick. There are other questions. What are the cells that you can attach in the category of DG categories? You start with DG categories that have only, so first of all, if you have a simplex,

you can take the free K module with those simplex. All right, if you start with just a simplexial complex, you start with the free K module generated by those simplexial complexes. This is gonna give you DG categories that have, for instance, one object zero and object one, and this K module and these simplexes,

and you start building from these ones. So it's essentially take the free DG categories generated by simplexes, and then you start attaching cells and building like that. Is it okay? More questions? Yes, I think so.

Okay, so essentially for the purpose of this talk, we will be taking this non-commutative spaces to be, sorry, this non-commutative motives to be what you get from DG categories up by forcing Nisnevich, by forcing A1 invariance,

and by forcing Poincare duality, meaning you force, sorry, you force this motif to be tensor invertible. So what you get is something formal,

but for the purpose of what we want to do, it's enough. So let me tell you what we want to do. So I just said that this construction is actually dual in this sense to the other talk, yes? Yeah, what about the A1 invariance property?

How is it expressed? Just this, I have functors from DG categories such that T tensor perf of A1 is perf of T. I force all functors to verify this property.

Okay, that's good. Okay. Okay, so why we did all this? So the only reason we did all this is just to have this diagram. And now it's completely, how to say,

setting is clear. We have the motivic stable homotopy of schemes. We have whatever this formal construction is, a stable motivic homotopy of non-commutative spaces. We have a functor, this functor, which just comes from the universal property

of this gadget. And by a joint functor theorem, it has a joint which is a symmetric lux monoidal. So all these are infinity functors and these two for free are symmetric lux monoidal. There's also a last question. Does not the last condition come for free?

What last condition? Can you go upstairs? I guess this- This one. Yeah. It comes almost for free, but you still have to invert the circle. So this, in this moral Wawatski theory,

you have these two circles, right? The circle, the topological circle and the algebraic circle. And in this case, what happens is that as soon as you invert the topological circle, the algebraic circle is invertible. And the reason for that is that this category has a semi orthogonal decomposition,

where this becomes just perf k plus perf k. So this is Bailenson's description of the semi orthogonal decomposition on perf of P one that makes this trick. So yes, you don't have to tensor invert all of P one.

It's enough to tensor invert the topological circle. So let's continue. So the second thing I want to say, it's this result. So this result was first, was proved by Tabuada.

I just kind of redid the computation in the setting, but this is Tabuada's result. So the result says that actually, if I compute the homes in non-commutative spaces, this is K-theory. Maps to one, this is K-theory.

So this is Schlichting and Waldau's, Schlichting's non-connective K-theory. Yes, so there's again a precision asked by Marc Levin. So what periodicity seems to be built in? What periodicity is built in? Yes, it's built in. It's built in from the fact,

well, it's built in because it's built in already on K-theory. Does this answer your question, Marc's question? That's sort of backwards, I mean. It's backwards, yes, I agree. I agree. But it seems it's in your description of P one. You killed the GM, so that, you said.

Yeah, right, because of this semi- orthogonal decomposition. Yes, right. Yes, yes. Okay, great. You're just making sure that that's what you were saying. Yes, yes, yes. Thanks, guys, thanks. And there's, again, yet another question, but that's good. There are so many questions. So by an anonymous, expected attendees.

So he says, when you are taking 3D G categories and simplices in this construction, is this the construction for a simplex delta N, take its image and have left a joint to a coherent nerve, C delta N, which is a simplicially enriched category,

then tensor V arms with K and take the associated complex and adult can. Yes. And then you have to do something else, because you have to force morality invariance. So there is the extra piece of that. This would only give them the cellular objects in the theory of the G categories. But as I'm working in morality theory,

I have to make one step further. So someone has a problem to read. I have a weird symbol. I think it's DNC inside when you, in the theorem that would have a low sigma infinity of T and inside it's DNC.

No, it's a one, it's a one, it's a tensor unit. Okay. This one is the tensor is a unit non-commutative motive. There's another question. What is the additional thing to make it more time variant?

It's already in that construction. Well, I have to invert more equivalences. So I have to look at the categories of modules. So I have to, okay, I have to take it important completion.

Does this answer? I have to take all retracts of an important morphisms. Again, another question.

Okay. What are examples of E1 and C contractible objects in SH and C? That's a very good question. I don't know. Maybe we can discuss about that later.

So where was I? I was about to tell you this result of, that describes the home spaces as K-theory. So a consequence of this is that now that we have this machine that goes from motifs to this gadget

and comes back, actually, what this result tells you is that if I take the unit, so this is one, again, this is a one, this is one. If I take the unit non-commutative motif and I send it through this adjoint M, what I get is a spectrum, is a motif ring spectrum

representing algebraic K-theory. And actually, a nice thing of this machine is that for free, I get the commutative ring structure on this spectrum. It's a way to get it just by playing with the machines. And another nice thing is that this M, this construction, because of this result, that it sends one to KH,

it lands inside modules over KH. So anything coming from whatever this is, lands inside KH modules. So let's go to motifs of DG categories.

So here's the construction we're gonna do. The construction we're gonna do is the following. Start with a DG category, which you see as a non-commutative space, some definition. You send it to SH, and then you take this OM to one,

and then you send it again through this M to SH. So in the end, all this construction, what is it doing? Let's pick a notation. I'm sending T to what I call, for this process, this parenthesis T.

So what is this explicitly? So let's look explicitly what this is doing. So if I take any smooth scheme in SH, so maps from the smooth scheme to whatever this is, is by a junction, is maps from perf to this OM. And this is just these maps. But I just told you that maps to the unit is K theory.

So this construction is sending a DG category T to an object in SH, which you can think of as a preshive of spectra, an object in SH that does the following. It takes a scheme, and it spits out the K theory of T tensor

by perfect complex on that scheme. It's a construction. And this is what we're gonna call the motivic realization of a DG category for the purpose of this talk. So let's just check some of the properties of these.

So the first example, if you take a smooth scheme over a base S, and if you construct this, look at what is this motivic realization of the category of perfect complex on X, what you get is just the following object. So X leaves over S.

You have the six operations. So this is due to the work of Sisinski, he agrees and Joseph, are you? So you can push forward this object KH that represents algebra K theory here. So push it forward. And the result is that these two things coincide.

So in a way, this is computing K theory of X, the global sections of K theory of X. So I will not well explain how this computation go. Instead, I will go to this singularity category story. So here's a setting of what we want to do.

I want to start with a scheme over X with a function. I want to look at zero locus of that function. And I want to look at the singularity category of that zero locus. So we will define this motive of the singularity category to be this construction I just showed.

So in a way, it gives me KTD of whatever I eat, whatever smooth scheme I plug in to answer this category. So this lands inside KH modules, and it gives me a definition. And this definition will be interested

until the end of this talk. It seems a complicated definition, but we can actually get some computations done. So let's look at, so it's just a first remark. I will tell you some properties of this construction. So the first thing is that if I take,

so this construction of MF that sends a pair to MF of that pair, it's actually locked monoidal. So meaning it sends, if I have two pairs and I multiply them with the functions being the addition of the two functions, it goes to the tensor product. So this is the Tom Sebastiani theorem I just mentioned.

Okay, and so in particular, the unit, so the point with the zero function goes to the unit, which is two periodic complexes. And because this is locked monoidal,

it tells me that for any pair, the motive of this pair is a module over this category. So there is an action of this, meaning two periodic complexes on this motif. And this just comes back again for free

because of the way we built this machine. And so in fact, so well, this is the action is what I meant by the first piece of the slide. So this thing of X0 is actually a two periodic KH module

because of this action. So can we compute, so this is how we're going to get the computation. So can we compute this object? So I will fix the following setting. So the setting is I have X living over a base scheme S with some map P. I will suppose that the genetic fiber,

so I will assume S is of the following kind. Either it's something of this kind, power series in T or power series in T with coefficients in FP or ZP, or even more generally some excellent trade. So in that case, with these hypothesis,

but you can think of one of these three examples, I'm going to consider the following setting. I have my X, I have my base scheme, my disc is like one of these is a formal disc. I have my punctured disc and I have my center of the disc. And then I'm going to consider the function given by my projection and the uniformizer.

So the uniformizer is just a choice of T or P here. And I have this set ring. And in this setting, I can make a bunch of constructions. They might seem a bit disconnected for now, but you will see how this comes into place.

So the first construction I'm going to build out of this setting is the cohomology of the punctured disc. So maybe I should just write this way. So what do I mean by cohomology of the punctured disc? Well, I'm just going to compute this object in sh, maybe I should write it in sh over sigma.

So meaning I take the unit object in motifs over eta, I push it forward, I push it back. And I get something I called motific cohomology of the punctured disc. So the first observation is that this is a commutative algebra object.

And the second observation is that I can give an explicit formula for this cohomology of this punctured disc. So it's not very different from what you would expect in topology from the cohomology of the disc. It's just average generated in degree zero and you have a generator in degree one, but shifted. It happens that here,

the shift has to be with a tape twist. So there is this extra piece of information, but essentially this is a circle. So I'm going to call theta this generator that lives in this state degree. And I'm going to use this algebra to say something about what I want to find is this category of sine x zero.

So how does this go? The first observation is that this cohomology of the punctured disc acts on the cohomology of the fiber, of the smooth fiber. So this guy acts in the cohomology of this guy by pullback, so there's a multiplication by theta. Another thing is that the cohomology of this fiber

acts under specialization on the cohomology of this fiber. So there is a map of algebras from one to the other by specialization. So I'm sorry if I cannot explain all what this means, but the upshot is that in the end,

I can combine the action of the circle with the action with this specialization to produce a map of algebras like this. But if you want, just keep in mind that there is an action of the punctured disc on the action or on the cohomology of the generic fiber. And then what we can also do is to repeat exactly the same thing,

but we work in the level of KU modules. So we replace the unit, somewhere I put the unit, by the KU module, the unit KU module. So the same thing happens, the same thing holds, I have again an action of the punctured disc or this KH version of the cohomology.

So I can now tell you the main, somehow the main technical result of this talk is this one is an explicit computation of the motif of MF as the motopy fiber of this action.

So this might seem a bit dry just to look at this way. And this is why I want to just briefly mention the main ingredient that goes into this proof. I think the main ingredient already can give some intuition. So the main ingredient, the main idea of proving these two things are equivalent

is actually to show that two certain algebras are equivalent. So what are these two algebras? On one side, we have the gadget I just introduced, cohomology of the punctured disc. On the other side, we have this thing of S zero.

So this is a category, but because this is a unit of the entire two periodic categories, actually it is a symmetric monoidal category. So in particular, when I take the motif, I get commutative algebra object. So the first claim is that the cohomology of the disc, of the punctured disc of the circle, and this motif of seeing S zero are the same algebras.

So this is an ISO of algebras. So this is the main piece of ingredient that allows us to prove this formula I described before.

And once you have this, once you have this theorem, this one follows just by playing with exact sequences. So I will not mention what this is, you can check notes after, but that's the main idea. And so using this result,

we can compute this, sorry, compute this motif of M F. And for the time left, I have, I'm sorry, how much time I have left? Oh, wow. Yeah, I have five minutes. Yeah. Okay.

You can take slightly more on it. So I will try to explain you very quickly what is the relation with vanishing cycles to this story. So I just told you how to define the motif of a digi category. If you don't want to,

you can forget all the details, the technical details, just think it's an object in SH, it's a KH module in SH. And now I have to explain you how the, from the side, the vanishing cycle side of the story, I can also produce a KH module in SH.

So very briefly, I will not have time to go into this as I expected, but this story of vanishing cycles essentially tries to study the following. If you have a family of algebraic varieties parameterized by over the line,

then you can try to understand the complex line, then you can try to understand how some cycles they generate once t goes to zero. So this is just a heuristic picture of what can be going on. We have cycles like these that collapses as the fiber moves to zero. And you also have the monodromy action

on these vanishing cycles. So I hope, I thought I would have some more time at this time. So I'll just skip this part and just tell you directly straight to the point. The point is that if I look at X zero, my fiber,

I can construct a sheaf called the sheaf of nearby cycles and another sheaf called the sheaf of vanishing cycles that captures exactly these kind of cycles like these that disappear when I go to a fiber with critical points.

So the main theorem is that I can do the following construction. So let's look at my central fiber and let's take for each point on my central fiber like this one, let's take a ball in CN and let's take a nearby fiber. So it's a fiber very close to the fiber at zero

and let's intersect this fiber with the ball. So I want to say something like this. So if I take a fiber at t small enough, I can take this thing called the muon fiber and as the point varies along my central fiber,

I'm going to have either nothing trivial cohomology or I'm going to have something like this as I move to the critical point. So essentially, if I just start taking the cohomologies of these fibers, of these small intersections,

as X varies, this forms a sheaf, this forms a sheaf on the central fiber. And this is called the sheaf of nearby cycles. This is the theorem you can find, this Milnor you can find is in this GA7. And you can also take the reduced cohomology and this is going to be called the sheaf of vanishing cycles.

So for the purpose of this talk, what is important is that you have an exact sequence where you have the cohomology of the central fiber, the nearby cycles and the quotient is exactly the vanishing cycles. So this is the upshot. So let's go back to the motivic side of the story.

So in this talk, we're going to have not up there, I gave you the, sorry, a betty side of the story, but we're going to use here the allodic side. So we have vanishing cycles, the sheaf of nearby cycles,

and this also has motivic versions. This is due to IU. And in fact, what happens, well, let's focus on the allodic side of the story. You have this exact sequence I mentioned before, where you have allodic, sorry, the cohomology of the central fiber,

vanishing cycles, sorry, nearby cycles and vanishing cycles. And all these come with the action of the Galois group. So the Galois group in this case is the Galois group of the punctured disk. And in particular, for, to make the theorem true, the theorem I want to give,

we're going to have to look at a particular subgroup of the Galois group. And it's called the inertia subgroup. I don't want to write it. Inertia subgroup. So we're going to...

be looking at inertia subgroup, it wouldn't be taking the homotopy fixed points for this inertia subgroup. And the way the previous part of the story comes in is this theorem of Deling. Let's say the following thing. If I look at the homotopy invariance of this,

of this, of QL sigma, what you get is the cohomology of the puncture disc. This is just a very refined way of saying something in topology you're perfectly used to, which is the following result. Think of the puncture disc as a circle, s1, and think of the choice of an algebraic closure of the puncture disc as the choice

of the universal cover of s1, meaning just a point. And think of sigma as c. And in this case, this side of the story is just telling that it can take fixed points for inertia,

but inertia is just automorphism of the universal cover, and taking fixed points, or derived fixed points, is just taking homotopy fixed points for the constant action, the trivial action on c. On the other side, this is the cohomology of the disc, which is this. So this theorem, if you look at what it's saying in the topological analogy,

it's just saying that the cohomology of the disc is just homotopy fixed points for the trivial z-action on c. So this is how to interpret this result. So another piece of ingredients we're going

to need for this story, the first one is this Helladic realization functor. So everything we built so far works in motifs. We're going to take the Helladic realization. And then there is this theorem of Vayub saying that the construction of vanishing cycles,

well, first there is a motific version, and second, they coincide under this realization. So with all this in place, I can tell you, finally, the comparison theorem. So the comparison theorem works like this. Let's start with the digi category.

For all the things I said before, I can lend, I can produce, this is the functor I called this. From all the things we said before, I can produce an object in sh, which actually lends inside k-h modules. And now I'm going to take the realization. And the realization

lands inside the realization of k-h. And there is a result of Joël Rieu that uses both periodicity and a gamma filtration to compute this realization. And it just says that the realization of k-h is just the two periodized Helladic homologies. Essentially,

the copies of a Helladic homology in all degrees, even degrees, with a shift by theta twist by n. So another way to say it is just the free algebra on the theta object.

So now that we have all this, I can tell you the main theorem. And the main theorem is this. The main theorem says, so let me go slowly here, the main theorem says the following. It says that we started with this category of singularities. We define this gadget called the motivic realization. So it lives in sh. And now

we took its Helladic realization. So now this is an Helladic sheaf over x0, actually. So I can take global, sorry, I should take global sections to make this work.

So on the other side, what do we have? Well, we have the vanishing cycles sheaf. I take homotopy fixed points. I take the homology, but then I have the two periodized.

And the theorem says that these two things are the same. So again, I will not give you the proof, but I will give you the main isomorphism that makes this work. So let me go like this. So as I said before, the main thing that made the first comparison that made possible

the computation of the motive of singularities was the fact that S0 gave us the homology of the puncture disk. So this we saw in the previous slide. And now I also saw in the previous slide, the algebraic version of the circle result that is the homology of the circle

is actually the homology, the invariance with respect to the z-action on C. So the combination of these two isomorphisms is what allows us to show these equivalence.

So everything follows from this plus some gymnastics with the exact sequence, the exact sequence that defines singularity category and the exact sequence that defines vanishing cycles. So I'm sorry if I had to go too fast here. I think I went really too fast.

Please tell me if you have questions or if you want me to go back. I had the survey of recent results ready, but I think I already went my time. So I apologize. Yeah. Okay. So maybe we'll see that for questions. So first of all, thanks. Thanks for the talk, Marco. And so we will go now to the questions if there are

questions. So I have a question. I would like to know a little bit more about the approach to the block conductor formula. Yes, this is one of the things I had in the

survey. So the approach, very briefly, very sketchy, the idea is the following. So if you are over C, let's take a family over C and you have the central fiber and smooth fiber, then you can compute this deling number. So this deling number is by

definition the dimension of the vanishing cycles of the piece of commodity of vanishing cycles. And you can compute it as the difference of Euler characteristics between the Euler characteristics of the smooth fiber and the Euler characteristics of the central fiber.

So but because of this difference of Euler characteristics, if you think the way how you define vanishing cycles, you define it by means of this exact sequence. So you have the cohomology of the central fiber, nearby fiber, and the excess is the vanishing cycles. So Euler characteristics sends exact sequences to sums. So in particular,

the difference between these two has to be the Euler characteristics of vanishing cycles. So this deling number is actually the Euler characteristics of the vanishing cycles. So this is the starting point of the story. So this works over C. So over C, this is known. So now you want to go to other disks like ZP. And in this case, you have to

correct the formula. So this is block conjecture. You have the formula is true, but there is a correction to be had, which comes from the presentation theory. And this extra term is called the Swann conduct. OK, so the approach that the Toan and Vizzozzi are trying to develop,

and they already have results for the block conjecture is, first of all, using the observation that the Euler characteristic of vanishing cycles is the same as the Euler characteristic with inertia invariance. This is because of the symmetric model function.

And because of the theorem that I just mentioned, the Euler characteristic of these and the Euler characteristic of Mf have to coincide. So the whole idea of the program is to compute explicitly the Euler characteristic of this Mf. So by Mf here, I mean the motive. And to show

that you get for free the left-hand side of the block conductor formula. So the new tool here, if you want, is the fact that you can

compute this approach, this number through Mf. So this is a theorem already when i is acting uni-potently. So this is essentially their program and the way they are tackling it. I don't know if this answers your question. OK, good. So there are other questions. So I read the first one by Tom Baughman.

Are there higher chromatic analogs of a relation between KH mod and SH non-commutative? So the answer is yes. I expect them to be, to exist.

I expect, so it's yes. So people, there are versions of non-commutative motives where essentially you can replace DG categories by other forms of DG categories or either

NDG categories or symmetric EL monoidal DG categories. And essentially you can build versions of non-commutative motives on this. And as it goes higher on the N, your priority should go higher in the chromatic tower also. But this is something, I mean, this is a work in progress, let's say,

with the Gabriele Vizzoli and Maru Porta. It's been in progress for quite a long time. We've discussed this last year also with Eldon and Elmanto. So now we kind of got somehow locked in this, but the idea is that there should be, yes, there should be. So if you have more, we can discuss this after this talk,

if you have more questions. It would be nice to talk about this. Sorry, so there's another question by Mark Levin. Is the, so it's in the main theorem, is the new or HL in the main theorem an et al shift or an object in SH of the central fiber?

Which one? So exactly this thing you have shown here, this new upper H I, the right-hand side. Where is that living? The thing inside, before you take the H star QL,

where's that thing living on the inside? That thing, yeah. New H I minus one. It lives on this edge of the central fiber. Oh, okay. So then the question is, do you get, do you have an isomorphism somewhere, maybe after making the right-hand side into a KH module before taking et al realization?

Before taking et al realization. So let me think. So you can make this new H I, this new H I, this is just a UBS. This is a UBS, yes. Yeah, so you can make it UBS. Yes. So I think, yes, I think this is possible. I just have to be careful about

this inertia invariance in the IU setting. But I think this is true. I think this is true. Oh yeah, okay. I can prove the theorem directly in motifs. Right. Great, thanks. Okay, and there is a last question. So someone wants to know,

someone wants to hear you say something about recent results. Oh, so I mentioned the blog, and someone asked the question about what happens when you have multiple functions. So I just want to briefly mention

the thesis results of Massim PP. You can find an archive. So the idea is that it extended part of the results I just mentioned, but to the case where you have multiple functions and they intersect all their zeros. So starting from some ambient scheme X with several functions,

he produced in his thesis a new definition of matrix factorizations and singularity categories in this context. He also extended the results of Orlov about his equivalence between singularity categories and matrix factorizations. And finally, he also computes explicitly this motif of MF, multiple functions,

using this result of Orlov and Burke and Walker that allows you to, if you have a scheme with multiple functions, this result allows you to pack all the multiple functions in a single, not a function,

a single section of a line bundle in some projective space. So essentially it allows you to reduce the problem of multiple functions to a problem of one function. And using this, he also gives a description, explicit description, of this motif of matrix factorizations with many, many functions.

So yes, so this is, I mean, I will not give you the details, but you can look at his thesis on archive. He posted the thesis on archive and it's very easy to, it's very well written. You can see the, you can follow it. There's a complementary question. Does the sequence have to be regular or can I just take derived vanishing locus?

I think it doesn't have to be regular. You can take the derived version. Yes. Okay. So I'm sorry if I went too fast at some point. So in any case, I'm going to post, I give you the slides.

Good. Thanks. So we'll put that on the YouTube page of your talk. Okay. So no more question. Yeah. So let's thanks again. Thank you Marco for a nice talk. And thank you for the, for organizing December School in these conditions.

It works. Yes. Okay. And so we, we meet tomorrow for at, at 1 PM for the next day. Okay. Okay. Thank you. Bye. Bye.