 much. It's a great honor to give this talk for Professor Kashiwara's 70th anniversary. So this is a joint work with Anthony Blanc, Berkhan Toen, and Gabriele Vizzozzi. And later today in the afternoon, Gabriele Vizzozzi will speak about a continuation of this work with the collaboration between Hinem and Berkhan Toen. So this talk is essentially an introduction to a project that is going on. And I will talk about a particular result that then later it's going to be used. So the talk is about motivicalization of digit categories of singularities and vanishing cycles. So I'll have to speak a bit about both sides. So in the end, we'll state a comparison between these two objects. So I'll have to define what are motivicalizations of digit categories and remind a bit about the formalism of vanishing cycles. So here's the plan for the talk. I will start with some motivations and I will state the main result in a preliminary form. Then I will make some reminds on digit categories of singularities and matrix factorizations. And then this is the core of the talk, the construction of these motivicalizations and then the comparison of vanishing cycles. So just a bit of motivation. So this category of singularities. So if we start with a scheme under some hypothesis, we have an inclusion of digit categories. From one side, we're going to have perfect complexes on the scheme, which are the compact objects in the right category with quasi-coherent homology. And this lives inside bounded coherent sheaves on Z. And so the first important result that underlies all this story is this theorem of Seach, then formulated in a different way by Thomason that Z is regular if and only if this inclusion is an equivalence. Okay, so this tells us that the difference between these two categories measures the existence and the complexity of singularities in Z. So there was a suggestion, a definition by Orlov, which takes the quotient of one by the smaller one. And this object measures the shape of the singularities of Z. Okay, so this is one of the objects we're going to be interested in. And we will be interested in a particular case, a particular situation, where we have a scheme, a map, a function on the scheme X, a function F, and our Z is going to be the zero locus of F. Okay, so in this case, we don't in general have F is not flat. So this product is going to be the dry fiber product. And in this case, I mean, in particular, when F is flat, this was the original version proved by Orlov. This category of singularities admits a description in terms of matrix factorizations. So what are these matrix factorizations? So these are pairs of vector bundles on X, and maps, delta zero, delta one, such that the composition is multiplication by F. Okay, so an example, the most generated case, if we take F equals zero, we get exactly two periodic complexes. Yeah, for the theorem of Orlov, it's F has to be flat. Flat. F. On X. Regular, maybe, in this original version. Yes, yes, yes. So in this case, for spec A, with the function zero, we get two periodic complexes. So, okay, so this is the case we are interested in. And then from another side, we have this, so we had this category of singularities. And another side, we have this vanishing cycle story. So this is just a picture. So the idea is that you want to study how you want to study the singularities of a fiber of a family on a certain point. And the idea is that you will study how cycles degenerate as you approach this particular fiber. Okay, so this is controlled by a sheaf. I note VF. And so the topic of this talk is how to relate this category, C and Z, to this sheaf VF. So the relation between these two constructions, we can, I mean, some people, there are some people here that started to explore this relation, just named Professor Kashiwara, Kepranov, Konsevich, and then later Saba, Cloud Saba and Saito. In the case of the formal power series over C, they give a relation between vanishing homology of F and the twisted form of the ramp homology, where we just twisted the differential by the differential of F. Okay, so this is, I'm not going to give a precise statement. I just want to picture the relation between the two. So, and more recently, we have results that use the previous one relating the periodic homology, secret homology of this category of singularities, with the two periodic version of this twisted-dram complex. Okay, so there are these two relations between singularities and vanishing cycles, and how a result, together with Antonio Blanc, Bertrand Touan, and Gabriele Vesosi, tries to give a motivic interpretation for these results, independently, without assumptions on the characteristics. So, I will state the result. So, we take A, an excellent and an excellent rate, with an informizer pi, and we take x over s, a proper flat map with x regular. Okay, yes. Then, we will identify two objects. So, the first object, it's going to be a q-alytic realization of this digit category of singularities. So, I'll have to explain what it means to take a q-alytic realization of digit categories, in particular of this one. And so, this construction, this realization is osomorphic, an ecologic shift to a two periodic version of this shift of vanishing cycles with respect to p. Okay? So, this is our result, and it will take me now some minutes to get to which one of these words. So, okay, so before starting a description of this result, I want to go back to Orlov's theorem for one reason. Because, so, Orlov's equivalence is an equivalent of triangulated categories between CZ and MF. So, for our machine to work, we need to observe that, in fact, both sides carry DG enhancements and actually two periodic structures compatible within the G enhancements. So, we need a version of Orlov's equivalence that takes these DG enhancements and two periodic structures in consideration. So, we have to compare and say they are the same also. So, okay, we have this lemma, which is inspired by results of Anatoli-Pregel over C. And essentially, the lemma says that these two constructions exist, can be enhanced to DG categories. So, there exist infinity functions going from pairs with X, flat over A, and F arbitrary. To DG categories, so here I mean the infinity category of DG categories, and I mean an infinity function. So, these two constructions can be enhanced. They are Lax monoidal. So, this category of pairs carries a tensor product given by convolution with the product on the fine line. So, these maps are compatible with this tensor product. And moreover, there is a Lax monoidal transformation from one to the other. So, saying that these two DG enhancements are the same and the compatibility with the tensor products also holds. In particular, if you take the pair A with the function zero, you get sin of A zero and Mf of A zero, which are both two periodic complexes. So, in the end, you get from this statement, from this machine, you get the compatibility between the two periodic structures. Actually, I go closer, considering the general trait, it's the snow addition law. How make convolution, additive convolution? It's a function doing A1 of the trait. A1 of the trait. Yes, F goes to A1 of the trait. Ah, see, so it gets two parts, kind of two dimensions. Yes. What is the meaning of Lax in Lax monoidal? So, it's a right Lax. There is a map from Mf or sin of Xf tensor sin of Yg to sin of X times Y over S, F tensor 1 plus 1 tensor g. Ah, there is a map. There is a map. There is a map. Oh, the fact. Yes. Each one carries a Lax structure. So, maybe I could, sorry. So, I'm the same for sin. Okay? So, in particular, if you plug in A0 on both sides, you get a monoidal structure on Mf A0, which is two periodic complexes, and the same for sin. And this comparison map says that they are the same. Okay? In particular, if we restrict to good enough pairs Xf, then on the monopic categories, so these are the g categories, on the monopic categories, we get the equivalence of our love. But it's still going to return to, because you have kind of two variables, you have trade and function, yeah? Okay, well, should I take it? In this case, there is no trade. Just fix A a ring, and you take function to take schemes over A, and functions, then you have the... But A is not a trade. No, A here is just a ring. I take a base ring. Okay. Okay. In the end, you're going to have to take a function over A, and you compose with the function informizer, and it goes to the A1 over the trade. Okay, so, all this to say that I'm going to use... We are going to use sin and Mf for the same thing over and over in this talk. Okay? So, every time you see one or the other, it means the same. Okay. So, now, I will have to describe you what are these motifs of DG categories. So, I start with... So, the goal is to have a DG category for A, and I want to say what is the motif of A. Okay? So, the first thing we have to say is what are motifs, and in this talk, motifs are going to be objects in the Motivic Stable and Motabic category of Moriel Wewocki, S-H of A. Okay? Okay, I'm not going to describe what the construction... And I just want to claim that this has a certain universal property. It's the universal place where you have this navage excision, and one invariance, and the tight twists are invertible. Okay? So, why do we need this construction of S-H? And the reason is because we need K-theory for this whole story to hold. So, in this construction, in S-H, there is an object, B-U-A, which represents K-theory, a motopy invariant K-theory, meaning that if you take a smooth scheme and you send it to this category S-H, then the ohms with a shift become a motopy invariant K-theory. Is B-U-A connected? B-U-A is... No, it's not connected. It's non-connective K-theory in the end. Is it maybe what I would call Z times B-U-A? Over a field, yes. Otherwise, you have, okay? So, some remark is that this object reflects the projective bundle theorem, so it doesn't see tight twists. Okay? So, where the tight twists, I mean P1 over A with infinity. Okay. So, we have this construction, we have this object, B-U-A. So, the construction of the motif of the DG category starts with the remark that the K-theory of X is actually... is a construction that depends only on the category of perfect complexes on X. So, the algebraic K-theory is the Waldhausen construction on perfect complexes on X. So, the construction we're going to do for any DG category is the following. We take a DG category and we're going to twist the K-theory by tensoring with this DG category. So, now the motif of X, the motif of T, sorry, is an object that represents the following thing. Maps from X, or image of X in SH to this guy is essentially the K-theory twisted by T. Okay. So, we... Yes? What was SH? SH is this category of motifs that has this proper... Okay? So, this object represents K-theory of T. We can make this assignment something which is infinitive-inventorial and symmetric-lux-monoidal also. So, there are several ways to do this. We can do this strictly and then go to the infinity categories or we can do a construction internal to the infinity category world. Okay. So, we have this construction. To every T, we assign K-theory twisted by T. And now to remarks, the first one is because of localization for non-connected K-theory, this construction preserves exact sequences of DG categories. Okay? And this is going to be important to us because SING is defined by an exact sequences, coherent, perf, and SING. And the second important feature is that this construction is lux-monoidal. So, on this side, you have the unit DG category. So, its image gives BU because it's just K-theory twisted by the trivial DG category. So, for any T, the motive of T is going to carry a canonical action of BUA. So, it's a module over BUA. Okay. And let me give you an example. This is just to mention that these kind of constructions of motives of DG categories has been used by Antonin Blanc. These is to define topological K-theory of DG categories. Yes? What is BUA? What is BUA? BUA is the object that represents a multiple invariant K-theory. BUA is a notation. Okay. So, now let's use this... We have this formalism. Let's use it to compute the motive of SING, perf, and db-co. Okay. So, the first step. We start with the scheme over A, quasi-compact, quasi-separated, and then we have the scheme over A. So, we have the category of perfect complexes on the scheme, easy DG category over A. So, we can take its motive by this construction. We are twisting K-theory by the perf X. At the same time, we also have a motive over X, which is BUX, which represents K-theory over X. And because we have push-forward for this construction of motives, these six operations, then we can take the image of BUX, and this leaves over S. And the claim is that these two constructions identify. And the reason is because just by a junction, we see that both represent K-theory twisted by perf X. By X, actually. So, this is the first step. Second identification we will need is the following. So, we take a scheme over S, sorry, and we take an open and a closed complement, and then you have a localization sequence of motives. And so, we start with BUX, and then you can push it and pull it back, shrink to Z, and push it forward, the same over U. And the claim is that the first term on the left is essentially the motive of DBCO. And the reason for this is because we were, I forgot to mention, we have to assume that X is regular in this case, so that K-theory becomes G-theory. And then, in this case, both X is regular, so U is also regular, and then K-theory is G-theory. And because of critical localization theorem and the visage for the closed immersion, these two guys identify with the G-theory with support. Okay, so we have an identification of the motive of the DBCO Z. And we also can describe the motive of perf X, in particular, of perf Z. It's the same thing, it's just something that leaves over S. Okay, so the first step now is we go back to this context where we want to work. So we have a scheme, we have a function, and we take the derived fiber product. So in this case, the derived fiber product, I-0 is an LCM up, so I is also gonna be LCI. Okay, so in this context, we can put together two exact sequences. So the top exact sequence is the one coming from the definition of sing, co-perf and sing, okay? And the construction of this motive preserves exact sequences. At the same time, because of the identification we just described, the term in the middle identifies with the K-theory with support. So we also have a localization sequence in the middle. And because we are now in this context where we have something, a family over A1, and we take the fiber over zero, the map is LCI, so the inclusion of perfect complexes on Z preserves, sorry, the inclusion of Z on X preserves perfect complexes. So we have this map at the level of perfect complexes, which essentially makes the diagram commute because it's also the map that is the first map on the vertical. Okay, so we have these two exact sequences, and the claim is that this map here, the I lower star is a homotopic to zero. And the reason is just because by drive-based change we have to compute the push forward I zero lower star. And this is also a motopic to zero because it's just the inclusion of the point zero in A1. And we have the resolution of A over AT given by AT, multiplication by T goes to AT and then A. So in K-theory, this becomes zero. So I have a question, by curiosity, if you write the core optimization sequence, I mean, you have this idea partially by I over star. Yes. What do you get? You cannot relate it to Z, to the because Z. Because the because Z is really K-theory with super. What do you get? Can you identify the categories that you get? Ah, I will think you just get, I will have to think a bit, sorry. Okay, so out of these, because this map is non-homotopic, what we get in the end is just, so these are triangles. So we get a triangle that goes from the co-carnal of the first map to the co-carnal of the second to the co-carnal of the third. So MZ, and here we have, this guy here is the co-carnal of the I over star because it's non-homotopic and then we get the last one. Okay. So we have this exact triangle of motifs and this is what we're gonna use to compare to vanishing cycles in the end because the two are gonna be defined by an exact triangle, okay? Okay, so now we're gonna take the co-alytic realization of this triangle. So, and here we need to fix X, an excellent trait and we take a closed point and a genetic point and we have this co-alytic realization for schemes over S of finite type. So in this case, we can go from S H, this construction of S H, S H we can tensor by Q and then we can go to Q-alytic sheaves. So Q-alytic sheaves here, I mean the inconstructible erlatic sheaves and then you invert L. So this is the construction of the Ling and Echdel which we can do in this infinity categorical sense. So there is this realization and the point is that because of works of Iub, Gaber, Sissiski, Degliz and Ivorah, we have this compatibility with six operations and taste twists, meaning the six operations exist on every, on all the terms, at every step and the maps are compatible. So, okay, so we have this realization so we can send the exact sequences to erlatic sheaves and in particular we can compute this erlatic realization of the rational BUX spectrum and by a theorem of Rieu, this realization identifies with a two periodic object which is the Q-alytic sheave QLX twisted by the roots of unit E, I and shifted by two I and the sum over all I. So this object we're gonna denote as QL beta. For the fact that it is a free infinity algebra, it's free infinity because RL, so sorry, it's an infinity algebra because BUX is an infinity algebra, RL is monoidal so the image is an algebra and it's set to periodic. Okay, so we're gonna use this notation and the consequence is that the sequence we had before, now we can realize it, we get an erlatic sheave which is also two periodic meaning that the first sequence before realization was a sequence of BU modules. So now what you get is a sequence of RL BU modules which are two periodic objects. Okay, so let's stop here for a while and let's go to vanishing cycles now the other side of the story. So in this case, I have to fix a bunch of stuff. So I have S, a trait, I fix a close point, a genetic point. I'm gonna fix a separable coser of K, a small K. I take S bar, the strict initialization along this point and then I'm gonna have a maximum or any final extension of BK and I take a separable coser and I get this eta bar. Okay, so this is the context and in this context we have an exact sequence of Galois groups, the Galois group of eta bar, the Galois group of sigma bar and the inertia group is the kernel. Okay, and in this context, we can define vanishing cycles. So we start with X over S and then we can pull back to S bar and we can define vanishing cycles as follows. So we take a QL sheep over X and then we can restrict E to the closed fiber, sigma bar. We can also restrict it, pull it back to X eta bar, the universal cover and then push it forward and we have an junction map between these two constructions. So the middle one is the construction of nearby cycles. So we complete the triangle, we get the definition of vanishing cycles and all these leaves in sheaves over sigma bar which are Galois, eta, equivalent. Okay, so this is the definition. So in particular, we can pick the sequence over sigma bar and push it forward to sigma bar. So now we have a sequence of Galois, eta bar, equivalent objects in sigma bar. Okay, so now we will apply this to a particular sheaf, E which is the two periodic sheaf over X, okay? Which is the realization of BU. So in this case, I'll write the sequence just in a different way, instead of writing vanishing cycles as the cofiber, I'll write it as a fiber, so we shift it by minus one and we have this sequence which is again Galois, eta, equivalent. And we have a compatible action, two periodic action of sigma bar. Okay, so we have this sequence and now we can take inertia in variance, meaning we have the inertia living in the Galois of eta bar so it can take a lot of fixed points with respect to this inertia, okay? So we get a sequence of a lot of fixed points of the previous one. The point is that the sequence lives in the same place but we have a compatible action of Galois invariance of a monopathic points of eta bar, okay? This guy is a trivial action but taking a monopathic points will give back the homology of inertia. So in the end, we get an action of the tensor product of the two algebras, the QL beta and QL inertia invariance, okay? So let me now state the comparison. We have, and the comparison is the following. So we take S is an excellent Ancelian trait. We take pi in an informizer and we take P, a scheme, proper and flat of an I type over S with X regular. So in this case, we're gonna have our map P that lives over the trait and then we compose with an informizer which you see as a map to a one over S. So everything lives over S in the end. So the point is that in this case where P is flat, the fiber X zero is automatically undrived, okay? So in this case, the theorem is the following. So under these hypothesis, there is a canonical Galois equivalent equivalence of hypercomologies between the hypercomology of the realization of sing and that these on what to pin variant points on the vanishing cycles with respect to beta, QL beta. Okay, induced by an equivalence between the two exact sequence we just constructed. One for sing, one for vanishing cycles. So I will try to sketch the proof of this statement. So okay, so the proof goes as follows. First, we reduce to S strictly local trait and then we do the comparison of the two sequences. So these are the two sequence we have to compare. One coming from sing, one coming from vanishing cycles. So here's what we're gonna do. We're gonna identify the terms of the sequences and then we have to say that the maps, the sequences come from the same, have the same origin. So the maps are the same also. So let's do the first strictly local case. So in the strictly local case, we started the sequence of vanishing cycles. We have to compute first inertial variants of the nearby cycles, which is the last term, okay? And this computation just by Galois descent in the strictly local case, I mean, in general, this computation would give J tilde ramified, but in the strictly local case, this construction just gives J. So taking a monopie invariant points just gives back pullback push forward straight on X. Okay, so this is just the first computation by Galois descent and the projective proper base change, okay? So now comes a more subtle point, which is the identification of the term in the middle, okay? So we have to compute a monopie invariant points for this push forward, okay? So the first thing we should notice is that the action that exists here is actually the trivial actual of I because this is just pulled back to the special fiber. So by a base change formula, this monopie invariant fixed points is just the object itself tensor by a monopie invariant fixed points of QL. So we have to compute a monopie invariant fixed points of QL and the computation can be done as follows. So first we observe that QL for sigma is essentially the image of is QL, so it's the pullback of, sorry, it's nearby cycles for eta, okay? So when we pass to a monopie fixed points, these things are still equivalent. So in the end, the point is that now the computation is reduced to the previous computation, which is a Galois descent statement that says that this is just pullback push forward on eta and sigma. Okay, so now we have to compute this and this by purity for the trait just gives back QL plus QL with a tape twist, she's about minus one. I think I'm not a note of purity for the trait. I mean, if you're asking for a good purity for this relation, I mean, this was an old. Yes, in SGA probably, yes, okay. I'm sorry. So the point here is that this construction, because we are working in this infinity categorical setting, gives back a new algebra structure on these amortopie invariance fixed points, because amortopie invariance fixed points is a lux monoidal construction. So we get that the algebra structure on amortopie fixed points is induced by this isle upper star, j lower star, which is also lux monoidal, is induced by the trivial algebra structure on Q eta, okay? So by transfer, we get an algebra structure on amortopie fixed points. And this is gonna be important to us. So now let me say this two pretty city, an important fact is that we have this two pretty city, both pretty city, when so when we got something twisted by minus one, minus one, which is what we got from the previous computation, you can twist everything and minus one, disappears, you just get one on the other term, okay? So we can do this computation. So the sequence that defines vanishing cycles just becomes this sequence, which has the same objects as the sequence coming from sing. And now we have to identify the maps, okay? So how do we compare the two maps? So let me go back to the previous sequence that was defining the motif of sing. So we got it from this combination of these two triangles. The point is that as this map here is not amortopic as we saw before, it comes, the map in the first horizontal map actually factors through the shift minus one of the last term. Okay, so in this case, this map theta by both pretty city, we can identify this previous shift by minus one by a tight twist and a shift by one. So what we get is a K-theoretic analog of the cycle last map of the pair Z and X as described by SGA foreign health and by Gapar. So we get this class and I have to describe now the realization of this class and what's the role of this class? So in SGA four or also Gapar's proof of purity, we have the cycle class map, which we can see as a class in H1 of eta with respect to theta object, which is sent by the exact sequence to H2 of L of S, okay? And the image of this gives the first term class of the conormal bundle of sigma in S. So the point is that if we start with this class, we can pull it back to X eta. So it leaves over eta, we go back to X eta and we can tensor with beta, okay? We get a map of two periodic complexes of two periodic shifts. And by a junction, this gives us a map as the one described there, okay? And this map extends to, by using universal property of three modules, so because this guy here is an algebra, we get a map of P over star BUX zero modules, okay? So the claim is that this map we get here is exactly the eladic realization of the map we got before on motives, okay? So we had the map theta on motives, you realize it and you get this map of chaotic shifts. And the reason for this is purity and the compatibility with the six operations, okay? So this map that we get from the sequence thing comes from the cycle class map. So at the same time, we had, so this QL, homotopy invariant points comes with the canonical algebra structure as I tried to explain, and which is induced by this function, I upper star, J lower star. From the algebra structure on QL eta. So the point is that now this homotopy fixed point is equivalent to this QL plus QL minus one shifted by minus one. And the claim is that multiplication by this element of T degree minus one, minus one, we can also see it as a map like this. And the claim, so following results of Rappapoch-Zink and Iluzila's Orgogoso and Gabor's uniformization, we get that this map here is essentially induced by the same class map, cycle class map, under this pullback, push forward, okay? So the point is that these two constructions, the construction of homotopy fixed points and the construction of vanishing cycles and the construction of Zink are both coming from this cycle class map. So in the end, we managed to use this to compare the two sequences. So the two sequences are induced by this class theta. So in the end, what we get is that so the sequences are, we get an equivalent between the two sequences, so the two terms, sorry, the two terms are gonna be equivalent, okay? So this is how we deduce the theorem. So step six is a reduction to a general excellent trade. So this was a strictly local case. So we need to use Galway descent for Zink and we need to use the compatibility of this QL realization of digit categories with push forwards, okay? So, but I will not go more on details on that. So just some concluding remarks. So we get this identification. From one side, you have the motif of Zink. From another side, you have this motif, the two periodic version of vanishing cycles, motor-by-fix points. So now you have something curious that we have to, I will explain in a while. So from one side, you're gonna have an action because the function we constructed Zink was Lax monoidal. So we have an action of Zink S0, okay? And Zink S0, we can compute by the exact sequence we, sorry, the motif of Zink F0. If we take the exact sequence we had, we had Zink and some other terms, you can show that this motif is essentially QL beta plus QL beta with the tape twist, okay? So because the construction is monoidal, Zink zero is a unit, this object inherits a symmetric monoidal, sorry, an infinity algebra structure, okay? At the same time on the other side, we also have an infinity algebra structure. So the claim is that these two, the equivalent we had before constructed from the sequences is compatible with these infinity algebra structures. Okay, so just another consequence of this theorem is that we managed to construct a sharing character from the K theory of MF to this homology of vanishing cycles, the two periodic homology. And that's it. So in the next lecture, Gabriele Bertolzzi will give, will explain how to use these results to approach Bloch's conductor conjecture. So thank you very much. So since your result is about the invariance of the inertia of vanishing cycles, why don't you formulate it in terms of the same vanishing cycle? Oh, I don't understand, sorry. Why don't you formulate the result in terms of the same vanishing cycle? You mean because the inversion, the action of the inertia factors? At the end, you are taking just the invariance in the inertia. Yes. So it looks like it's more reasonable maybe to approach the problem here, the same vanishing cycles. The same vanishing cycles are much simpler. You understand? But in this case, so when I take the inversion variance, the action of the inertia will factor through the taming inertia, right? To me sometimes it's the question about why you formulate your statement about vanishing cycles. It looks like it's more a statement about tamed vanishing cycles, more than the inertia. I think the state, it should be a more general statement, maybe you agree with me. It should be that before taking heladic realization, the multiple seam should be the same thing as a tamed vanishing cycle to Bayou, in the same sense. So what is the obstruction of doing this at the equation? The obstruction is time. And also that more mathematically, the obstruction of Bayou is not the motive, it's a diagonal movement. So you have to make sense of this. But I think this is both the same. If you just consider the time. Yeah, but it's a matter of time, as you said. I understood the question. If you just consider the time, you can do these in motifs, at least in et al motifs. Yes.