 Okay, thank you very much. So I'm gonna talk about the joint work with Grigori Garkusha. And the main idea of this work is to construct an explicit fiber replacement of a certain spectra in the stable material model category. So in this talk, we're gonna work over the Batesfield key, which is infinite and perfect field of characteristic difference from two and we'll be interested in the stable motivic model category, right? We're gonna need two descriptions of this category. Either we can think about this as a category of P1 spectra or the category of GM S1 by spectra. And the main target is to construct explicit fiber replacements of certain spectra in this stable motivic model category. So the main example, the main motivating example is the algebraic cavodism spectrum, MGL. That's the spectrum that consists of the spaces which are the tom spaces of the tautological vector bundles of rank I. So tau I here is the vector bundle over the Grasmanian I, where Grasmanian I is the co-limit of Grasmanian of subspaces of dimension I in the space of dimension N when N goes to infinity. So then there is a tautological vector bundle over this Grasmanian and then the tom space of this tautological vector bundle is the space in the algebraic cavodism spectrum, MGL. And let me first formulate our main result for the spectrum MGL and then I'm gonna talk a bit about similar spectra that looks similar to MGL that also consists of the tom spaces of certain vector bundles. That's what we are calling the motivic tom spectrum. So let me start with the construction with the explicit construction of the fiber replacement of the spectra in the stable motivic model category. So for this construction I need several steps. The first step, I need to introduce the notion of omega correspondences. So if U and X are smoothed varieties over K, I'm gonna introduce omega correspondences from U to X as a group point of the certain objects. So the objects look like this, these are the heads. Use the X where the map P from Z to U is finite, flat and locally complete intersection morphism. And Z here is some scheme over K. It not need to be reduced, it's just a certain scheme over K that's with a locally complete intersection map to U, which is finite and flat, right? So these are the objects and isomorphisms in this group where it adjusts the usual isomorphisms of these heads. If we have two such heads, Z and Z prime, an isomorphism between them is just a map that can be used with both projections and both maps from the top of the heads to the variety X. So that's the group point. And then for every smooth variety U, we can associate the nerve of the group point core omega from U to X. And this is gonna be a pre-chief of simplicial sets. So that's the first subject I need to introduce. Then we can construct a fiber replacement from the subject just in two more steps. So the second step I need to do, I need to apply the single or simplex construction C star. So C star and core omega of U X, that's a simplicial object that looks like this. So it's zero simplices are just nerve and core omega U X. It's one simplex. One simplex is this N core omega of U cross delta one over K into X. Two simplices are N core omega of U cross delta two K, X and so on. So then that's again, it's gonna be a simply show pre-chief and delta one and delta two here. I just algebraic simplices. So delta one K is just a spectrum K X zero X one, where zero plus X one is equal to one. Delta two K is the algebraic two dimensional simplex and so on. So we apply this simplexial construction to this nerve of the group void of the omega correspondences. Excuse me, Sasha, there's a question for you. What does the omega stand for? Omega here is just a reference to the co-boardism. It's really, I mean, it's just for some historical reasons. There's no real reason to call it core omega. That's just how we call it. Okay. Okay, thank you. So the next step, we need to make it an S one spectrum. So we're gonna construct a final replacement in the category of GM S one by spectrum rises naturally as the GM S one by spectrum rather than P one spectrum. So here we constructed just the spaces, the simplexial pre-shifts. So now we need to make an S one GM by spectrum. So first of all, we construct the S one spectrum and it's constructed also in a standard way. So to every U, we just associate the spectrum C star and core omega U X smash S. And that's a spectrum that consists of the following things. C star and core omega U X smash S one C star and core omega U X smash S two and so on. Right, instead of X, we can put a simplexial scheme as X plus smash S one, X smash S two and so on. And then we get this construction that makes it a C star and core omega X of U into a simplexial scheme is also a simplexial pre-shift. So then the sequence of simplexial pre-shifts naturally becomes an S one spectrum. And then the last step, we need to make it into S one GM by spectrum where, so it's a GM spectrum of S one spectrum. So it consists of the following things. So the first element is the aspect from C star and core omega U X smash S. The next is C star and core omega U X smash GM smash one smash S and so on. The next gonna be C star and core omega U X smash GM smash two, smash S and so on. Where GM smash one here is a simplexial scheme. Also that's just a cone, simplexial cone of the inclusion of the point into the pointed variety GM. Sasha, a question for you again. Yes. Would you mind describing the structure maps in step three? Oh, the structure maps in step three. Okay, so we have C star and core omega X, sorry, U X smash SI smash S one goes to C star and core omega from U to X plus smash SI plus one. So basically this map is constructed in as the following sequence. So first of all, we have a map into C star and core omega U X plus smash SI, smash S one. And then it goes over here where, yeah, the idea here is that the C star and core omega X it's covariant with respect to the second argument. So when I have any simplexial scheme and I put it on the second argument, I automatically get a simplexial reshift here. Okay, great, thanks. Okay, right. So here I did the first step. So here I got the S one GM by spectrum and this is everything we need to know. I can formulate the following theorem. Yes, or I didn't give a name to this. Let me call this by spectrum C and C star N core omega U X smash G smash S. So then our main theorem tells that this S one GM by spectrum C star N core omega X smash G smash S is locally equivalent to a motivic fiber. So it's not a fibrant replacement itself, but it's locally equivalent. In particular, it has the same homotopy groups as the fibrant. So it has the same local homotopy groups as the honest motivic fiber replacement of the spectrum X smash MGM. Or more precise, to make it a fibrant replacement, what I need to do, I just need to, I just need to make it a fibrant replacement what I need to do. I just need to put every step of my GM spectrum, of my S one GM by spectrum. I just need to apply the fiber replacement in the local model structure without any A one occurrences. All right, so if I do this on every step, then this S one GM by spectrum will be there, will be motivically fibrant and will be weakly cool into the spectrum X smash MGM. Right, so in particular, if we wanna compute say, sheaves of homotopy groups of the spectrum MGL, the sheaves of homotopy groups will be the same as, so A one homotopy groups of the spectrum X MGL computed on local Hensel and U will be just homotopy groups of this A one S one spectrum C star and or omega of U X smash S. Okay, so that's the main application and similarly we can do a similar result for other spectra that are similar to MGL so other examples where we can state similar results are MSL and MSP. So these are the spectra for special linear co-bordance that sublactic co-bordance. But let me talk a bit about how this result is related with the theory of frame correspondence. Basically, this result is just an application of the techniques that are available to us thanks to the theory of frame correspondence. So the frame correspondences were introduced by Wojewódzki, developed Garkus and Pani. So here let me look at a bit more general situation. Suppose E is a Tom spectrum and by Tom spectrum, I mean the spectrum that consists of Tom spaces. So E has spaces E0, E1 and so on, where EN is a Tom space of some vector bundle VN where VN is a vector bundle of rank N over XN, right? So the main examples of this Tom spectra are suspension spectrum, X plus dash T X plus dash T square and so on. The second example that we've already seen is just the X plus dash MGL or similarly, we can think about X plus dash MSL and X plus dash MSP. But we need to be a bit careful here because naturally there arises T square spectrum but again similar things can be done for these two spectrum. But let me mostly concentrate on these examples on the suspension spectrum of a smooth variety and on the just MGL spectrum. Then we can, if we have such a Tom spectrum, then we can introduce the E frame correspondences from U to X as just the home set in the category of pointed misnaved chiefs from U, MSP, MSN, into X plus smash E N. And by Lemo Wojcicki, this set of homes in the pointed category of misnaved chiefs has the following geometric description. So all these homes can be described using the following geometric data. So first of all, that's a close subset into affine space A N over U, which is finite over U. Second ingredient here is an etal neighborhood of Z in the affine space. And the next is the map from the etal neighborhood W into the vector bundle V N such that Z is the pre-image of the zero section Q. And also the second part of the data is just a map from etal neighborhood to the variety X. So this map phi here that defines Z inside V, we call them a framing. And that's the frame, the E frame correspondence that arise with respect to the Tom spectrum E. So yes, so this sets of data describe this morphisms up to a certain equivalence where the data Z, W, phi, F is equivalent to Z, W prime, phi prime, F prime. If W and W prime are two etal neighborhoods of the same close subset Z and all phi and phi prime, F and F prime coincide on their common refinement on W cross A and U, W prime. So in this data, we can just shrink our etal neighborhood W and the correspondence stays the same. So then that's the description of this home set. Excuse me, just a quick question, Sasha. So does this also holds without A1 invariance? That's, here we don't discuss any A1 invariance. So this are just, so here we just talk about the pointed Nisnevich sheaves. So there is no A1 invariance here so far. Okay, thanks, thanks. Okay. And then if E is just a suspension spectrum, it's just point T, T square and so on. Then this frame, the E frame correspondences from U to X are just usual frame correspondences from U to X. Okay, when we consider just a suspension spectrum here instead of vector bundle VN, we just have a trivial vector bundle over a point. So then we can use their theory of frame correspondences to use this E frame correspondences to construct a fiber replacement for the material spectrum E. So again, we can do this in two steps. First of all, we construct an S1 spectrum MEX that again, I need to apply the C star construction to this pre-sheaf frame E from U to X and then make it into an S1 spectrum, right? And I didn't say what is frame E here, frame E from U to X is just a co-limit when N goes to infinity of frame N, E from U to X where I can embed frame N, E from U to X into frame N plus one, E from U to X is just, I just need to embed A and U inside A and plus one U and then instead of W, I take W across A1 and so on. So there's a natural stabilization process that makes the level of this frame correspondence bigger. Right, so then I take the co-limit with respect to N, apply the simplicial construction C star and make it an S1 spectrum. So then what I get, I get this S1 spectrum MEX and then we can make it into G spectrum using the same construction as we saw before. It's a S1 GM spectrum consisting of MEX, MEX smash GM smash one and so on. So that's gonna be an S1 GM by spectrum and then the general theory of frame correspondences tell us that MEGX is locally equivalent to a motivic fiber replacement of X smash E. So MEGX is locally equivalent to the motivic fiber replacement of E smash X and this motivic fiber replacement looks like this. Again, I just need to apply local fiber replacement. So MEX smash GM, smash I, F here is a fiber replacement of MEX smash GM smash I in local stable Nisnevich stable, S1 stable Nisnevich model structure. So in particular, again, if we interested in the, yes, it's locally equivalent to this one, sorry, and this one, this S1 GM spectrum is motivically fibrant and is equivalent to X smash, sorry, smash E. So in particular, if we interested in the homotopy groups, then the homotopy groups can be computed explicitly as the homotopy groups of the corresponding S1 spectrum. So that's the general result, it works for every Tom spectrum. So it works for the suspension spectrum, it works for MGL and so on. And next for MGL and some others Tom spectrum, we can further make a reduction and make this construction a bit simpler. So let me show you the steps of the reduction we needed to do in order to obtain the result for MGL. So the first step, that is also a general one that works for every Tom spectrum is the following. Instead of considering the frame correspondence, we can consider an object, which we call normal frame correspondences. And that's the following thing denoted as frame and E tilde from U to X. So that's the set of the following geometric data. So first of all, again, we have a variety U, we have affine space A and U, and we have a closed subscheme of A and U. So Z inside A and U is a closed local space. We have a fully complete intersection subscheme and the map from Z to U is finite and flat. Then again, we consider a tall neighborhood of Z inside A and U. But now instead of the whole map from W to the vector bundle VN, we just construct a map. We just need to consider a map from W to the base XN of the vector bundle. So here VN is the vector bundle over XN from the definition of the Tom spectrum. Here we consider the map from the et al neighborhood Psi, the inclusion I. And then instead of framing here, we have the isomorphism between the normal bundle of Z inside A and U and the induced vector bundle by this map Psi I from Z to XN, right? And then the second piece of information is the map from Z into X. And then we have a similar equivalence relation where we can shrink a bit et al neighborhood W when we shrink et al neighborhood W. The normal bundle will not change and we just need for this isomorphism fight to stay the same when we shrink the et al neighborhood. So here what we did instead of having the, having a whole set of equations that define Z inside W, we just remember one piece of information. We just remember how these equations give us an isomorphism between the normal bundle of Z inside W, which is the same as normal bundle of Z inside A and U and the actual induced bundle on Z that's induced from XN, right? So that's a normal frame correspondence and as a side remark, it looks especially nice where XN is just a point. Right, so where VN is just a trivial vector bundle of rank N, XN is just a point. Then what we have here, this data is just equivalent to the following data. We have Z, which is finite flat over U. It's a locally complete intersection sub-scheme of A and U. And then we have a trivialization of a normal bundle of Z inside A and U. Right, let's say X is also a point here. So then we have just locally complete intersection sub-schemes of A and U with a trivialization of the normal bundle. And that it's really very similar to what's known in topology as frame cavortism introduced by Pintragan. So that's the first reduction step we need that instead of the whole framing, we just keep track of the isomorphism on the normal bundle of the supports. We're gonna call Z here, the support of the framed correspondence. So it turns out that instead of the whole frame correspondence, we can just consider the isomorphism of the normal bundle of the support with the bundle induced from VN. Right, so. So then the theorem says that if I construct in the same way the normal frame motif E of X, which is just C star frame E tilde of X C star frame E tilde of X meshes one and so on, then M E tilde. So there's a natural map from M E of X to M E tilde, right on the level of frame correspondence. And when we have a frame correspondence like this, then it defines Z as if I consider not just a closed subset defined by the equations phi by the whole closed sub-schemes Z defined by phi, then it defines Z as the whole closed sub-schemes in Z defined by phi, then it's gonna be a locally complete intersection, it's gonna be finite flat, automatically gonna be flat over U. And then this phi gonna induce the isomorphism between normal bundle of Z inside A and U with the bundle induced from VN. So we have naturally a map from a usual frame, E frame correspondence to the normal E frame correspondence just by forgetting some structure. Then it induces a map on the level of S one spectra and it turns out that this map is locally in a coolance. So it turns out that for any E frame spectrum we can simplify our construction like this. So that's the first step, it works for every term spectrum. And now let me show you other steps that work specifically for the spectrum MGL. So for E, it goes MGL, we can simplify the construction further. So when E is MGL, then VN we have here is just a total logical vector bundle of rank N, right? So this normal frame correspondence we're gonna give us the isomorphism of the normal bundle of Z with a pullback of the total logical vector bundle. And then it turns out that we can get rid of this information as well. So we can introduce another object which I call embedding N from U to X that consists of the following data that consists of embeddings of Z inside A and U which is locally complete intersection. So that the projection onto U is finite and flat and then just a map from the support Z into X, right? So here then we have a natural map from normal frame correspondence from U to X into this embeddings and UX that just takes all this data Z, W, Z, Psi, Phi and F and just forgets everything keeps only Z and F, right? So then we can show that this thing after we apply the C star construction is an equivalence. So C star frame and E from U to X into C star embedding N from U to X is an equivalence of simplicial sets for any smooth affine scheme U, right? So for the case of MGL, we can forget about the framing and the only thing we need to keep track of is the support actually. And then the last step for MGL, we can also forget about the embeddings of Z inside the affine space, right? So there is a natural map from embedding and UX into the set of core omega, UX where we take this embedding data and just forget the embeddings. So we have, we just keep Z, U and X. So we need this map peak here to be finite flat LCI map from Z to U. So that's what I started with the set of omega correspondences, right? And then it turns out that when I apply the simplicial construction again, so here I map to the set of, to the zero, zero simplices of the nerve. So when I consider a map from embeddings and into nerve of core omega of UX then it turns out that this is also, is a weak equivalent of simplicial sets, a smooth and affine. And then this is how we arrive at the result I started with. So we start with, we start with the general construction of a final replacement for every term spectrum MEG of X. Then we reduce it to the spectrum that we get using the normal framed correspondences. And then for MGL, we first of all reduce this normal frame correspondence to the spectrum that can be constructed using this embeddings and then to the spectrum that we construct using this omega correspondences. All right, and another remark here is that embeddings and UX is representable by a smooth scheme. So that's gonna be the smooth part. It's gonna be a LCI part of the Hilbert scheme of A and U and then C star, then the spectrum C star embedding X, G, S that we can construct from this data will be a local model for motivic fiber replacement of X mesh MGL which is representable just by co-limits, directed co-limits of smooth, simplicial schemes. So I guess it's a good point to stop here. That's what I wanted to tell you about. Okay, thanks. Thanks a lot, Sascha. Thank you. Wonderful. I'm gonna press a whistle. Let's see, I'll do some questions. A little question, can I ask? To what extent does this tell you about the various P1 suspensions, the maps on the P1 suspensions of things like MGL or MSL, like negative or positive suspensions? So naturally these things they live in the GM S1 spectrum. So it's not clear, but does it tell you about the P1 suspensions? Mark, if you positively suspense Tom Spector, everything works. Yeah, but on the negative ones, it's unclear. Negative ones are unclear exactly. Oh, okay, good. Okay, great. Thanks. Any other questions? Just maybe this is a question from many times. So in this remark, you described the say take X to be correct. So take local model for the Mach-Evic factor replacement of MGA. And it is representable as you described by smooth, superficial, coordinates of smooth, superficial things. Can we just work in with this model proof consolation theorem for this by Spector? The GM consolation? Yeah, means not taken back to original kind of omega MGA correspondences. I'm not sure if you can. So your question is, can you get the GM consolation directly from this description? Yes, yeah. I don't know, it's hard to say. It's not clear to me. Okay, so that was the question. And I got an answer. Thank you. Thank you. Thank you, I can wait a little bit longer for possible other questions. And maybe I will ask one more question. Could you take that a little bit up? Say, okay, say for the step three. In the step three, you require U is to be smooth and defined. Could you comment why it is enough? Means that typically one should expect that U should be taken to be local Hymns alien. And in your case it is smooth and defined. The point is why it works for smooth and defined because actually we are able to construct an explicit homotopy between. So if you have the same Z over U but you have different embeddings of Z inside an affine space, then you can construct a homotopy that drags one embedding into the other embedding. And for this thing to work, it's just enough U is enough to consider a smooth and fine U. Uh-huh, okay, very good. And is it kind of the same principle which works for the previous step? Means for the step number two? Yeah, for the step number two, you mean the step for normal frame correspondences. This you have shown to me the step number one, but go a little bit down there. Yes, step number two. Yeah, here it is here. The step number two goes from the normal frame correspondences to the embeddings. Yes, again, the reason why it works over the smooth affine U is that we're able to construct an explicit homotopy that works over smooths and affine U. So the point here is that when we talk about this normal frame correspondence for MGL, then what we have here, we have just a total logical vector bundle, VN. And so the framing here is an isomorphism of a normal bundle with a pullback of the total logical vector bundle. And every vector bundle is isomorphic to the pullback of the total logical vector bundle over the affine variety. So actually turns out that we can just, if we have two isomorphisms to the pullback of the total logical vector bundle, then again, what we can do, we can produce a homotopy that sort of transforms one into another in a consistent way so that it makes this as cool as possible. Okay, thank you. Thank you, Sasham. Okay, we have some more questions for you, Sasham. I can read them. So Tonya Nala asks, what part of the geometric part of ultrubatic cobaltism can be represented by these methods? What part of? Of the geometric part of ultrubatic cobaltism can be represented with these methods. So geometric part, do you mean two N and diagonal? I guess. Okay, so actually locally, we can compute everything using these methods because if you have a AB homotopy group of A1, say MGL smash X on U, U is local zillion, then using these methods, this is gonna be equal to just by A of C star and core omega of, sorry, I guess I need to write A minus B here, right? Because we have a simplicial sphere of dimension A minus B and here we have GM smash B, smash U into X. Yeah, the point here is that, so GM smash B, smash U is not local Henselin anymore, but because everything here, all the sheaves we have here, they are framed sheaves. And for the framed pre-sheaf, which is A1 invariant and with some additional properties at hold here, we know that it coincides with this sheafification on the open subsets of affine space. So actually this computation here of the A1 homotopy groups, it works not only for local Henselin schemes, but also works for all the open subsets of affine space. So every A1 homotopy group can be explicitly, can be computed as the usual, simplicial homotopy groups of this. So sorry, I need to write down S1 spectrum. So when U is not local, we don't have this result, but when U is local Henselin, we can get this description. Okay. So maybe we'll specify a little bit. The previous question, not of mine, but you try to answer it. Say, substitute U to be speck of a field going on in this field. When U is what? U is speck of a field. Okay. And say X is a point. So you answer on the question of the person before mine, as I understood you answer it like this, that you take this pi a-b of this simplicial set. And so this is the answer, yeah? Yeah, this simplicial spectrum, but it's omega S1 spectrum starting from the first. So we can actually rewrite it like this. You can say it's a-b plus one of this simplicial set C star N or omega of GM smash B K plus smash S1. So then that's an explicit simplicial set that computes the promote A1 homogeneity of MGL. Yeah, good. Then there's two more questions, at least two more questions, Sasha. So one of the questions says, is this the non-borel-more-bordism of X? Is non-borel-more-bordism of X? I'm not sure what's exactly the question. So maybe that's because the map to X is not proper. I think that's the question. Oh, what's the difference with the algebraic-cobordism of this construction? Yeah, the point is that Z here, first of all, Z is not a variety. Z can be a scheme. And also, yes, the map from Z to X is not proper. It's just any map. So Z here is a scheme. And the only thing we need from Z is that it's the map from Z to U is a locally complete intersection map and it's finite and flat. Okay, so that's good. Then another question from Ola Sande. Could you explain why you need to take the nerve of core omega? Oh, that's because when we forget about the embeddings here, the point is that we need to keep track of the isomorphism between the tops of the heads because it's not gonna work if I just take, say, an isomorphism class of the heads of core omega. Just, we'll not get a wicked clue here. So, because when we forget about the embeddings here, we still have to keep track of the isomorphism on the top of the heads. Okay, great. Then Kirsten asks, if you can say something more about the relation with Hilbert schemes. Oh, yeah, so the relation with Hilbert scheme is pretty straightforward. So when X is a point, say, then this embedding N of U into the point, that's just the, by definition, that's the sub-scheme of a Hilbert scheme that consists of LCI schemes. And it's known that this part of the Hilbert scheme is smooth. So this embedding, so U goes to embedding N from U to K is just represented by a sub-scheme of Hilbert scheme of A N, that then the sub-scheme of Hilbert scheme of A N is exactly this sub-schemes of Hilbert, the sub-schemes that form the Hilbert schemes that are locally completed intersection sub-schemes. Okay, so I, seems that wraps up the question round. So thanks a lot, Sasha, and also thanks to all the attendees. Wonderful to have you. And we'll see you all tomorrow. Thanks a lot indeed. Thank you. Thank you very much, Sasha. Okay. Lidry, how to say our pleasure in French? Our pleasure? Possibly, it was on pre, something like that. Yeah, yeah, that's it. Do you see, I'm improving, Sasha, during my stay in Digital Paris. Very good. At least I promise to learn more French words during the two weeks. That's it. You did. Okay.