 Thank you very much Paul once again to organize this for the party to speak on this topic and in the second lecture I described the category SH frame needs of K and proved the main theorem of the second lecture which stated that this category is equivalent to the category SH of K due to Mariela Lewowski. So then I might remark that the category SH frame needs of K is translated category in a natural way and I formulated lemma 32 which is a simple lemma that found this event bar big frame. I'm usually in this triangulated triangulated equivalencies of triangulated categories SH frame needs of K and SH of K. So this category SH frame needs of K maybe looks a little bit artificial the main purpose of this lecture of the first half of this lecture is to show that it is not artificial at all so it can be constructed it is in fact coincide as we will show with the category of local objects for a localization function so but before to do that let me recall this formalism of localization function so given a triangulated category SH we define a localization in tau as a triangulated endomorphism L maybe I will pronounce L tau to tau together with a natural conformation eta such that eta evaluated on the morphism eta x coincide with the morphism eta of L x for any x in in tau and eta induces anisomorphism L x to L of L of x we refer to L as a localization function in tau such a localization function determines a full subcategory care of L whose objects are those x such that L of x vanishes an object x is called L local so the morphism eta x from x to L of x is an idomorphism we write we will write tau log for the full subcategory of L local objects the phanta alien uses a phanta L bar between tau modular care of L to tau log let's in with the full embedding and let you know from tau to tau modular care the factorization function the formula simply is well known and in fact can be proved pretty clearly namely the following is true the phanta L bar between tau modular care of L and the phanta yeah the phanta which takes value in L log and the phanta you compose this in which works like that we first embed a T log tau log into tau and then project to Q project like Q tau usually your strangulated equivalences secondly the phanta L from tau to tau log and the inclusion the full embedding from tau log to tau I let and write a joint of each the other so this is a general formalism simple formalism and now we would like to apply this general formalism to our specific situation for that let's consider the phanta M bar big frame between from SH of K to SH frame is of K with a phanta from the construction to 0.7 so it's in fact explicit formula is this if you take an object E then it takes its one should first take its front tutorial cos co-fibre and triplacement H A then apply the construction C star frame and then apply stabilize in Jn direction so this is explicit formula for M bar big frame so take this phanta and set M big frame to be the composition firstly projection from SH means of K the projection P to SH of K and then apply the phanta M bar big frame which lands in SH frame is of K so we will get this phanta which we call M big frame and the phantas M big frame and M bar big frames are called big frame more big phantas in these ledges. Bonner, excuse me, Bonner can you see the question in the Q&A? No, it means a sort of tangential question is the following one of the true for frame correspondences. Let me maybe try to answer this question at the very end. Okay so we just keep the question. Thank you. Yeah, yes, sorry. So the definition let I be the full-impedient and P be the canonical phanta and set frame motive to be the this composition the composition of three phantas so firstly we project from SH means to SH of K then we apply M bar big frame and we land in this frame and then we take the full-impedient high so we land in SH. That explicit high frame motive cofee So, in theory, it comes with a T-constructed interpellation in the bicep. So, in theory, it comes equipped with a phantom transformation eta that takes a bispectron T, two morphisms in s-h, n-th of k, determined by this zigzag. So, the left morphism is the cofibre replacement, and the right-hand side morphism is taken from the construction 2.7, I think. So, in the second, from the second lecture. So, this second morphism is, I recall, is a magnetic appearance, and the morphism beta-e is a level-wise local appearance. This is the key proposition. The poly-inertions are true, because they are frame-motive, and eta is a localization in s-h, n-th of k. Secondly, the kernel of a frame-motive coincides with the kernel of p. And kernel of p, as we know, is generated by cones of invariance, or cones of projections of this form x cross a1 plus 2 sigma of x, and we need to take cone of that, and shift of this cone. Then, this is regarding the second point. And third is the most important, that this full-sub-category i of s-h frame is of k, is the full-sub-category of l-frame-motive local objects in s-h, n-th of k. So, like you'll say, that kind of first time, or maybe the second time, I'm giving you the proof of lecture. So, a sketch of the proof of the assertion 1. Recall that f is just p pre-composed with i. And also recall that the Fanta m-bar big-frame is quasi-inverse of the Fanta f by theorem 2,3. So, consider the chain of Fanta isomorphisms. All isomorphisms in this chain by, in fact, equalities. And so we write down what explicitly was this l-frame-motive. So, it is this, this, then the second copy of l-frame-motive, this is this. We merge p and i, and we will get f by the very definition of f. And then this Fanta is isomorphic to the identity of s-h of k. And so, therefore, we have this equality. And, but the last equality is the very definition of the Fanta l-frame-motive. So, we prove the first assertion. Both now the second assertion. Take e, which is in the kernel of e, which is in kernel of p. Then e is in the kernel of this composite Fanta, of this composite Fanta. But this Fanta is l-frame-motive. So, therefore, e is in care of l-frame-motive. Take now e, for another inclusion, take now e, which is in care of l-frame-motive. So, write down this explicitly. And then e in care of this, moreover, is in kernel of this composite. The first two p-compose with i gives us the Fanta f. And the composition of f and m-bar, big frame, side with the isomorphic to the identity of s-h of k. Therefore, we have this equality. And the last Fanta is the Fanta p. And therefore, we have this equality. Therefore, we start with another e, which was in kernel of frame-motive. And we prove that this e belongs to the kernel of p. So, this proves the second assertion. I would like to say that the key, kind of the most difficult assertion, is the assertion 3. So, because it computes the category of h-frame-motive, local objects, s is the category of i, s-h frame, is of k. Typically, it's very difficult to compute explicitly the category of local objects. So, let's start with the proof. For the Fanta f-frame-motive material, one has this inclusion. Because the last Fanta, so it is i composed with m-bar, big frame, and composed with p. But here there is this i, which implies this inclusion. So, for each key, we claim that this category is contained in the full subcategory of l-frame-motive local objects. Indeed, take an object key in this category and continue there. Note that both object e and l-frame-motive of e are in this full subcategory, s-h frame means of k, i of s-h means s of k. And recall that the arrow and recall that the arrow at the e becomes an isomorphic in s-h of k. This is due to the fact, which I recall you about, that this morphine is a Mach-Euf-Führer. This was formulated in the second lecture. So, taking back to the proof, yes, and recall that this arrow becomes, so this arrow becomes an isomorphic in s-h of k. By theorem 2, 3, the arrow at the e is an isomorphic already in i of s-h frame means of k. Thus, e belongs to this category of local objects. So, on the other hand, if e is an object in s-h means of k, which is a l-frame-motive local, then this morphine is an isomorphic. But also, e is an isomorphic. But also, l-frame-motive of e always belongs to this category. So, above, we checked that we have this inclusion function. And now, we see that function l-frame-motive restricted to this subcategory lands in this category. Even if we do not restrict this function, this function just lands in this subcategory of s-h means. So, claim is that these two functions are quasi-inverse of each other. And this is very simple exercise. Just one should use the function of the transformation data for both compositions. So, this was the proposition. Now, let me formulate the corollary. It is the formal corollary of the proposition and the lemma and the formal lemma 3-3. And corollary says that the formula is true. So, the function l-bar-frame-motive from this quotient category and the function p-pre-compulsive with the full embedding are usually related categories, related equivalences. And secondly, the function l-bar-frame-motive and the function full embedding function in are left and right adjoint of each other. So, since s-h means of k, the quotient category s-h means of k, the kernel of p. But the definition of p coincides with s-h of k. This corollary and proposition 3-6 yields the following theorem, which is the main theorem of the first half of the lecture today. So, the theorem tells us that the function m-bar-big-frame between s-h of k to s-h and the function f from s-h means of k. So, usually inverse triangulated equivalences. In fact, this does nothing else as exactly the statement of the theorem 2.3. But nevertheless, secondly, the function m-bar-big-frame and the full embedding function are left and right, and the categories s-h means of k and s-h frame means of k are the left and the right adjoint of each other. And I recall that i is the full embedding function. And eventually, the category i of s-h frame means of k is the full subcategory of h frame mod local objects in s-h means of k. So, I would like to summarize the results of this part of this section as follows. We start with the local stable chromatopic category of strips of s-frame spectra, s-h means s-1, which is also closed symmetric monoidal triangulated. Then we stabilize s-h means s-1 of k with respect to the under-front smashing with jm-mash-1. During this, we arrive at the triangulated category called s-h means of k. Then we apply the explicit localization factor l-frame motivic to s-h means of k. We compute the quotient category, or equivalently saying the full subcategory of corresponding local objects is the full subcategory i of s-h frame means of k. And we prove that s-h frame means of k is equivalent to the classical Morale-Weiwowski stable motivic-mattupi theory category s-h of k. So, let me maybe repeat once again that we introduced this factor, which is a localization factor, from s-h means of k to itself, and computed its local objects as this explicit full subcategory in s-h means of k. So, this means that this category s-h frame means of k, which is not at all an artificially constructed category, but this is the category of a frame motivic local object. So, also, I would like to take a look at here, and I would like to stress that this factor and bar in the frame, it is given by an explicit formula using the infinite s-h frame of e-c. So, we take e to this, and this is the explicit A1 localization factor on this category. Very explicit A1 localization factor. Okay, so the first part of my talk is over, and I promised at the very first lecture that I will give some application, and so I pass into the second half of my lecture, which were the course of theorems 1, 1, 1, 2, and 1, 3. But I will begin with the sketch of the theorem 1.3. And for that, I need to recall something, namely, there is so-called additivity theorem in this business. Namely, it sounds like this. For any smooth schemes u and y, we have canonical morphemes of simplicial sets from here to this product, just induced by projecting from y prime is joint union with y2 to y prime in frame nodes, and projecting from y prime is joint union y double prime to y double prime in frame nodes. And the theorem states that this morphemes of simplicial sets is a weak equivalent of points of simplicial sets. So this implies that the single space of this form is special. Now, in fact, we could replace the smooth scheme y by any smooth simplicial scheme. For instance, take a smooth scheme x and consider a simplicial scheme, a simplicial object in frame node, which is x times the rest 1. So, remark, it is quite easy to check that pi0 of sister frame u, x times the rest 1 is the zero, so it consists of one point. So this simplicial set is connected for any x which is smooth. That the single space of this form is very special. And hence the single space is very special. Hence the single S1 spectrum of this form is an omega S1 spectrum. But this S1 spectrum, just topologically coincide with this spectrum. So we take the suspension spectrum, the S1 suspension spectrum of the simplicial scheme x times the rest 1. Then we apply the construction sister frame, we get again an S1 motivicus S1 spectrum, and then we evaluate it on u. And the result, as I told, is this single S1 spectrum. So the latter S1 spectrum is an omega S1 spectrum. Hence the consequences. So firstly, the canonical morphism from the zero space of this spectrum to the infinite loop omega infinity S1 of sister frame of sigma inverse. S1 of X tensor S1 is a weak equivalent for each u and x which are smooth over three, particularly the equality holds. Namely the homotopy, misnevych homotopy sheath of this motivic space coincide with the misnevych corresponding misnevych staple homotopy sheath of this motivic S1 spectrum. So let us apply this theorem 13. I recall that theorem 13 states that for any x and for any field extension k capital over k, here k is infinitely perfect. They are the ordinary homotopy group of the simplicial set frame delta dot k capital comma S tensor S1 coincide with the weight evaluated on k. With the A1 homotopy sheath of the pi suspension of X plus smash S1 evaluated on k. On the left hand side, we have a group on the right hand side. So the statement of theorem 13, which we supposed to prove. First, I recall that this canonical morphism of bispectra is a motivic equivalence. This was formulated in examples in the second lecture in sample one. So this is the motivic equivalence. And from other sides, this spectrum is in SH frame means of k. So this feels that firstly, let me be a little bit slow. Let's denote by E this spectrum, bispectrum. Take, it's a 1 with 0 homotopy group, homotopy sheath. S was stated in the second lecture for each bispectrum E from this category. It's with 0 A1 homotopy sheath coincide with the Nisnevich sheath of 0 weight of the spectrum E. But since my spectrum E is of this form, it's with 0 spectrum E of Nisnevich sheath in a very easy way. One should take just this suspensions, motivic suspensions spectrum, S1 suspension spectrum of X1, Z1 and apply the construction C star frame. This will be exactly the way 0, by trivial reason, by the definition of C star frame construction. Of the bispectrum E. So, therefore, as I explained, there is this equality. So A1 homotopy sheaths of weight 0 are computed as Nisnevich homotopy stable homotopy sheath of this spectrum E of 0. On the other side, we have proved that the Nisnevich sheath coincide with, on the previous page, we have proved that there was this formula in rectangle that the Nisnevich sheath coincide with the stable homotopy Nisnevich sheath coincide with the ordinary Nisnevich sheath of this motivic space. Particularly for any field K capital, which is an extension of the field K, in this theorem K is also infinite and perfect. This I already mentioned. One has this equalities, so this homotopy group of this space coincide with this stable homotopy group of this S1 spectrum, but on the other hand this homotopy group coincide by due to this equality coincide with this one evaluated on page. This is the end of the proof of theorem 1.3. Now let me prove theorem 1.2. By theorem 1.3, we have this equality with finite coefficients as well. Yes, because we have this integral coefficients and therefore we have as a consequence with finite coefficients as well. And somewhere I missed the complex numbers, right. And also there is a deep theorem due to Mark Levine saying that the A1 homotopy sheath of way zero, we find that of the by suspension spectrum of the by suspension spectrum of this form with finite coefficients and evaluated on point, so evaluated on the complex number C coincide with the ordinary stable homotopy group of the space X plus suspended one time with finite coefficients. This is exactly the statement of theorem 1.2. Now let us go to theorem 1.1 and again, due to theorem 1.3, we have this equality this time without finite coefficients. But this time f is point. And this is again Mark Levine theorem, which says that there is this equality. That on the right hand side, we have the first stable homotopy group, sorry, stable homotopy group of S1 group. So remark one can show that. In fact, there is a weak equivalent between. Of two spaces from one side. Simplicial set frame delta dot C, come as one and from other side, the omega infinity sigma infinity in the usual sense. Of S1 talk, let me do not discuss this today. The last I would like to discuss is the single theorem, material equation of the single theorem, proof of single theorem, material equation. I'm not so I will profit in this same formulation as I have written in the first lecture, as I formulated in the first lecture. There is more general statement. Yeah, but let me skip that. So, I will give the proof. And the four. Yeah, so I will start with the end of the proof. This is the material, this is the material space on the right hand side. Sister frame of X number one. And on the left hand side. Omega infinity, sigma, omega S1 infinity, omega gm infinity of sigma gm infinity sigma S1 infinity of X plus smash S1. And this exact is the exact of local equivalent case. This is how one can read the statement of the material equation of single theorem. So, let me start the proof. So, the material S1 square. This one. And this one. Both local wise. Omega S1 square wise. So, firstly, let's take a look on the left hand side. On the left hand side, we applied the material looking fun. And for the left hand side is automatically local wise. Omega S1 spectra. It is even section wise. Omega S1 spectra. And let's take a look on the right hand side. The right hand side is, in fact, on the page one. It is, in fact, section wise. Omega S1 spectra as well. Let us go to the page one. And I will show you. Yes. This is the formula. This is a weak equivalent of simple spatial space. Just for issue x in smooth of case. So, even section wise. Local wise more over. Omega S1 spectra. Since the spectrum E, which is this one, is in SH frame of K, the arrow from its weight, zero S1 spectrum to the Omega GEM loops of E is a stable local equivalent. This is the property. One of the properties of the spectrum from this category, that it's Omega infinity GEM loops. Fanta is computed by taking E of node. So, this is a stable local equivalent. From other sides, this morphine is a stable local equivalent of material spectrum. On the other side, both the spectra are local wise Omega S1 spectra. Thus, this morphine of material spaces is a local equivalent of material spaces. Now consider the arrow this matific space to consider this arrow. As I have shown you appealing to the page one, this arrow is a scheme wise weak equivalent. Thus, the composite arrow is a local weak equivalent of matific spaces. What eventually we have? So, this is a local weak equivalent of matific spaces. Eventually one says so we have this morphine which is local equivalent of matific spaces. Then we have this equality. This is equality. This is just notation. The renautations for e was taken to be this spectrum. Once again, this arrow is a local equivalent of matific spaces as we already have checked. This is the equality. Just the equality. And there is this arrow which is scheme wise. I claim that it is scheme wise of matific spaces. So, on the right hand and on the left hand side we have this, the composition of the two fountas of formula groups fountas. But inside we have by suspension spectrum of X plus Mesh S1 and we have we applied T covering construction to this his covering comes to this founta to this by spectrum. And as I mentioned above this arrow inside is a matific equivalent. Therefore, it is an isomorphin in S H and therefore applying this founta we get seriously an isomorphin of matific S1 spectrum and from another side since this fountas are formula groups founta we will get scheme wise of matific spaces. So, this arrow this arrow is a scheme wise of matific spaces and this arrow is a local equivalent of matific spaces. This is the sigal theorem. So, we proved the matific version of the sigal theorem not in full generality but in very much reasonable generality. So, thank you very much. This is the end of my third lecture this is the end of my most I am very grateful to organize and to participants for giving me opportunity to to perform this this series of lectures. Thank you. Thank you very much Vanja from all of us for a very nice lecture series. So, we can start the questions. Maybe I can ask the first question for you. So, the theory of framed correspondences is a subject that's now in a great shape much thanks to many papers that you've been written but I would like to ask you how you see the how the subject will develop in the next years. How the subject will keep developing over the next years. So, I think that there are quite a lot of questions trying to attack by using this technique and also we all know that many people including very young people use this technique and part of them use the internet category technique to develop this frame business from slightly different viewpoints. And maybe I would like to mention just one conjecture which is kind of a favorite of mine. So, one could replace frame correspondences which we could call right now by rational frame by regular frame correspondences because we have used just regular functions by rational frame correspondences. Rational frame correspondences are defined in the very second page of the Wiewowski notes on frame on frame correspondences and sorry also right conjecture. So, they look like that frame rational of and instead of using regular functions we can use rational correspondences as I thought. So, then we have this nice error is regular functional function and therefore we have this error and it is stated, yes, this frame so it is formulated in the second Wiewowski notes on frame correspondences which are now available on his great page that this space is always looking omega p1 infinity sigma infinity p1 of X plus. I liked very much this conjecture and I think it would be wonderful to prove such a conjecture because this okay, so you need this space is perfect material looking of of X plus. This is one conjecture I would like to see to be proven in the future but also you could see that several times several times I built to the beautiful theorem and deep theorem due to Mark Levine and I would be very much happy and some other people would be very much happy to prove to avoid Mark Levine result namely for that there is only one ingredient which is topological one which is absent namely one need to show that constellation theorem holds of the topological of the context realization of the of the script term of the form frame of sigma of X plus yes, and then if I context point with that as was explained more or less as people know from the as was explained in session and that the list can be replaced by all the terms of the biosphere can be replaced by take complex point of that and one should expect and from Mark Levine theorems there is how to say there is a reasoning for that that there is a constellation theorem I mean topological one for taking complex point that the previous column S1 is the ordinary S1 loops of the next one which sounds like how is it called doltom theorem for frames in topological frame context or doltom theorem for complex this would be one to prove to my mind this is first very much expected and secondly it is a nice application and maybe the last what I would like to say that that the cdh context so this frame business should be developed I think to cdh context maybe widely for the second right now maybe I will stop on expected questions there are more questions maybe I will fix them over the summer and put on the web I don't know like that okay thanks a lot Vanya for sharing all your insight on this subject are there any other questions doesn't seem to be the case so then let me thank you for wonderful lecture series so thank you very much thank you very much I would say not very easy to be a lecture but as I noticed it's more I think more difficult from what have to be an organizer so sure absolutely so I have seen samples of other people I can imagine something like this yes and they told me about the experiences so coffee is very essential absolutely sure otherwise the mind is in foggy okay before you go Vanya maybe do you have time for a question Vanya or does Ura is asking if the notes you are available somewhere sure sure I could send this note if there is IGS IGS source this conference source where could I send my file then I will do that okay but so I am ready to do that so am I ready that there is an IGS source where on my file on the YouTube of your channel of your video we can put your file in the description so you should send the file to me and then I will transmit to the IHS okay I will send the file to you sure thank you very much okay it's Elizabeth and just to tell you that generally the notes are online with the video at the same time so if you send them just now it will be online tomorrow or the latest I think maybe okay I will send a little bit later but today okay thank you very much yeah wonderful