 the title of my second lecture, a local approach to SH of k. The purpose of this lecture is to give another approach to a stable motivic mathematical category. Precisely, we define a triangulated category of frame by spectra SH frame means and prove and prove that it recovers the triangulated category SH of k. The main feature of this new category SH frame means of k is that its construction is generally local in the sense that it doesn't use any kind of motivic equivalences. In other words, we get rid of motivic equivalences completely, making SH frame means of k more amenable to FTC calculation than SH k of Marov-Leyovsky. So let me start with some details. Consider the category of by spectra in the category m dot, where m dot is the category of motivic spaces. Motivic spaces means point of motivic spaces means point is implicit in its name which is on smooth. And here gm smash one is the mapping cone of the one section spec plus of the map spec plus to gm plus in the point of motivic spaces. So this map takes the point plus to the point plus and take non-distringing point in the spec of k to the point one in gm. More precisely gm smash one is the push out of the exact diagram of this form in which i is the point of the superficial set delta of one. So the interval, the superficial interval is the base point one. So one could draw a picture how does it look like but let me skip this because this is not quite essential. The category of by spectra comes equipped with a stable projective local module structure defined as follows. Firstly, by a work I apologize. It is well known that the category of point of motivic spaces comes equipped with the projective local monoidal module structure in which weak equivalencies are just as much local weak equivalencies. So a map of motivic spaces is called weak equivalence in this structure provided that it is stock wise weak equivalencies of stock wise weak equivalence. Stabilizing the module structure in s one direction we get the category spectra s one of k of motivic s one spectra which is equipped with a stable projective local monoidal structure where weak equivalencies are maps of spectra inducing isomorphisms on shifts of stable homotopy groups or in different ways induces induced stable weak equivalencies on stocks of the motivic s one spectra. Sorry. Now stabilizing the module structure on spectra s one of k in the gm slash one direction we arrive at the stable projective local module structure on the category of by spectra. It strangulated category if triangulated homotopy category is denoted by s h needs of k and it is this category is triangulated homotopy category s h needs of k is the very basis for all our further definitions. So the nearest aim is to define this category s h frame needs of k as a full subcategory of s h needs it will be a full subcategory of s h needs. So but to do that I need to recall few definitions concerning frame pre-shifts, frame shifts and so on namely definition. A pointed frame pre-shift f on smooth of k is a contravariant factor from frame plus of k to the category of pointed sets. A framed misnevy shift on smooth k is a frame pre-shift such that its restriction to the smooth of k even is misnevy shift. A framed pre-shift f of a billion groups on smooth k is a contravariant factor from frame plus of k to the category of a billion groups to the category of a billion groups. It is called radiative if f evaluated on the empty scheme is the zero group and f of the joint union of smooth schemes is the product f of x1 cross f of x2. A really new point with frame pre-shifts is the following one. A framed pre-shift f of a billion group is called stable if for each smooth variety x the morphism from f of x to f of x induced by the suspension morphism sigma x from x to x. I recall that there was such suspension morphism defined in the previous lecture that the induced morphism between f of x and f of x induced by sigma x is the identity. A framed misnevy shift of a billion groups on smooth of k is a framed pre-shift of a billion groups f such that its restriction to smooth of k is a misnevy shift. Finally a framed pre-shift f is going to be invariant or even a a1 invariant if for x and projection a1 cross x to x the induced map is a bijection or it is an isomorphism of groups provided that a pre-shift is a pre-shift of groups. A framed misnevy shift is a one invariant if it is a one a1 invariant as a framed pre-shift. So these are all standard definitions about framed pre-shifts and as I stress the only non-standard and new is this one that framed pre-shift f of a billion groups is called stable provided that this induced map sigma of a star x from f of x to itself is the identity map. Now the very key definition namely the definition of the category sh framed mis of k. We define sh framed mis of k as a full subcategory in sh mis of k consisting of those by spectra e satisfying the following conditions. Firstly, first h-matific space eij of e is a space with framed correspondences. That is a point it's a pre-shift shift defined on the category of framed correspondences frame plus of k. The structure maps this is the second condition in both direction in this one direction and in the jfm smash one direction respect or preserve in different way respect a framed correspondences. The third condition is this for every j integer j which is non-negative. We can consider the s1 spectrum e star comma j and the requirement is this the framed pre-shift pre-shifts of stable stable hematopic groups by the star of this s1 spectrum of this material response spectrum are stable, radiative and a1 and v. And the final condition the final requirement is called consolation theorem and it sounds like this for every integer j which is at least zero the structure map between the j's s1 spectrum and in the home from gm naive in the home from gm smash one to j plus plus one once a native s1 spectrum is a stable local appearance. Objects of the category frame means of k are called framed by by spectra. We should stress that the definition of this category s h frame means of k is local in the sense that its morphisms are computed in the category s h means of k and has nothing to do with the category s h of k. Here is the main theorem. There is the natural front of f which is the identity on objects. So we first go to s h means of k using the full embedding and then projects to project projects to s h of k which is the identity on objects is an equivalence of categories. Its quasi quasi inverse is given by the big frame material motive functor denoted like this which is described in the paper four. But also another is quasi another quasi inverse functor to the functor f is given by a functor m bar big frame constructed below. So in the lecture today I will use I will construct this functor and I will use this functor as a quasi inverse for the functor f. So I will prove that it will be quasi inverse to the functor f. So before proving this theorem let me show some immediate immediate inventive advantages of the category s h frame means of k of the category s h frame means of k. The first given by spectrum e we shall also write e in the following form e of not e of one and so on and so on where e of j is the motivic s one spectrum with spaces e of j with the subspeed i is equal to the space e ij for all i equal to zero. And we also call the s one spectrum e of j the j's weight of e. So here is the first very nice really very nice property of frame by spectrum. If e and f full frame by spectrum then amorphous f between e and f f 30 f 30 e and 30 f is a stable motivic equivalence in the sense of Marilyn Wojewski if and only if the induced morphism of nisnevic sheaths of stable homotopic groups the induced motion of this form are isomorphisms in each weight q. There are four stable motivic equivalences between frames by spectrum inside with a naive level wise local equivalences between them. So next what I would like to stress is the following property. Usually it's very difficult to compute a one homotopic sheaths of a motivic spectrum or a motivic by spectrum. So the next property the next property says that the motivic homotopic sheaths of a frame by spectrum 30 e are computed in terms of ordinary nisnevic sheaths of stable homotopic groups for weighted s one spectrum of e namely let e be written in the form e of north e of one and so on be a frame by spectrum so a by spectrum in s h frames and p q e to integers. Then the a one homotopic sheaths of the q d e are computed as follows firstly if q is at more zero then the a one homotopic sheath is by index by q of 30 e coincide with the prime minus q's nisnevic sheath of the spectrum e of let me write e of q with this shtuka where q with this shtuka is the modulus of q. I just don't know the name of this vertical bars so and the if q is greater than zero then the stable the a one homotopic sheath with index by q coincide with the following one one firstly should take the s one spectrum e of zero then compute the nisnevic sheath of homotopic groups with index p minus q and then use the operation take the operation minus q and this operation means the q's contraction of the sheath of the sheath by nisnevic p minus q of e of naught. The third property is as follows let e be e of naught e of one e of two and so on be by spectrum in s h three nis of k and let e f be e naught subscript f e one subscript f and so on be a by spectrum obtained from e by taken a local stable fiber replacement of each weight e of j in the stable local projective model structure on motivic s one spectrum then the spectrum e f is a local motivic fiber. Here I would like to stress that this kind of property is very often happen happen for many uh by spectrum we have seen we have seen say this kind of property was used yesterday uh in the lecture by uh Sasha Meshkov sorry so the nearest my aim uh is to describe this uh found the m bar d frame between s h of k and s h frame nis of k and which is supposed to be a quasi inverse uh to the fund f which i this which is identity on the object which i described above this is the nearest my aim so some notation for a smooth scheme x write t star frame of x four three four uh frame delta dot cross bar of x and i recall that frame delta bar uh cross bar of x uh are the motivic spaces i used um i'll say i advertised in uh in the first my lecture uh saying that this is uh kind of the very key construction uh in the stable uh motivic hammer to p t unit so uh it is appointed uh simplicial framed shift and so in different terminology it is uh appointed framed motivic space frame motivic space replacing x uh with a simplicial objects in frame not of k uh and taking the diagonal we get appointed framed motivic space c star frame of x dot finally if a is a filtered k limit of k smooth simplicial schemes a i's then define c star frame of a as the filter of k limit of c star frame of a i's and here is first i'll let us say a key construction uh suppose a by spectrum e suppose a by spectrum a consist of motivic spaces a ij which are filtered limits filtered limits of simplicial smooth space schemes we then take the key star frame uh of a ij at every entry and get a by by spectrum which we call c star frame of a let me skip uh how to write down the uh structure maps they're written down uh in an obvious way so let me skip this uh finally take this by spectrum and stabilize it in the gm direction in the standard way they the stabilization stabilization uh is called the infinity gm smash one of c star frame of a but typically uh i will call it i mean i will pronounce uh keep the infinite c star frame of a uh because uh i will not use stabilization in s one direction i will use only stabilization in gm smash one direction so it can be checked that the by spectrum c star frame of a and tip infinity c star frame of a are subjected to conditions one two three of definition two two so since uh most probably you have forgotten already about those those conditions let me show you take back and show you those conditions the conditions uh are these ones so i will take back so yeah so this uh by spectra has subjected conditions one two three uh or definition two two so therefore we still cannot say that they are framed by spectra but nevertheless the significance uh of this general construction is reflected in the following examples for each small scheme x we can take the by spectrum uh the by suspensional spectrum of x plus and then apply the construction c star frame and it turns out that uh this by spectrum will be already a frame by spectrum so it will be in s h frames more generally so what i should maybe stress here that uh particularly to prove this property we need the consolation theorem uh for a one invariant uh radiative uh and sigma stable sorry sorry we need a consolation theorem for frame models but i didn't speak about maybe i should stress yeah sorry maybe i should stress that uh this spectrum uh is in the paper four is denoted as m g frame of x okay so more generally uh let a be a motivic s one spectrum such that every entry a i of a i filtered a limit filtered a limit of face most simple skills then take firstly the m suspension spectrum of a and then apply the construction c star frame to this suspension g m suspension spectrum of a and the result uh is a frame by spectrum so and third example is the more general one so you could take uh a by spectrum a such that every entry a i j of a is a filter at the limit of smooth simple skills then this spectrum it in need of c star frame of a uh is again a frame a frame by spectrum moreover the canonical morphine of by spectrum is a stable motivic agreement this will be used though and uh it turns out that each frame by spectrum each frame by spectrum is aomorphic to the one uh of this form maybe i should say also that uh if you take the island mark between spectrum uh by spectrum h of d hd the reward c one it will be in this category as well and many other uh spectrum if you take i don't remember exactly the notation maybe like this so the spectrum uh uh yeah like this is also isn't uh s h uh frame miss okay and many other interesting spectra uh are in this category say the best spectrum which was described yesterday by uh Sasha Neshetov is in this category as well sorry uh now i'm ready to um define to construct this phantom m bar big frame from uh sh of k to s h ring miss okay namely uh choose a factorial coefficient replacement uh in the projective model structure uh on by spectrum then uh each uh ec by spectrum ec consist of motivic spaces ec ij which are filter at the limits of simplicity smooth space k therefore by the example three one has the property this uh by spectrum it infinity c star frame of ec uh uh is a frame by spectrum so we define uh m bar big frame uh of e uh s the t to infinity c star frame of ec so this construction in is factorial in by spectrum and uh let me show that it takes motivic equivalences uh to stable uh to local equivalences to naive level wise stable local equivalences of by spectrum for that consider the stable material equivalences of uh by spectrum then uh the induce morphism between curly c and curly uh curly is curly is c curly fc is again uh a stable motivic equivalences of by spectrum then by the example three these two morphisms are stable motivic equivalences this is because uh each this uh uh that this spectra has this property that its entries filter at the limits of the initial smooth scheme and the same for fc so so these two morphisms are stable motivic equivalences by the property three and we put right down a simple diagram and conclude that the induced morphism between big frame motive of e and big frame motive of f is again a stable motivic equivalence so maybe let me uh write down the diagram because otherwise uh so uh i think here you did if we see rule of c this is uh a motivic equivalence then the big frame so these maps are of the type alpha and they are stable motivic equivalences this is a stable motivic equivalence as well uh and so uh since the diagram can be used therefore this is a stable motivic equivalence this is a stable motivic equivalence but this by spectra this by spectra yeah this by spectra uh are framed one so they are in s h frames of k therefore the property one of by spectra shows that the morphism and big frame is in fact a naive level was wise stable local appearance and uh thus the sponsor uh m big uh m bar big frame converts so maybe i should put here bar so m bar big frame uh converts stable motivic equivalences to naive level wise local stable equivalences uh and in this way we could eventually this fun time so now we are ready to prove hearing people's problem we are ready to prove hearing to three uh saying that the fun fact that the fun fact uh which goes this way and reaches identity on objects uh is an equivalence of 30 days and m bar big frame is in fact in squadron inverse so firstly uh let me prove that the fun fact f is fully faithful in winning so for that take two by spectra uh e and e prime uh which are framed uh and compute and compute uh in s h frame means between e and e prime this home uh is computed in fact by the very definition in s a means of k so we need to replace e by its uh for five and replace replacement e c and we need to replace e prime by its uh local level wise uh five and uh replacement e prime f uh then compute home uh of by spectra from e c to e prime f and uh no doubt uh by sorry by the naive uh chromatopy by naive uh chromatopy equivalence and from other side from other side so here is the computation of uh the homes in s h frame means but from other side uh if it descends to uh s h and compute home we'll try to compute home in uh between e and e prime is in s h k uh s h of k uh then uh we have to do the pooling we must uh we have to do the pooling we have to replace e again by uh a cofine is cofine and replacement s e and we have uh to replace uh e prime uh by its uh motivic five and replacement but uh as we know by the property three motivic motivic uh five replacement of e prime can be computed as e prime f which is used here uh therefore we can cancel this m and use just e prime f for the compute for our computation and we see uh that uh these two formulas are identically the same so this formula and this formula sorry these are identically the same formula therefore this map uh is an item of so we proved that the front of f is fully patient what is uh remained to prove the theorem uh it's remained to prove uh that the founder that the essential image uh of uh the founder f coincide with uh s of s h of k so for that consider uh the founder m bar big frame constructed above uh and consider also the the exact equivalent the exact uh this is the cofine replacement and this is uh the motion uh alpha uh both errors are stable motivic equivalences so the fact that the left error is stable material it is even low-wise uh local motivic equivalence but the right coincide uh level uh uh is a motivic stable motivic equivalence due to the example three so we have proved that each uh motivic by spectrum e uh is uh isomorphic in the category s h uh to the spectrum uh of the form m bar big frame of e and also uh due to example three uh we know yeah as i uh told uh above that this founder takes videos uh in this category so we have proved that uh e is isomorphic to the founder uh to uh m bar big frame of e and hence the founder m big frame uh is quasi inverse to the founder so the theorem is proved so i have uh several more minutes and uh okay so i will continue let me describe one more nice property uh of the category s h frame n is of k namely you know that by uh s h one uh of pre-disabled motivic amount of category of s1 spectra there is a canonical pair of drawing funtons the suspension founder and the uh omidor loop founder uh in the dream direction uh so we've heard the following results which is the consequence of theorem two three and uh theorem two one from the word four namely the it's very easy i will say in the following way it's very easy to describe uh the founder the omidor loop uh infinite omidor loop founder uh in terms of the category s h frame mist namely one should take uh a frame by spectrum e and send it to the zero weight e of zero and this is the founder this very very convenient uh uh in turn the composite founder which is the uh so the suspension founder we would like to describe the suspension founder but again in terms of this category and it takes uh a motivic s1 spectrum e to the frame by by spectrum of this form where e c uh is the co-fibon replacement uh of e so we need to take uh the co-fibon replacement e c of e then we need to take the gm suspension by spectrum of that and eventually apply the construction sista frame and what we will get we will get the suspension interpretation of the suspension founder uh using this category so what i would like to say a little bit more uh is as follows uh we would like to compute for each motivic by spectrum so for each motivic s spectrum s1 spectrum b the omega infinity loops so the composition of these two functions omega infinity loops of the infinite suspension fund of the infinite suspension spectrum of b and uh it has a very nice description namely we need to replace b with the co-fibon replacement bc and apply the founder sista frame sista frame so sista frame of bc this is a green ns1 a motivic s1 spectrum and this canonical morphism which is written here will be stable local appearance of motivic s1 spectrum so this means that uh this computation of omega infinity gm sigma infinity gm of any s1 spectrum b is given locally in a very easy way a simple way so i have five more minutes so i will maybe say what the next lecture is supposed to be about uh and let me uh make uh it begins with the following remark that the category s h frame is of k is in fact the triangulated category in a natural way namely the shift fund of one is the shift fund in the category s h nth of k so and distinguish triangles in s h frame nth of k are those which are distinguished in s h nth of k and theorem two three states came to three states the the function uh earth and m bar big frame uh are usually inverse equivalences of categories s h framings and s h of k but one could say a little bit more namely the following simple lemma is true the fantasy earth and m bar big frame are usually inverse equivalences of triangulated categories of s h framings and s h of k and so the final point i would like to stress uh is uh this is uh something for the future for the my last lecture i would like to stress that at the very first glance the category s h frame nth of k looks a little bit artificially constructed but in fact i will show in the next lecture that this category uh is not at all artificially constructed it is a category of local objects for uh uh localization fund uh which will be called h-frame motivic uh this is a localization counter uh in the category sorry in the category s h nth but let me stop uh on let me stop and say that this is the end of my lecture today okay many thanks indeed and let's thank uh ivan uh for the lecture any questions or comments it seems that there are no questions and let's thank uh the lecture again and we'll meet at half past three perry's time for the lecture of gonzala