 So this is now the final lecture of this lecture series, and I want to give some applications of the results. So let me first summarize what you have proved so far. So again, let us fix a perfectoid field k, as always. And so we had introduced its tilt. So let me recall again that an example would be something like take a periodic field and then join all powers of the uniformizer at all square p-power roots of the uniformizer, and then its tilt in this case is just some of the similar thing in equal characteristic. And then we had introduced this notion of perfectoid k-algebra, k-algebras, and we had the theorem that the categories of perfectoid k-algebras was equivalent to the category of perfectoid k-flat-algebras, which was called the tilting equivalence, which sends any perfectoid k-algebra r via this fontan function to our flat, which is the inverse limit over some of the p-power map of r, where this somehow is a multiplicative. So the transition maps are all multiplicative but not additive, and so one has to do a little bit to define the addition here, and note in particular that there is a map from r-flat to r, somewhere by projection to the last coordinate, which again is multiplicative but not additive, which I denoted by f-maps to f-sharp. And so somehow the deepest theorem that we have proved so far was that if r is the perfectoid k-algebra and s over r is finite eta, then s is also perfectoid, and this induces, and so we can apply the tilting function to s, and this induces an equivalence between the finite eta r-algebras and the finite eta r-flat-algebras, sending any s to s-flat. And in fact we have, this implies a little bit more, so we know an almost version of the faulting's almost purity theorem on this side, and so we deduced it on the other side here. So in particular we see that under the tilting equivalence somehow the finite eta r-covers behave in a nice way, but we want to generalize this somehow to another morphism of the whole eta r-sides, and so what we did last time was that we introduced the category of perfectoid spaces, so we have the category of perfectoid spaces of k. So locally this is of the form, the eddy spectrum associated to some perfectoid k-algebra and some ring of integers inside, so we're r is perfectoid, and so we had introduced, so we have the also somehow tilting extents to spaces giving an equivalence of categories, x-maps to x-flat, and what we did last time was that we defined a notion of etymorphism of perfectoid spaces and which was not so obvious because some of the perfectoid rings are always reduced, so we cannot just say that etymorphism is some morphism which satisfies the infinitesimal lifting criterion, but there is some way around this and so what we get from last time is the corollary that there exists a eta r-side, x eta r, and tilting induces equivalence of sides between the eta r-side and of x and the eta r-side of its tilt, and so somehow if x was just a point, this would just mean somehow that the absolute gyro-group isomorphic and now we have extended this isomorphism to spaces. But somehow we have to relate these etal-tropoi of perfectoid spaces now to etal-tropoi of more classical spaces in order to make any use of this result, so we have somehow to compare them with the etal-tropoi of classical rigid analytic varieties and so we have also somehow the proposition that if y is a locally Nussirian edict space over k, so this means these are the edict spaces that Huber usually works with so there are some Nussirian conditions imposed on the ring here and x is a perfectoid space and we have a morphism f from x to y then this induces a morphism of sides x eta r to y eta r which basically means that we can pull back an etal-map to y to an etal-map to x and we basically have proved this last time. So somehow these etal-tropoi perfectoid spaces somehow have the expected property that somehow they are functorial even with respect to some objects in this larger category and now we need the following result. So let x be a perfectoid space and xi i and i somehow be a filtered inverse system of Nussirian edict spaces so Nussirian is locally Nussirian plus quasi-compact and quasi-separated edict spaces over k then we write that x is similar to the inverse limit of the xi so now first we have to give some maps let x maps to xi be a map to the inverse limit to the inverse system then we write that x is similar to the inverse limit of the xi if first of all on topological spaces induces an isomorphism topological condition and secondly we need a condition somewhere on the rings and it's enough to impose a rather weak condition which is that for all x and x with images xi and xi the map from the direct limit of the residue fields these map to the residue field of x if this has dense image so there's a similar definition where once somehow also requires the inverse limit to be still edict to be still locally Nussirian and in this case Huber proves that the etal topos of x can be written as the inverse limit of the etal topo of xi and some of the same theorem stays true here we have the following theorem that x is the inverse limit of the xi then the etal topos is the inverse limit of the fiber topos taking the inverse limit of topo is a rather technical thing and so makes use of this notion of a fiber topos but don't worry about this too much I should say that this is the topos this is the associated topos limit limit of the fiber topos somehow fibered over this index category i yes, yes okay and in fact we need also a preposition about this notion of being the inverse limit namely if you have such a situation and you have to some xi some etal morphism of Nussirian etic spaces then as I said you can take the fiber product which again is a perfectoid space and this then is also just the inverse limit over the j which are bigger than i of y times xi this will be important in a second and some of the corollary also of the theorem somehow what does it mean that something is an inverse limit of this fiber topos if f i over xi etal is a sheave with pullback fj to xj etal and f to x etal then you can just compute the homology of f by taking the direct limit over the j which are at least i of xi I shouldn't call this of the homologies on the xj and another corollary of the theorem someone characteristic p there are some cases I mean I recall that there somehow you can take any Nussirian etic space and pass to a perfectoid space just by taking the completed perfection of all the rings and some of this doesn't change the teresky topology and the analytic topology and the etal topology because Frobenius is clearly inseparable and this is some incorporated in this theorem so assume that x can be written as such an inverse limit and all transition maps xi to xj induce isomorphism, homomorphisms xi homomorphic to xj and purely inseparable extensions on completed residue fields then the result of Huber says that all of these topoi are equivalent and hence some are also the inverse limit is equivalent to all of these in my case I think the reference I want to use is this condition but I check sorry the site, yes the topo, sorry but it should be true for the site and so we want to use all of this now in an example to see what all of this really means concretely and so we do it in the example of Toric varieties so let me shortly call the definitions let k be some field and the Toric variety over k is a normal separated scheme of finite type over k finite type over k with an action of a split source g on x and a dense open subset such that there exists somehow a dense Torus orbit on this variety so a dense open u and x and a point such that if you take the Torus orbit of x then this is isomorphic to u yes sorry I should maybe say t is more isomorphic to u via acting on x yeah I mean it's probably better to take the point as a part of the datum but I don't want to and so the nice thing about Toric varieties there's an example is that you can just take something like a projective space or also you can take a product of projective spaces and many more things and the nice thing about them is that they can be classified completely combinatorially and most things about them translate into combinatorics and so let me recall this classification of Toric varieties and for this let n be a freer being group of finite rank a strongly convex polyhedral cone in n is a subset of the form or let's say in n is better is a subset of the form sigma which is some of the cone generated by finite number of elements where the x i are all elements in this lattice n and such that sigma contains no line no line through origin and second part is a a fan in n tensor r is a collection of such cones it's a collection of big sigma of such cones sigma such that it's stable under faces stable under faces and if sigma and tau are in sigma then also the intersection n is a phase of both of them and so for example you can take n z squared you can take a picture and you can take the cones that come from this subdivision so you have three two-dimensional cones three one-dimensional cones and one zero-dimensional cone and we can associate to sigma a Toric variety x sigma and this goes as follows we glue it yes yes yes yes I guess I wanted to contain one maximal-dimensional cone to make it sort of canonical so some are not contained in a contains cone of dimension we can associate to sigma a Toric variety x sigma and this goes as follows so do I want this in fact well let's see what happens so first for each sigma in each cone we have its dual which lives in m tensor r where m is the dual of n and we set u sigma to be the spectrum so I guess yes I just want sigma non-empty the spectrum of this thing so that's some schema finite type and tau is the face of sigma one gets an induced open inversion from u sigma from u tau into u sigma and so somehow to get this functoriality we really have to do the sterilization procedure and so somehow this allows us to glue everything together and get this variety x sigma and note that if you take the cone which is just zero then the sigma dual will be everything and hence this is just a spectrum of k of m so this is a free a billion group so this is really just a torus so it's a split torus so let's call this t and note that t naturally acts on on x sigma and there is a natural base point inside here so one and t gives a point x in here and then this really becomes somehow the open dense orbit that we wanted to have and then the theorem is that any torus variety is of the form x sigma and if we fix the space point x and x then sigma is a unique fan in the co-characters of t tensor r okay so this part is probably rather well known I should say one more thing somehow that if you have an element in u then this gives rise to a function now on this open subset t and tends to a rational function on a variety u and we let chi to the u be the rational function be the associated rational function on u on x sigma and but we will need some a few statements about divisors on toric varieties and the nice thing about them that they as well are classified in a combinatorial manner and so that's the following so definition of proposition so there are some some divisors which you can describe combinatorially so that chi and sigma be the one-dimensional cones then we want to first associate to each such one-dimensional cone some veil divisor on on the historic varieties and namely the open subset corresponding to chi after a change of coordinates maybe it's isomorphic to the f i line times g m to something or d minus one maybe is a better notation and that d i in x sigma be the closure of 0 times g m some of the open subset where the first coordinate is non-zero exactly this open dense torus orbit and then now we have something of co-dimension one divisor yes yes sorry sigma is a unique extension here if fix get a unique asomorphism that's right okay so we find that a chi veil divisor is an element of direct sum c times d i and then it's true that any veil divisor is equivalent to a chi veil divisor and if you fix the generator of this line tor i then for the global sections of and d is the sum of a i d i then h naught of x sigma with values in o of d is direct sum overall u and m such that the scalar product of u is v i is at least minus a i of k times the rational function chi to the u okay and so this is somehow what we need about direct varieties and now we want to define some of the attic spaces and perfectoid spaces versions of these varieties and so let us fix a fan and assume now that k is a first and non-archimedian field then we define an attic space which I write curly x sigma add because it's not in general the edification some of the the analogification of the scheme x sigma which is glued out of u sigma which is this thing so somehow if the torque variety was just f i in space then this would not be all of f i in space but just the closed unit ball if x sigma is a n then this curly x is maybe what you call b n so it's the closed unit ball meaning that if you for example consider it's k valued points then it's the set of triples x1 up to as a set of triples x1 up to xn and k such that the absolute value is the most one but if x sigma is proper which by the way is equivalent to the condition it's completely combinatorially again that sigma covers n tensor r then this curly x add is really just the space associated to x sigma and in our case a perfectoid field we define curly x per as being glued out of u sigma per which is the space where now we somehow join all of the p power roots of the coordinates that we have so we allow denominators in p and then we have the following comparison results by the way I should have set in this example that I gave of this specific fan the associated varieties would just be two-dimensional projective space okay so we have the following theorem so we take the perfectoid field with tilt okay flat and then we have some of this perfectoid version the stoic variety over k and this tilt to the same thing over k flat secondly we have this perfectoid version of this space is basically the inverse limit of the usual versions where this transition map phi is induced by multiplication on p on m then we get that in fact if one takes the attic thing over k flat and it's some underlying topological space and it's just the inverse limit over phi of the topological spaces associated to the thing over k so in this sense some of the thing in characteristics stoic variety in characteristic p can be seen as a pro cover of the same stoic variety in characteristic zero in fact this is not only true on topological spaces but it's also true on etal topoi take the etal topos associated then it's the inverse limit of this fiber topos with the integers of so this relation that this is a pro cover of this is true on topological spaces and etal topoi and we will need the following one more statement so for any open u dance there somehow so over equal characteristic mixed characteristic somehow with pre-image v on the other side we have a commutative diagram of topoi somehow that you can you have this projection map somehow from year two inside you have the open sub topos u etal and then there's a map here because again you can write v etal somehow as an inverse limit of the topoi here so in particular somehow if you take just the pn over this equal characteristic field then in some sense it's equal to the inverse limit over phi of the projective space over k where this map on coordinates is just given by p's powers and so this induces a projection map somehow pi from the pn over k flat to pn over k which somehow does not really exist but somehow at least morally exist and for example it exists on topological spaces and some of this is on coordinates given by sending such a tuple to the sharp representatives somehow that's the relation between the projective spaces over the two fields unfortunately this is not completely functorial so this identification somehow depends on the choice of coordinates on pn so on characteristic p there is this canonical way of passing from this variety to a perfectoid space but in order to make this uncharacteristic zero from a variety to a perfectoid space you have to make some choice okay so this basically follows directly from what we have proved and we also need the following proposition that we can compare the homologies of the two historic varieties now so assume that k is a separate average variable closed then the homology of the historic variety and L is invertible on p in k no no no no in k flat I mean so if you take the homology of this shift z what L to the m then this somehow via this projection map goes isomorphically to the homology and for the proof just note that so because of this equivalence of topois that the one is the inverse limit of the others you only have to see that the transition maps in this tower induce isomorphisms which is now something truly uncharacteristic zero and you can just somehow check this by hand okay and for this application for every conjecture we will need an approximation lemma and this is the following so a proposition assume that historic variety is proper and let y in x sigma over k be a hypersurface and let us choose a small neighborhood so associated rigid analytic varieties or edict spaces a small open neighborhood then there exists oops and there exists the hypersurface z in some of the space on the other side such that it's contained in the inverse image of the small neighborhood of y so some maybe some short explanation for what this is supposed to mean it's a hypersurface so it's of column engine 1 uh yes uh let's see let me give an example and continue and then we will see what happens so we can consider inside of p2 over k we can consider something like well this is a hyperplane where sum of the coordinates is 0 and then we somehow have p2 over k flat which is the inverse limit of the tau where now we here have these transition maps and so we have pi inverse of y inside here and so what is pi inverse of y so we can first compute somehow it's the inverse limit of the pre-image into all of these p2 over k's and so inside here we have uh pi inverse of y which is now given by the much more complicated equation where we raise to the p's powers here and so on at each stage this equation gets more complicated and so pi inverse of y some of the inverse limit of these the underlying topological space the inverse limit of the underlying topological spaces and so somehow as a subset of p2 over k flat it has somehow a curve of infinity degree so some kind of fractal this pi inverse of y and this maps to y which is something very nice but and some of the lemmas says that if we so this is something very bad so this inverse image but the lemmas says that if we take a small neighborhood of this then we can't find a curve inside somehow we can approximate this okay and so let's give the proof so so the divisor y is equivalent to the t-veil divisor d which is the sum of a i d i and uh we take a section on x sigma over k of o of d defining y and so we want to approximate somehow this section by section on the other side okay and uh for this we use this approximation lemma that I had proved somehow two lectures ago and so in a slightly variant of it it's a slight variant of it so we consider the graded ring which is the direct sum overall which may have piece in the denominator of dh1 okay with values of j times d so someone on this perfectoid signal it makes sense to somehow take of j times d where j has piece in the denominator and so it can also be written as direct sum over the j's times the completed direct sum in the as a banner space of the u and m p inverse such that somewhere in the notation that I uh reduced of k times k2 that R be its completion so somehow for the obvious k not sub module such that this is open and bounded and then if we form it still then it still just gives somehow given by the similar construction over over the tilt k flat so for this we use somehow that the this combinatorial datum d here this is purely combinatorics somehow and hence transfers to the tilt and so we may assume that somehow our neighborhood y tilde is given by the set of all x and x sigma k at such that the absolute value of f is at most epsilon so because somehow everything is defined by the tori construction everything has an integral model and hence it makes sense to state this inequality here and uh so now we use this approximation lemma so we approximated by g and r flat such that somehow the set of all points in x perf sigma k flat such that the absolute value of g of x at most epsilon is exactly somehow under tilting the set of all here such that the absolute value of f of x at most epsilon it's curly x yes I mean it's the same because it's toric somehow it's proper I assumed in the proposition that my varieties yes no I mean I consider it yes yes I mean isn't it somehow true that some of this x this toric has an integral model over k naught and also some of this d has o of d has an integral model and so somehow you trivialize ah it's not a line bundle I guess I can make this assumption without destroying everything I can just write smooth everywhere and then it should be ok ok so then we have this and then take that to be the set where ah so now so g is also homogeneous of so g lies in h naught of x perf sigma k flat of d somehow because we have this homogeneity constrained somehow in this approximation lemma so a priori is only the completed direct sum of k flat times k to the u over a certain u but we can approximate g by element of uncompleted direct sum and then if you take a high enough g power then this lies in fact in somehow we'll have no more p power roots of the coordinates in its expression so it lies in sigma k flat at of with ways in o of p to the n times d and then we take that to be the locals where g to the p to the n is equal to zero ah ok and maybe let's have some break now for 15 minutes it's not so easy to write down so the approximation algorithm is I mean there's sort of an explicit algorithm computing some of this approximation but it's rather complicated to carry it out ok so by the way it's of course enough to assume that the ambient variety is smooth and doesn't matter whether the hyper surface is smooth and so we get something corollary that if so assume again x sigma is proper and smooth and that y in x sigma over k be a set-theoretic complete intersection or set-theoretic is enough somehow and again we take a small neighborhood yes that's all I want then there exists complete into sorry complete intersection yes sorry ok then there exists that in on the other side so you can assume it's irreducible and it's the same dimension as y such that it's contained in the pre-image of y to the and yes that's just by approximating each of the hyper surfaces and might even be that after this the thing is too large dimension and then we just cut down a little bit more and to take an irreducible component at the end sorry in the end I will assume it's projective and then somehow because the intersection in the characteristic zero is non-empty you can cut by an ample line bundle a little bit more and see that the intersection of these devices will be non-empty and then somehow the same intersection will have to be non-empty in characteristic p because everything is combinatorial and since this intersection cannot be empty ok so generalizing from projective space to troic varieties is a little more subtle than I thought ok so finally we will talk about the weight monodromic conjecture so let us first recall some things about the representation so let K be a local field so GK is absolute Galois group and we have the inertia subgroup and let me also fix the geometric Frobenius and and Q is the cardinality of the residue field which is finite and residue characteristic is P and I take a prime L which is not P ok so recall that the profile quotient the inertia I K is given by the homomorphism T L from I K to T L twisted by 1 so which is the this is the inverse limit over the L to the nth roots of unity so this T L is the inverse limit of certain homomorphisms T L N and how are these defined we will recall it so if Pi in K is a uniformizer and you choose Pi in L to the nth roots of Pi then if you apply sigma to this thing then it will be some root of unity depending on sigma time which is just the T L N homomorphism times Pi to the 1 over L to the n and now we have Grotendieg's quasi-unipotent theorem which is the following proposition let me write it on the right that V be a Q L bar representation of G K so we have homomorphism from rho from G K to G L V then there is an open subgroup I 1 subset of I K the inertia such that for all sigma in I 1 rho of sigma is unipotent then there exists a unique operator N from V to V minus 1 say such that for all sigma in this open subgroup rho of sigma is given by the exponential of N times T L of sigma so some locally this inertia will act through its well quotient and do this via unipotent operators which somehow give rise to a new potent operator by taking the logarithm and say in order to ignore some twists I will fix an isomorphism Q L 1 was 1 Q L and then N then N is really a new potent operator of V itself it commutes with the geometric Frobenius up to the factor Q somehow this follows from uniqueness of this operator and some other thing looking at how this commutes with the action of the Frobenius operator and now we have the so called monodromy filtration which is the following so some very great generality so that V be some finite dimensional vector space and N a new potent operator then there exists a unique separated and exhaustive decreasing filtration fill INV subset V such that the following two properties are satisfied first of all the operator maps the filtration step the ice filtration step into the I-2 and decreasing so this is larger than no it's too difficult for me and N to the I is an isomorphism from the Gru IN of V to Gru minus N and so in fact one can write down this filtration very explicitly so it is a direct sum over I1 minus I2 equal to I of the kernel of N to the I1 plus 1 intersected with the image N to the I2 and so the conjecture is now the following so let's do the lean that if X is a proper smooth variety over K and V is the tachromology of X over K bar with coefficients in Q bar then for all integers J the J squared of V is pure of weight I plus J as a representation of the absolute gyro group of this finite field, the residue field meaning that all eigenvalues of Frobenius on the thing are real numbers of the correct weight and so I want to explain one possible way of looking at this weight monogrobin conjecture in terms of zeta functions because some of it is maybe not so clear what some of the integration behind this is in terms of all functions so let X over Q be some variety I mean over any number field or global field be some proper smooth and let us consider V which is the is a tachromology group with coefficients of Q bar and then we have the zeta function or the L function associated to this homology group and it's defined as a product of all places of local factors so B 1 2 places of F and so let me recall the definition of V is the finite place of residue characteristic not equal to L with local field K then some of this local factor it's just defined as the determinant of 1 minus Q to the minus S times the Frobenius the inertia and variance of this representation and if X has good reduction at P at V then we can apply the wave conjectures and this says that all poles of this local factor have real part I over Q and in particular except for finitely many factors the whole L function is absolutely convergent when the real part of S is greater than I over 2 plus 1 and some of the weight monogamy conjecture says that somehow all the other factors don't destroy this this behavior so weight monogamy implies that all other factors Q to the minus S ah sorry okay sorry all other factors contribute no poles of real part well not even bigger than I over 2 but in particular not bigger than I over 2 plus 1 and in the sense if one quantifies somehow overall possible proper smooth X and somehow the weight monogamy conjecture the behavior is that these other local factors do not contribute other poles and in equal characteristic the linear could turn this argument somehow into a proof of the weight monogamy conjecture so well for the Riemann Z function I guess you have absolute convergence for real part greater than 1 so we have the theorem of the linear that if K of equal characteristic and you have X over K which is proper and smooth and defined over a global field function field then the conjecture is true and some of the argument that the linear gives in real 2 is by somehow establishing this expected property of all local factors somehow by using what we know about L functions in over function fields and in subsequent work by Ito and Terrasoma independently they showed that one can get rid of this something that is defined over a function field by some approximation argument and so but in mixed characteristics the conjecture is basically wide open so it's known in dimension at most too and in context context basically which are related to more varieties where one can apply some arguments and so I want to prove another following theorem that I take a local field of characteristic 0 and let Y be a connected proper smooth variety over K such that Y is a complete, that's the complete intersection in a projective smooth troic variety X sigma okay then the weight monogamy conjecture is true for X for Y so so for the proof we will somehow use this tilting machinery and so we let we somehow first have to produce a perfectoid field and for this we just adjoin all of the p-power roots of the uniformizer and complete and let K flat be the tilt so K flat is a completed perfection FQ and power series over Pi flat and so this, let me call this K prime and then we have that's the absolute gara group of K prime bar over K prime or SEP isomorphic to the absolute gara group of K bar over K which is a subgroup of the absolute gara group of K and so the weights and monogamy of GK representations can be seen by restricting to G K prime because this extension is purely it's totally ramified and hence somehow the geometric Frobenius will survive and its pro-p and hence the L-addict inertia will survive this tower and also we have the following pictures somehow we have y inside of historic variety over K so take its base change to K and basically what we have to do is we would like to take this projection of Pi so we have to first go to the X-paces and take the pre-image of Y but again we have seen that this is not something good but so let us take first some small neighborhood here then take the pre-image here and then we will have something here so what do we have there so it's a serum of uber that there exists open neighborhood such that the coromology is the same as the coromology of this maybe you have to go to the completion so first there is this implicit serum that the coromology doesn't change when you go to the from the algebraic side someone to the rigid side and then somehow you can take a small neighborhood and the coromology still doesn't change so somehow you should imagine some small tube somewhere around Y and so you have this approximation result which tells us that there exists Z inside of this pre-image which is irreducible and and such that the dimension is the dimension of Y and we let we resolve singularities by an alteration so that prime should be projective and smooth and so what we get is a GK isomorphic to GK flat equivariant map from the coromology of Y with coefficients in Z mod L to the M to the coromology of Z prime again with coefficients in Z mod L to the M and I should say that we can assume that by looking back at the proof of this approximation lemma is defined over K prime or if you could even assume if you want to cite deline's result directly that it's defined over a function field because now we have this freedom to change the equation slightly okay yes I don't know I haven't read the proof so it's in this paper on finiteness for direct image chiefs on the tar side of rigidity varieties okay and and hence somehow by taking the inverse limit over all all M and tendering with QL bar we also get a map with coefficients in QL bar and the lemma is now that for in the top degree this is an isomorphism so let's check this so so we have the coromology of Y which maps to which has a map from the coromology of X sigma so let me just write shorthand here and this on the other hand is again the same as the coromology of X sigma by this proposition that I stated that's in the tilting the coromology gets identified and this maps to the coromology of that prime and we have this map and so these are both one dimension to QL bar vector spaces so if the map is not an isomorphism it's a zero map so we have to exclude the case that it's a zero map so we would have a zero here and by this see that there's also a zero here so we have to exclude the case that the restriction map from the coromology of historic variety to the coromology of that prime is not the zero map and just note that if you take the first turn class of an ample line bundle to the dimension of Y's power so Y L is an ample line bundle on X sigma K flat has non-zero image so somehow what you should think is that somehow morally there's some or a retraction somehow back to Y and so this that prime can be seen somehow as an alteration of Y along this map somehow and what this lemma says is that some other degree of this of this alteration is non-zero so somehow it's it does not go to a closed sub-scheme or something like this so somehow we have covered our mixed characteristic variety by some of the variety in equal characteristic now and now the Poincare radiality pairing shows that this becomes the direct summand of this so H1 of Y is the direct summand coromology of that prime and here we have Deline's theorem and it's clear that if you have a direct summand of something which satisfies the weight monotomy conjectures and so does this direct summand and hence we are done okay and that's the end of this lecture series