 Thank you and thanks a lot for the invitation to speak at this at this day, it's a great honor to speak to speak in this event for Barry Mazer. And especially it's a great pleasure to speak on a subject of flat comology. Of course Mazer was one of the pioneers of using flat comology in the arithmetic, especially in his paper on the Wasawa theory for elliptic curves and in relation with Selmer groups and we'll see some points of contact between what I'm going to discuss and what Barry has thought about over the years. So the subject within flat comology that I'd like to discuss is purity, purity phenomena and this is in fact a flat comology analog of what was Grottenich's Absolute Comological Purity Conjecture which was subsequently proved by Gabber and let me just recall a statement of a letter to put the flat comology statement in perspective. So Grottenich's Absolute Comological Purity, now a theorem of Gabber who gave multiple proofs, the first one in 1994, says that if one has a regular Noetherian local ring, regular local ring R with maximal ideal M and if one has a finite etal commutative R group scheme, G, whose order is invertible in R, then the etal comology of this regular local ring R with supports in the maximal ideal and coefficients in this group G which could be for instance Z mod N with an invertible in the ring vanishes for incommological degrees less than twice the dimension of this regular ring R and in fact there is a sharper version which also describes the top degree twice the dimension and above twice the dimension, also vanishes and okay, stated here in the local setting, but if one has a global regular scheme and of course this kind of result says that etal comology classes can be extended over closed sub-schemes of large enough co-dimension in comparison to the degree of the comology class, hence the name Purity and the result for flat comology that I'd like to discuss is an analog of this for flat comology and it has together us with anything else that I'll be discussing is joined with Peter Schultz. So here we have, okay, so we want to drop the assumption that G be etal and that it's order being invertible in the ring and in fact it also turns out to be good to drop the assumption that the ring is regular, to allow that the ring is complete intersection that it's singular but its singularities are complete intersection. So the statement more precisely is that if one has a local complete intersection notarian local ring, R would maximal ideal M and as before, well, we have a commutative finite flat R group scheme G of any order, perhaps of order equal power of the rest of characteristic which probably is positive, then the flat comology which supports the maximal ideal of this local complete intersection ring R and with coefficients in G, flat comology now vanishes in comological degrees less than the dimension of the ring. Okay, so in comparison to Grottenbeek's conjecture here there we were having twice the dimension, now it's only the dimension I'd like to begin right away by giving an example which shows why this occurs in fact it occurs already in the talcomology because we have complete intersection singularities or the order which is invertible on R one still only has the dimension rather than twice the dimension but let me give an example which somehow captures the phenomenon and also in the end are responsible for this vanishing as we'll see in a moment. So firstly by the comological characterization of depth in commutative algebra we have that the comology with supports of this local ring R and with coefficients in the structure sheaf or GA in other words vanishes for in comological degrees less than the depth which in this case is equal to the dimension of R because R being a complete intersection is of course called Macaulay and so if our R is over is an Fp algebra then we get that the comology integrates less than the dimension and with coefficients in say Z mod p or alpha p also vanishes because Z mod p and alpha p admit art and tri are exact sequences for instance like so and by taking and by taking the associated comology along exact sequence from the vanishing supplied by depth we get the vanishing of flat comology with Zp coefficients in degrees less than the less than dimension likewise for alpha p and so this flat comology purity phenomenon that we'll be discussing is really somehow governed by depth in the end at least in positive characteristic and we'll see that in fact in mixed characteristic it's also governed by depth but before proceeding to mix characteristic let's just begin with the case when the carnality of G is invertible in R after all it's not I mean it's in that case it's different depth of R like so one always has so for notary and local ring the comological characterization of depth as in SJ2 is that one looks at this local comology group and it vanishes precisely in comological degrees less than the depth of the notary and local ring and so because our ring is complete intersection depth equals dimension we get we get this and so we somehow in the end we see that our flat comology phenomenon is real coming from local comology it's not enough from UP it's not enough to know the depth reserved for very historic singularities inside it's just from the intersection yes yes I mean then it is also not so difficult to prove for yes okay I mean in this theorem it's really it's important that it's a complete intersection rather than just con Macaulay which we don't see in this example but I mean in fact the theorem fails for con Macaulay and okay it's not all the phenomena that are witnessed in this example but at least we see that it must be dimension rather than twice the dimension okay so now in the case when the carnality of this group is a unit in the ring it was known and this is a result of Gabber who deduces it from from his absolute comological purity here we're dealing with complete intersection before we have we had regular rings after passage to completion our complete intersection becomes a quotient of a regular ring by a regular sequence and by some sort of local left theorem which says that on comology on local comology there's isomorphism in in the range up to the dimension roughly of the ring that we get we get this vanishing from the absolute comological from the absolute comological purity result in the in the regular case in fact there are also other ways to prove it that's also shown by Gabber and the low comological degree cases of this of this of this theorem settle conjectures of Gabber more precisely the cases in comological degrees two and three give the following more more concrete geometric consequences if one has for instance in comological degree two we get that if one has a local complete intersection no theory on local ring r with maximal ideal m of dimension of dimension three precisely three then the Picard group of the puncture spectrum of r has no no non-trivial torsion line bundles so so if dimension is free then the vanishing is in degrees two and two and lower and this local comology I mean this sense this H2 with supports in the maximal ideal for for mu and then somehow translates into into H1 and lower for the puncture spectrum and H1 and lower for a puncture spectrum is sense this and really torsion the Picard group and so from from this vanishing we get we get this we get this statement and just just for comparison let me recall that in SJ2 Grotendig showed the Grotendig-Levschetz theorem that under the same assumptions except the dimension was supposed to be at least for the Picard group of the entire puncture spectrum was shown to be zero so this torsion requirement is not needed if dimension is at least four was was known since since the sixties but there's a there's a refinement in dimension three that allows to get rid of torsion well that shows that there's no torsion and the second the second point conjectured by Gabber in fact in the in the similar in similar vein is purity for the Brow group of in the case of complete intersection singularities more precisely if we have a Noetherian scheme X and a close sub-scheme Z of X of co-dimension at least two oh sorry of co-dimension at least four such that such that the local ring of X at at at Z is a is a local complete intersection for any for any point Z that lies on this close sub-scheme Z then then the comological Brow group of X namely the et alcomology group with coefficients in GM the multiplicative group and torsion coefficients is insensitive to removing this close sub-scheme this close sub-scheme Z the kernel and co-kernel of this kind of pullback map again one somehow can replace GM by mu N are governed by H2 and H3 of comology with supports along Z then by working locally one immediately reduces to this kind of statement and the vanishing of this comology for the degrees two and three with mu N coefficients then gives by globalizing this this global statement of insensitivity of the Brow group upon removing a closed a close sub-scheme or sufficient large co-dimension as long as the singularities as long as the singularities are defined. Okay and so the the because this this case when the carnality of G is a unit in R is is already is already known as settled the main remaining case the main the main case of the theorem is when the carnality of G is is a power of P where with P being equal to the residue characteristic to the characteristic of the residue field of this local complete intersection ring R and this in this residue characteristic is positive and it's and it's this main case that we'll discuss we'll discuss in the rest and what they'll do is I'll briefly summarize the method of the proof overall of this main theorem and then I'll discuss in more detail the aspects of that the parts of that that touch mostly with that have most intersection with with Barry's work over over the years. So since we're we're discussing somehow the bad residue characteristic it's it's no no surprise that that it's a perfectoid methods that allow this advance in the end and in fact there is a part of what we prove is a purity for flat-comology phenomenon in in in the setting of perfected rings and we reduce the main theorem to this to this latter one so let me give let me give a statement let me give a precise statement and for that let me just recall the definition of a perfected ring a commutative algebra type of definition of a perfected ring so a commutative ring a is perfectoid if it is of the following form if it is a quotient of the bit vectoring of a perfected p-algebra for an implicitly fixed prime p by a principle ideal generated by by by by an element by an element psi so it's okay so for so okay so for some here B is some perfected p-algebra the Frobenius and the morphism of it is is a nice morphism perfected p-algebra B and some C which in bit vector coordinates well okay it has its bit vector coordinates C0 C1 so on and this this element C1 is required to be a unit is required to be a unit in a perfected p-algebra in a perfected p-algebra B that's a condition that's just a definition of what what kind of rings will be will be considering let me tell right away that this ring B in fact is determined by a and in fact one can give a definition purely in terms of a rather than B and more precisely this ring B is nothing else than the than the tilt of a in other words it's a inverse limit perfection of the a modular modulo p if one takes a Frobenius and the morphism of a mod p and takes the inverse limit then then then that's that's just our B and B is not your definition B is not complete necessarily complete related to the zeroes coordinate so what happens is that after you complete B related to the zeroes coordinate of this vector you must get this but in general you can have you can start from B which is not complete and then it is not exactly the case okay okay thanks so the C0 added completion of B is just a tilt although I don't I think in the definition one doesn't need to require that BB C0 adequately complete. Yeah you can see that when you complete you've got the same set. Okay yeah thank you all right and this ring then and this ring sorry W I think you're about to tell us. Yeah W is just a it's just a bit it's just a bit ring p typical bit ring of B and in fact it's usually while it's also the can be this I mean it's just notation in the end but somehow it's also the contains period ring a and of this perfected ring a just by definition the bit vector ring of the tilt of the of the perfected ring a okay and another basic fact that is that this element psi is a non-zero divisor automatically that in the end comes from from the assumption that the first bit vector coordinate be a unit okay let me just give two basic examples to have something to have something concrete in mind so firstly if a is an fp algebra then then being perfectoid is the same as being perfect well and in this case C is chest P which in bit vector coordinates is 0 1 0 0 0 and so that in particular satisfies this assumption bit vectors of perfecting module of P is just that perfect ring again and for the converse there I mean for the converse direction of the perfected is of this form is slightly less slightly less trivial but it somehow follows from the fact that any element in the kernel of the of of this map which which satisfies this condition in fact is a generator is a basic property one one proves and the second perhaps more relevant example to what we'll be discussing is that where one has for instance Zp takes out all P power roots of P and then one has some some free variables X1 up to Xn takes out all P power roots of these variables and then we're considering complete intersections with some how we want to quotient by some equations f1 up to fm and when takes out P power roots of these equations as well and and one periodically completes in the end and one assumes here to be able to make sense of these I mean where where where the fi are somehow just for the for the sake of example there are monomials in P and X1 up to up to Xn so that we can take so that we can take P power roots for instance if our if our complete intersection was just Zp and X1 up to Xn module f1 up to fm and the f1 up to fm happen to be monomials like so then some other perfected ring that will reduce to in eventually is some ring of this type of this of this sort of of this sort of form okay and so the main theorem then reduces to the following perfected statement purity in a perfected setting which says precisely that if we have a perfected ring A and we have a closed a closed sub scheme Z contained in the locus where P equals zero in the speck of A such that there exists an A regular an A regular sequence so we basically assume that the perfected A has enough depth along along the Z we assume that there is an A regular sequence A1 up to AD and in A that vanishes on Z so somehow informally the depth the depth of A along Z is at least is at least D that's that's our assumption then in this in the situation the flat comology with supports in Z of of A with with coefficients in any find locally free commutative group scheme of P power order vanishes for in comological degrees less than less than this depth bound less than less than this D so okay so here G is commutative finite locally free A group of P power order okay and in practice this this closed this closed subset Z is just somehow I mean our initial complete intersection will be covered by a by a huge perfected cover in the Z just the pre-image of the maximal ideal will construct that cover in such a way that it has that along Z we have enough depth which will follow from the complete intersection assumption and the enough depth in A and so from from this theorem in the end when one deduces that theorem although I mean there's something in the reduction that I'm definitely not explaining that but this is the key perfected statement that that that one wants and to get this perfected statement we use classification results for such find locally free group schemes over perfected rings these have been shown first in the positive characteristic setting namely in the setting of perfect FP algebras by Bertie Loh and Gaber Bertie Loh did perfect relation in case and Gaber built on his method then by by by Ike allow and Peter Schultz in in mixed characteristic case and there's also ongoing work of unshoots and LeBron who give a prismatic interpretation of the construction anyway the result says that the such g g g g over a g fired locally free group schemes commutative of people or order of a perfected ring a they are classified by their do the modules in particular one can associate them one can associate to them do the modules in the setting of prismatic homology and this do the module denoted by M of g is a module over over a nth of a is an a nth of a module of projective dimension at most one and it's equipped with a Frobenius and semi-linear Frobenius and for Shibong for Shibong endomorphisms and explicitly somehow it's given by the formula as x1 on the prismatic side of this of this perfectoid of of our group g against the prismatic structure sheaf just just like in crystalline do the ne theory in positive characteristic such such find locally free group schemes one can associate them their their do the module given by x1 in the in the in the crystalline site against the against the structure sheaf and evaluated at the terminal object of the of the crystalline site the same thing is happening here in mixed characteristic and this gives an equivalence of categories between between g over a find locally free commutable people order and such such linear algebraic linear algebraic data and in terms of this classification we take this classification and we show the following the following the following formula yeah that's right so in fact I'm not I'm not saying but there are the following there are the following relations that that are required to hold this part is part of the condition work sees a generator of the of the we already seen there is just generator of of the map from a info a back to a of that's rejection and it's okay in prismatic terms is an orientation of the corresponding of the corresponding prism okay and so in terms of this classification we have the following formula for our flat homology that we're interested in with supports in z of the ring a and and coefficients in a find locally free group scheme of commutative commutative with of the power order is given by the homology again with supports in z of of the ring a in of a and coefficients in in the in the in the module and then one takes a fishy boom invariance of this of this of this of this right hand side in homotopical sense so and take some mapping fiber of for shipping minus minus identity so this this formula whose whose profile to some extent discuss in in the rest it allows us to deduce this this this perfected purity and and let's let's see how I mean the point is that since we assume that the depth of a along z is at least d then the depth along z of the in for a is at least is at least d plus one I mean the element psi was a nonzero divisor always and so because a is a quotient of a in for a by this nonzero divisor psi if along z a has depth at least the one has this regular sequence and we can add as a zero tellment of that regular sequence this element psi and one gets depth of a in at least d plus one and so now once one has this depth at least d plus one one once vanishing of this comology in low comological degrees firstly by using by using exact comology sequences one can get rid of for shibong invariance I mean that that's it somehow in the right end at right at the right edge of the exact triangle and afterwards once we'll get rid of that this m of g is of projective dimension at most one over a in so it's so it's a quotient of essentially I mean of projective modules over a in for each of which is a direct someone in the free module and so one one one therefore by the massage reduces to to free modules but one once vanishing in degree one higher and that's where one higher what that's where one higher comes comes from in the end in this fact that size of nonzero divisor and from this key formula it's really it's the depth again that gives that gives that gives that gives a purity of course it is the it is finally presented module of projective dimension less than 0.1 yes yeah that's right all right and so in in the in the in the in the rest of talk I'll discuss the steps that go into proving this key formula star and particularly steps involve proving some new properties of flat homology some some of these properties in some in some special cases of some of these properties have already been considered the major for instance by major for instance in his it was our it was our theory paper on of elliptic curves and on particularly nice feature okay well a nice feature about about the proof of this of these of these new properties is that they actually involve passage to flat homology of simplicial rings and use homotopical techniques in the end and of course it was Mazer as well who pioneered the homotopical methods in in algebraic geometry and then arithmetic with his with his work on at all homotopy joined with my Clarkton so the steps for proving this key formula star is to first to first show it in a case one is perfect in positive characteristic here one uses crystal in do the net theory classification like that which was established by Bertel and Gabber and also written by law and this case is significantly simpler but one one shows it first and then uses this case to derive new properties of FPPF homology FPPF homology with coefficients and finite locally free group schemes commutative and this this derivation in the end it's somehow by reduction to perfect rings and one uses simplicial techniques to get more to get more robust formalism to do those reductions we'll discuss to some extent and finally one uses this one uses these new properties to settle this key formula in general here the techniques that go in are again this classification and the formalism of the Arctopology of Bertel and Matthew so let me perhaps write out more precisely some of the inputs are P complete Arctopology more precisely P complete Arct descent I'll discuss a bit later more precisely what this means for the functor that takes a perfectoid ring A and associates to it the FPPF homology of A with coefficients and finite locally free commutative group scheme of P power order and this P complete I mean the Arctopology is a recent work of Bertel and Matthew that we use that we use okay so let me then proceed to giving those these new properties of FPPF homology that we show along the way okay so for this we'll fix a ring A we'll fix a ring R sorry and commutative finite locally free group scheme G over this over this ring R this will be our coefficients of the flat homology we consider and the first theorem that's used in the proof is excision for flat homology the statement more precisely is that if we have a finitely generated if we have a finitely generated ideal I inside this ring R and we have a map from S to S prime of R algebras and in fact they can be allowed to be simplisher R algebras even this I mean for a moment one can ignore this but in the proof in fact it's very important of simplisher R algebras such that such that this map is an isomorphism along the along the ideal I such that derive the reductions modular powers of this ideal agree for S and S prime for every positive integer N then okay so in the setting the FPPF homology of S with supports in this ideal and coefficients in this group is the same as that of S prime let me just give a concrete example in which this type of situation occurs in which it's particularly useful so for instance in the main theorem in the setting of the main theorem where we have an Aetherian local ring and its maximal ideal then when we consider a flat homology with supports in the maximal ideal of the ring R with coefficients in this group scheme it doesn't change if we replace if we replace the ring by its completion by its maxatic completion in fact it's a statement in special cases that was also used in the wasala theory where flat homology occurs for instance the ring R could be a localization of a number field that's in prime ideal and with okay so alright the second the second property of FPPF homology that I'd like to mention is FPQC descent precisely FPPF homology with such coefficients agrees with FPQC homology and this gives like more robust descent properties larger spectral sequences more precisely the statement is that the functor which associates to an R-algebra S simplitial R-algebra actually the FPPF homology of S with coefficients in G satisfies descent for the FPQC topology and concretely this statement means concretely this means that the FPPF homology of this ring S with coefficients in G is the same as the FPQC homology of course FPQC topology is large for instance that of a field so one needs some cutoff cardinal but that's just the inequality so for every large for every large cutoff cardinal kappa for every large cutoff cardinal kappa we have that the FPF homology can be computed in FPQC topology and in particular if one has a large cover of this S flat cover then one can compute the FPPF homology of S in terms of the check nerve of that cover which is how this result is useful in the in in practice okay and the third property is is descent for yet another topology but after restricting after restricting S is descent for a FPF homology in the P complete arc topology more precisely here one assumes that this group is G that is group G is of P power order implicitly fixed prime P and then one considers a the functor which associates to a perfectoid ring to a perfectoid R algebra S the FPPF of S with coefficients in G this this functor is okay so this functor satisfies satisfies descent for for for a P complete arc P complete arc topology and more precisely what I'll explain the moment what that means what that means more precisely is that if one has a map from a perfectoid S to a perfectoid S prime such that such that for every for every map from S to evaluation ring periodically complete valuation ring of rank one for every such map there exists a diagram there exists another periodically complete valuation ring V prime and a completion of a diagram like so such as such as a diagram commute and this map is finite is faithfully flat in other words okay and this V prime is also a periodically complete valuation ring of rank one in other words one the P complete arc topology is chest topology here in on perfectoids for which any any point of S valued in a periodically complete valuation ring of rank one lifts to such a point of S prime except possibly after enlarging the valuation ring so in particular it's kind of topology insensitive to of course one just for defining it one does not need to restrict the perfectoids and so it's insensitive to reduce structures it's somehow any faithfully flat cover for instance is such cover and so the statement is then that the concrete interpretation of this statement is that the FPPF of this S with coefficients in G can be computed in terms of the cover as the as a derived inverse limit of the FPPF of the check nerve be completed a check nerve of the cover and the point why this is useful is that in practice because the topology is is somehow so weak then in practice one can choose S prime to be a product of valuation rings product of periodically complete valuation rings with even algebraically closed fraction field the nature of the topology allows such covers then once we have this descent result to prove the key formula we can both sides of the key formula satisfy therefore descent for this very weak topology and so effectively we can replace this ring A this perfectoid A by a product of valuation rings by forming suitable hyper cover in the P complete topology and in that case we can somehow almost check it by hand this formula that's roughly how the argument for the key formula goes based on these new properties of flat topology so to which extent is it essential to use the arc topology rather than the previous I think what V topology or the slightly the one where you use not rank one at any so is it for the reductions that you do is it enough to use the V that is you can do things for I don't know I think like in the end one wants these valuation rings V to also be perfectoid and if one has a valuation ring of some really large rank then I mean perfectoid need to in particular be okay I mean in short I don't know right away but I suspect I suspect it's important and in any way it's certainly very convenient for this P complete topology whether one can get away with the V the V descent I not sure okay alright and so the proofs of these okay so in the end we see that it suffices to prove maybe just this statement with that excision and so it's these properties of FPPF topology that in the end give the key formula and so how do we prove this based on the key formula and positive characteristic that we assume there that's a step one so the proofs reduce to the case 1s and s' are perfect FPP algebras okay I mean initially these s and s' need not be even FPP algebras so certainly it has I mean reduction precisely what is what is happening here in the reduction is arg this well if one just looks at a at a functor which computes after inverting P then this functor and here the can be taken to be a topology because after inverting P the group which is of P power order by decomposing into primary parts this is functor so that's why it's descent for the arctopology which is a slightly slightly stronger topology than the P complete arctopology this is a result of bat and matthew which I mean some of the consequences of this of the sarg descent already shown by by Gabber and Fujiwara one also uses deformation theory in a crucial way more precisely the statement from theory that one uses that if one has a square zero thickening square zero thickening S of a ring S bar both simple actually secretly with ideal with ideal I square zero ideal I then the flat the FPPF comology of S with coefficients in G is computed in terms of the FPPF comology of S bar with coefficients in G in terms the cofiber of this map is the r-home r-algebra r-home from the pullback along the identity section of the cotangent complex of this flat group scheme G over A r-home into a shift of the kernel by one like so and in fact it's for this deformation theory where the complete intersection assumption in the main theorem becomes really important because to control this r-home one wants this ideal I to be perhaps free over S bar in practice to be able to control this comology this is a perfect complex supported in the degrees zero and minus one but if the ideal I is some arbitrary ideal this would be for a conmocholering then one couldn't control this deformation here sufficiently sufficiently finally and the third the third input sorry over r I mean r is my base fixed base sorry this should also be r I mean it could also write over S it's not this is somehow object in the drive category of r modules because my group scheme is defined over my basering r and in fact I write in this way so that I don't have to discuss I mean S will be like simplisher learning in the end and so I don't really want to get into what is our home over simplisher learning it's just okay so these two ingredients give some of the reductions that go into the proof of this of these new properties of flat comology but really they by themselves don't allow us to pass the positive characteristic as we want to reduce the perfect p-algebras and the third input is a certain p-addic continuity formula that I'm about to state which which is another property of FPPF that we show so the third in these inputs to the proofs is the following p-addic continuity formula here we assume that G is of p-power order I'll just write it informally like so that G is killed by a power by a power of p and then for a simplisher our algebra S for which for which the 0th homotopy group is Henselian along p so if R is just the usual algebra would know our homotopy then this assumption just means that this S is Henselian along the ideal p for instance so for instance S could be a discrete and pedically complete ring we complete our algebra so under these assumptions the flat homology of S with coefficients in our group G of p-power order is computed in terms of the reductions of S modulo powers of p as the inverse derived inverse limit of derived reductions of S powers of p still with coefficients in G and it's this formula that in the end allows us to pass from something of mixed characteristic really of after inverting p's of characteristic 0 to something that is really a thickening of an fp algebra in the end and it's through this that we eventually go to perfect fp algebras in our reductions and in the remainder of a talk I would like to give a brief sketch of what goes into proving this theorem which somehow will also be illustrative of what goes into proving these properties of fp-pf homology okay so okay so first I mean certainly for simplicity one could assume that S is discrete let's say but nonetheless when it's even in that case these derived reductions they're not really discrete rings unless say is p torsion 3 so it really is convenient to have this extra generality of S being a simplicial ring but to be able to talk about that let me just briefly define what fp-pf homology of a simplicial ring with coefficients in G actually is so first one defines fp-pf side and map f of simplicial rings so these are just simplicial objects in the category of rings say one can think of them like that for just for simplicity say is flat is flat if the following two conditions are met if the map on zero homotopy groups is flat and on higher homotopy which are modules over the homotopy over a homotopy in in degree zero the natural base change map is an isomorphism so the young homotopy map and homotopy group of prime is just the base change of the hand homotopy group of S along the map by zero in other words that in higher homotopy nothing is really happening the whole thing is kind of governed by what is happening in pi zero which where the map is required to be flat and a flat map f is faithfully flat or respectively of finite presentation if the map on pi zero is so then we get the notion of fppf fppf cover in particular a faithfully flat map and we get a site we get an fppf site on simplicity rings that's right I mean I think it's not really a problem like yes somehow really it's an infinity category that's behind this but everything in the end depends on the homotopy category one takes this drive tensor product but the property of the base change being flat or not it really only depends on homotopy group so it's I mean I think it's not so abusive to say it's actually a site but one uses for instance equivalent spectral sequence for the drive tensor product of simplicity rings to show that its notion is stable under base change which perhaps is not immediately obvious okay it's certainly stable under under composition that is clear from the outset and so one defines the fppf homology of S with coefficients in G by chiefifying by okay by chiefifying sufficiently many times transfinately many times just like in Lurie's book on hierarchal post theory the functor which sends a simple shell ring S simply shell R algebra S to the mapping space of to the S point space of G which can be thought of just as a simple shell abelian group and so one places a simple shell abelian group in non-positive homological degrees and then one chiefifies that functor in the infinity categorical kind of sense and so one gets the homology possibly also in positive degrees. Okay let me just give a brief summary of the very basic properties of this process and then we'll proceed to this argument for Piada continuity so firstly in negative degrees nothing happens just like if one forms fppf homology of usual ring coefficients with coefficients in a group or in a shift and in degree zero it's just nothing else but the S point space of G itself. So in negative degrees nothing happens and in positive degrees this one can show meaning it is not it is not immediate from definitions and the fact it's specific to G being a finite locally free group scheme which is the definition of fppf homology and fppf homology okay so in positive degrees this is actually nothing else than just fppf homology of the Pi zero with coefficients in G so this fppf homology of simplisher rings somehow splices this simplisher will be in a group with homology that happens in positive degrees but splices in the ways that's amenable to formalism it's amenable to manipulations in particular taking those R limbs and so for instance one property that one has is that Poznikov towers behave well with respect to this process if one has our S and mapped to simplicial R-algebra SN, SN-1 is such a tower for instance given by Poznikov truncation of the original S the assumption is that the truncations in homological degrees lesser or equal on N are insensitive when passing to SN then the fppf homology of S with coefficients in G is just the inverse limit of the fppf homologies of SN with coefficients with coefficients in G with respect to a tower and it's this property that allows us to get rid of the homotopy somehow in some of the arguments when it's desirable to do so okay so let me get back to this periodic continuity now we assume that SSP Henselian and we want and G is of people order and we want that the R-gamma SG is the same as our limb of our gamma modular well, right properly and there are tons of product formula for H0 and lower I suppose we have lower because ring is simplicial you wrote G of S but this should depend on S if you have a quasi isomorphism simplicial ring this should not change your R-gamma G of S when G is not smooth it's not so natural to evaluate G naively I mean there's a map from this to that and it's a quasi isomorphism it's an interesting isomorphism on homotopy and like Pi H-negative I of this is the same as H-negative I of that which is then really official resolution of something that so if you have a coefficients like mu p or alpha p which is important so you can have a section of this on some scheme which is important but you can resolve the ring of this by smooth things so then you have this simplicial ring with smooth things and then you take G of this but when the group is like alpha p or mu p there are no non-trivial sections so the morphisms when I'm writing G of S it's really a bit disingenuous because it's really morphisms in the infinity category of simplicial rings yes it's not the naive morphism so that's somehow the trick that that's the thing that I'm hiding really that the objects are the naive ones but the morphisms are subtle so you always have to use this yes okay and out of time let me just give the basic basic step is to build starting well first one reduces to chested spy zero by using this postnikov tower argument both sides behave well with respect to postnikov towers because of this fact and then the main some whole step is to replace is to build is to build an in-depth ppf hypercover of S by some SI such that each SI is phenzelian as S was and in addition in addition it has no non-split non-split ppf covers just like one builds local rings in the ppf topology by taking limits overall I mean then one can also for any scheme for any for the ring S one can build in the ppf cover which has no non-split ppf covers itself and if one alternates sets with phenzelization one can make this such that the ring is phenzelian and one continues on the to do the hypercover of the sort this computes the argama of this in terms of the argamas of the SI and for the SI somehow one computes more or less by hand I mean for instance then for SI one has that the argama SI of G because has no non-split covers it's just G of S then one wants to kind of show the same for S modular ppf ppf to the end and then there's only one uses a big array sequence one uses a passage to derive periodic completion somehow derive both the last law anyway this amount of time so I won't enter into details there So I want to go back to a question of of course the G of S there I would have thought of just a simplicial but it's currently not I mean it's in some infinity category is that right? Yes it is a simplicial abelian group as stood as a simplicial abelian group in somehow the infinity category of simplicial abelian group so it's not really like saying that it's a concretely like a physical simplicial abelian group that is not well defined but it's up to somehow But I think a simple approach by hand so to speak is perhaps to resolve your firefight group scheme as usual as a two term complex smooth things and I suppose that for the smooth things some of the manipulations at least if you work at all locally there is no problem of lifting so it should be possible to work more naively like to define G of S at least in the using a time topology as more naively I don't know maybe the world theory needs you but I believe that you can in many cases one can simplify at least for the coefficient Yes in fact we use this kind of resolution heavily in the proofs so for instance the proof of this periodic continuity formula perhaps one can still use it using just by getting a resolution from the outset and without entering the formalism What do you say the proof of? Of this periodic continuity formula I mean it's relatively soft in terms of simplicial techniques that enter and perhaps one can really carry it out from the outset just taking just working in terms of this resolution but for the later ones especially for the excision it is really convenient to have the simplicial rings and especially to have this formula for deformations with respect to ADL I mean it is true that our coefficients in the under are sufficiently simple just a 5 file group scheme or a complex of smooth groups I believe that but also in your continuity thing of course you are using this fancy kind of homology of Luri where you even have to iterate infinitely many times because of this negative homological degree so I'm not so sure about my question but my question was so you write Arlim but I suppose it in good cases so the good cases for example you have an azalean that's a usual azalean ring let's say the p-torsion is even bounded then the situation is like a peak approximation is much better not only Arlim but it is more essentially not exactly essentially but I think that in every homology the projective system will be essentially constant yes in fact it's this I mean we use this type of valkyrie thing in the end of this proof when we have this Arlim and we want that on pi zero for instance so here we have just G of Si which is concentrated negative non-positive degrees and on modular p-torsion we then also show that it's concentrated non-positive degrees we take the Arlim but we still need to show that the pi zero is satisfied with the glypholic condition so that we don't jump off into H1 and to do that we use the l-kick type of argument because the cotangent complex of G is killed by a power of p because it's a and so somehow it's the same type of which is there in this step but the Arlim in general is not essentially constant at least we don't know that okay even so I suppose if you start from a usual language I think it's reasonable to expect that it is