Thank you and thanks a lot for the invitation to speak at this day.

It's a great honor to speak in this event for Barry Mazur. And especially it's a great pleasure to speak on the subject of flat cohomology. Of course, Mazur was one of the pioneers of using flat cohomology in 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 cohomology that I'd like to discuss is purity, purity phenomena.

And this is in fact a flat cohomology analog of what was Grothendieck's absolute cohomological purity conjecture, which was subsequently proved by Gaber. And let me just recall the statement of the latter to put the flat cohomology statement in perspective.

So Grothendieck's absolute cohomological purity, now a theorem of Gaber who gave multiple proofs, the first one in 94, says that if one has a regular noetherian local ring, regular local ring R with maximal ideal M

and if one has a finite et al commutative R group scheme, G, whose order is invertible in R,

then the et al cohomology 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 in cohomological degrees less than twice the dimension of this regular ring R.

And in fact, there's a sharper version which also describes the top degree, twice the dimension and above twice the dimension, it also vanishes. And okay, it's stated here in the local setting, but if one has a global regular scheme, and of course this kind of result says that

et al cohomology classes can be extended over close sub-schemes of large enough co-dimension in comparison to the degree of the cohomology class, hence the name purity. And the result for flat cohomology that I'd like to discuss is an analog of this for flat cohomology.

And it has to give us with anything else that I'll be discussing is joint with Peter Schulze. So here we have, okay, so we want to drop the assumption that GB be et al,

and that it's order be 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 a complete intersection that is singular, but its singularities are complete intersection. So if one have a statement more precisely is that if one has a local complete intersection,

not even local ring, R with 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 restive characteristic, which probably is positive, then the flat cohomology with supports in the maximal ideal of this local complete intersection ring R and with coefficients in G, flat cohomology now vanishes in cohomological degrees

less than the dimension of the ring. Okay, so in comparison to the Groenitz 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 et al cohomology because we have complete intersection singularities or the order of G is invertible on R, one still only has the dimension rather than twice the dimension. But let me give an example which somehow captures a phenomenon that also in the end are responsible

for this vanishing as we'll see in a moment. So firstly, by the cohomological characterization of depth in commutative algebra, we have that the cohomology with supports of this local ring R and with coefficients in the structure sheaf or GA in other words,

vanishes for in cohomological 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, cone Macaulay. And so if our R is an FP algebra, then we get that the cohomology in degrees

less than the dimension and with coefficients and say Z mod P or alpha P also vanishes because Z mod P and alpha P admit Artin-Schreier exact sequences, for instance, like so and by taking the associated

cohomology along exact sequence from the vanishing supplied by depth, we get the vanishing of flat cohomology with ZP coefficients in degrees less than the depth and dimension likewise for alpha P. And so this flat cohomology purity phenomenon that we will 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 mixed characteristic, let's just begin with the case, with the case when the cardinality of G is invertible

in R after all it's not, I mean, it's in that case, it's different than, yeah. The depth of what? Of, I mean, depth of R. Like, so one always has, so for a noetherian local ring,

the cohomological characterization of depth as an SJ2 is that one looks at this local cohomology group and it vanishes precisely in cohomological degrees less than the depth of this noetherian local ring. And so because our ring is complete intersection, depth equals dimension, we get this.

And so we somehow in the end, we see that our flat cohomology phenomenon is still coming from local cohomology. It's not enough for mu P, it's not enough to know the depth observed for various toric singularities and so on. It's just for complete intersection. Yes, yes, I mean. But then it is also not so difficult to prove for complete intersection.

Yes, okay, I mean, in this theorem, it's important that it's a complete intersection rather than just calling Macaulay, which we don't see in this example. But I mean, in fact, the theorem fails for 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 cardinality of this group is a unit in the ring, it was known. And this is a result of Gaber, who deduces it from his absolute cohomological purity.

Here we're dealing with complete intersection before 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 Lefscher's theorem, which says that on cohomology, on local cohomology, the isomorphism in the range up to the dimension

roughly of the ring, we get this vanishing from the absolute cohomological, from the absolute cohomological purity result in the regular case. In fact, there are also other ways to prove it as also shown by Gaber. And the low cohomological degree cases of this theorem,

settle conjectures of Gaber. More precisely, the cases in cohomological degrees two and three give the following more concrete geometric consequences.

If one has, for instance, in cohomological degree two, we get that if one has a local complete intersection, not even local ring R with maximal ideal M of dimension three, precisely three, then the PICAR group of the puncture spectrum of R

has no non-trivial torsion line bundles. So if the dimension is three, then the vanishing is in degrees two and lower. And this local cohomology, I mean, this sense, this H2 which supports the maximal ideal for mu and then somehow translates into H1

and lower for the puncture spectrum. And H1 and lower for a puncture spectrum is then really torsion in the PICAR group. And so from this vanishing, we get this statement. And just for comparison, let me recall that

in SJ2, Grothendieck showed, the Grothendieck-Lefschetz theorem that under the same assumptions, except the dimension was supposed to be at least four, the PICAR group of the entire puncture spectrum was shown to be zero. So distortion requirement is not needed if dimension is at least four, was known since the 60s.

But 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 point conjectured by Gaber, in fact, in a similar vein, is purity for the Brau group in the case of complete intersection singularities.

More precisely, if we have a Noetherian scheme X and a closed sub-scheme Z of X of co-dimension at least four such that the local ring of X at Z

is a local complete intersection for any point Z that lies on this closed sub-scheme Z, then the cohomological Brau group of X, namely the et al. cohomology group with coefficients in gm and the multiplicative group

and torsion coefficients is insensitive to removing this closed 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 cohomology

with supports along Z. Then by working locally, only one immediately reduces to this kind of statement and the vanishing of this cohomology for the degrees two and three when mu n coefficients then gives by globalizing this global statement of insensitivity of the Brau group upon removing a closed sub-scheme

or sufficiently large co-dimension as long as the singularities are right. Okay, and so because this case when the cardinality of G is a unit in R is already known, is settled, the main remaining case,

the main case of the theorem is when the cardinality of G 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 residue characteristic is positive. And it's this main case that 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 method that touch mostly with, that have most intersection with Barry's work over the years. So since we're discussing

somehow the bad residue characteristic, it's no surprise that it's a perfectoid methods that allow this advance in the end. And in fact, there's a part of what we prove is a purity for flat cohomology phenomenon

in the setting of perfected rings. And we reduce the main theorem to this latter one. So 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 width vector ring of a perfect Fp algebra for an implicitly fixed prime p by a principle ideal generated by an element xi.

So for some here B is some perfect Fp algebra, the Frobenius endomorphism of it is an isomorphism.

Perfect Fp algebra B and some C which in width vector coordinates, well, okay, it's width vector coordinates xi zero, xi one, so on, and this element xi one is required to be a unit in a perfect Fp algebra B.

That's a condition, that's just the definition of what kind of rings we'll 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 tilt of A. In other words, it's inverse limit perfection of the A modulo p. If one takes the Frobenius endomorphism of A mod p and takes the inverse limit, then that's just our B.

And B is not, in your definition, B is not necessarily complete relative to the zeros coordinate. So what happens is that after you complete B relative to the zeros coordinate of this vector, you must get this. But in general, you can start from B which is not complete and then it is not exactly the tilt.

Okay, okay, thanks. So the C0 at the completion of B is just a tilt. Although I think in the definition one doesn't need to require that B be C0 adequately complete. Yeah, you can see easily that when you complete

you've got the same set. Okay, yeah, thank you. All right. And this ring, sorry? W, I think you're about to tell us. Yeah, W is just a with ring,

p typical with ring of B. And in fact, it's also can be this, I mean, it's just notation in the end, but somehow it's also the Fontaine's period ring A and of this perfected ring A. Just by definition, the width vector ring of the tilt of the perfected ring A.

Okay, and another basic fact is that this element psi is a non-zero divisor automatically. That in the end comes from the assumption that the first width vector coordinate be a unit.

Okay, let me just give two basic examples to have something concrete in mind. So firstly, if A is NFP algebra, then being perfected is the same as being perfect.

Well, and in this case, C is just p, which in width vector coordinates is zero, one, zero, zero, zero. And so that in particular satisfies this assumption, width vectors of perfecting modulo p is just that perfecting again. And for the converse, I mean, for the converse direction that ever perfected is of this form is slightly less trivial,

but it somehow follows from the fact that any element in the kernel of this map, which satisfies this condition, in fact, is a generator is a basic property one proves. And the second perhaps more relevant example to what we'll be discussing is that where one has,

for instance, z p takes out all p power roots of p. And then one has some free variables, x one up to x n takes out all p power roots of these variables. And then we're considering complete intersection.

So somehow we want to quotient by some equations f one up to f m, and one takes out p power roots of these equations as well. And one periodically completes in the end. And one assumes here to be able to make sense of these, I mean, where the f i are somehow,

just for the sake of this example, they are monomials in p and x one up to x n, so that we can take p power roots. For instance, if our complete intersection was just z p and x one up to x n,

modulo f one up to f m, and the f one up to f m happened to be monomials like so, then somehow the perfecting that we will reduce to eventually is some ring of this type, of this sort of form. Okay, and so the main theorem then reduces

to the following perfected statement, purity in a perfectoid setting, which says precisely that if we have a perfected ring A,

and we have a closed sub-scheme, z contained in the locus where p equals zero in the spec of A, such that there exists an A regular sequence.

So we basically assume that the perfectoid A has enough depth along the z. We assume that there is an A regular sequence, A one up to A D in A that vanishes on z.

So somehow informally, the depth of A along z is at least D. That's our assumption. Then in this situation, the flat cohomology with supports in z of A,

with coefficients in any finite locally free commutative group scheme of p power order vanishes for in cohomological degrees, less than is depth bound, less than is D. So, okay, so here, G is commutative finite locally free A group

of p power order. Okay, and in practice, this closed subset z

is just somehow, I mean, our initial complete intersection will be covered by a huge perfectoid covering the z. It's just a pre-image of the maximal ideal. We'll 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 this theorem in the end, 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 perfectoid statement that one wants. And to get this perfectoid statement,

we use classification results for such fine locally free group schemes over perfectoid rings. These have been shown first in the positive characteristic setting, namely in the setting of perfect FP algebras

by Bertiello and Gaber. Bertiello did perfect valuation in case and Gaber built on his method. Then by Ikelau and Peter Scholtzen in mixed characteristic case. And there's also ongoing work of Anschutz and Lebra

who give a prismatic interpretation of the construction. Anyway, the result says that such G over A, G fired locally free group schemes, commutative of P power order over perfectoid ring A,

they are classified by their duodone modules. In particular, one can associate them, one can associate to them duodone modules in the setting of prismatic homology.

And this duodone module denoted by M of G is a module over A-inf of A, is an A-inf of A module of projective dimension at most one. And it's equipped with a Frobenius and Semilinja-Frobenius and Ferschibung endomorphisms.

And explicitly somehow it's given by the formula as X to one on the prismatic side of this perfectoid of our group G against the prismatic structure sheaf.

Just like in crystalline duodone theory and positive characteristic, such finite locally free group schemes, one can associate to them their duodone module given by X to one in the crystalline site against the structure sheaf and evaluate it at the terminal object of the crystalline sites.

The same thing is happening here in mixed characteristic. And this gives an equivalence of categories between G over A, finite locally free commutative of P power order and such linear algebraic data. And in terms of this classification, we take this classification and we show the following formula.

Yeah, that's right. So in fact, I'm not saying, but there are the following relations that are required to hold this part of the condition.

Work C is a generator of the, what we already seen there, it's generator of the map from A info of A back to A of that search action. And it's, okay, in prismatic terms, it's an orientation of the corresponding prism. Okay, and so in terms of this classification,

we have the following formula for our flat cohomology that we're interested in with supports in Z of the ring A and coefficients in a finite locally free group scheme of commutative of P power order is given by the cohomology, again, with supports in Z of the ring A-inf of A

and coefficients in the Deodonia module. And then one takes a Fershebung invariance of this right hand side in homotopical sense.

So one takes a mapping fiber of Fershebung minus identity. So this formula who's proof out to some extent discuss in the rest, it allows us to deduce this perfective purity. And 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 A-inf of A is at least D plus one.

I mean, the element psi was a non-zero divisor always. And so, because A is a quotient of A-inf of A by this non-zero divisor psi, if along Z A has depth at least D, one has this regular sequence and one can add as a zeroth element of that regular sequence this element psi and one gets depth of A-inf at least D plus one. And so now once one has this depth at least D plus one,

when one's vanishing of this cohomology in low cohomological degrees, firstly, by using exact cohomology sequences, one can get rid of Fershebung invariance. I mean, that's it somehow at the right edge of the exact triangle. And afterwards, once one gets rid of that, this M of G is a projective dimension

at most one over A-inf. So it's a quotient of projective modules over A-inf, each of which is a direct sum and a free module. And so, one therefore by the visage reduces to three modules, but when one's vanishing in degree one higher, and that's where one higher comes from in the end,

in this fact that psi is a non-zero divisor. And from this key formula, it's really, it's the depth again that gives the purity. So of course, it is finally presented module of projective dimension less than or equal to one. Yes, yeah, that's right.

All right, and so in the rest of our talk, I'll discuss the steps that go into proving this key formula star. And in particular, these steps involve proving some new properties of flat cohomology. Some of these properties in some special cases

of some of these properties have already been considered by Maser, for instance, in his Iwasawa theory paper on elliptic curves. And one particularly nice feature, okay, well, a nice feature about the proof of these new properties is that they actually involve

passage to flat cohomology of simplicial rings and use homotopical techniques in the end. And of course, it was Maser as well who pioneered the homotopical methods in algebraic geometry and in arithmetic with his work on Atala homotopy joint with Michael Artin. So the steps for proving this key formula star

is to first show it in a case when A is perfect, in positive characteristic. Here one uses crystalline duodone theory, classification like that, which was established by Berthelot and Gaber

and also written by Lau. And this case is significantly simpler, but one shows it first. And then one uses this case to derive new properties of FPPF cohomology.

FPPF cohomology with coefficients and finite locally free group schemes, commutative. And this derivation in the end, it's somehow by reduction to perfect rings. And one uses simplicial techniques

to get more robust formalism to do those reductions. We'll discuss to some extent. And finally, 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 arc topology of Butt and Matthew. So let me perhaps write out more precisely, some of the inputs are pre-complete arc topology,

more precisely pre-complete arc 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 cohomology of A

with coefficients and finite locally free commutative group scheme of P power order. And this P complete, I mean, the arc topology is a recent work of Butt and Matthew that we use. Okay, so let me then proceed to giving these new properties

of FPPF cohomology that we show along the way.

Okay, so for this, we'll fix ring A, we'll fix ring R, sorry, and commutative finite locally free group scheme G

over this ring R. This will be our coefficients of the flat cohomology we consider. And the first theorem that's used in the proof is excision for flat cohomology.

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 simplicial R algebras even. This, I mean, for a moment one can ignore this, but in the proof it in fact is very important, of simplicial R algebras such that, such that this map is an isomorphism along the ideal I, such that the derive the reductions

modulo powers of this ideal agree for S and S prime for every positive integer N. Then the, okay, so in this setting,

the FPPF cohomology 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 a notarian local ring and its maximal ideal, then when we consider a flat cohomology which supports the maximal ideal of the ring R

with coefficients in this group scheme, it doesn't change if we replace the ring by its completion, by its max-adic completion. And in fact, it's this statement in special cases that was also used in the wasawa theory where flat cohomology occurs, for instance, the ring R could be a localization

of a number field at some prime ideal. All right, the second property of FPPF cohomology that I'd like to mention is FPQC descent.

More precisely, FPPF cohomology with such coefficients agrees with FPQC cohomology and this gives more robust descent properties, more larger Liré spectral sequences. More precisely, the statement is that the functor

which associates to an R-algebra S, simplicial R-algebra actually, the FPPF cohomology of S with coefficients in G satisfies descent for the FPQC topology

and concretely, this statement means concretely, this means that the FPPF cohomology

of this ring S with coefficients in G is the same as the FPQC cohomology. Of course, FPQC topology is large, for instance, that of a field, it involves any other field, so one needs some cutoff cardinal, but that's just the inequality.

So for every large cutoff cardinal kappa, for every large cutoff cardinal kappa, we have that the FPPF cohomology can be computed in the FPQC topology and in particular, if one has a large cover of this S, flat cover, then one can compute the FPPF cohomology of S in terms of the check nerve of that cover,

which is how this result is useful in practice. Okay, and the third property is descent for yet another topology,

but after restricting S, is descent for FPF cohomology in P-complete arc topology. More precisely here, one assumes that this group is G, that this group G is of P power order for implicitly fixed prime P,

and then one considers the functor which associates to a perfectoid ring, to a perfectoid R-algebra S, the FPPF cohomology of S with coefficients in G.

This functor is, okay, so this functor satisfies descent for P-complete arc topology, and more precisely what,

I'll explain in a 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 for every map from S to a valuation ring, P-adically complete valuation ring of rank one

for every such map, there exists a diagram, there exists another P-adically complete valuation ring V prime, and the completion of a diagram like so,

such that the diagram commute, and this map is faithfully flat. In other words, okay, and this V prime is also a P-adically complete valuation ring of rank one. In other words, the P-complete arc topology is just a topology here on perfectoids,

for which any point of S valued in a P-adically 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, I mean, of course, when just for defining it,

one does not need to restrict the perfectoids, and so it's insensitive to reduce structures, somehow any faithfully flat cover, for instance, is such cover. And so the statement is then that the concrete interpretation of this, the sense statement is that the FPPF-comology of this S with coefficients in G

can be computed in terms of the cover as a derived inverse limit of the FPPF-comologies of the checkner of P-adically completed, a checkner of the cover.

And the point why this is useful is that in practice, because the topology is somehow so weak, then in practice, one can choose S prime

to be a product of valuation rings, product of P-adically complete valuation rings with even algebraically closed fraction field. The nature of the topology allows such covers. And so 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 perfected A by a product of valuation rings, by forming suitable hyper cover in the P-complete arc 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-comology. So to which extent is it essential to use the arc topology rather than the previous, I think, called V topology or the slightly, the one where we will use not rank one, 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, when one sees 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- Well, you get only perfectoid ones from that. Okay, I mean, in short, I don't know right away, but I suspect it's important.

And in any way, it's certainly very convenient for this P-complete arc topology. Whether one can get away with a V, with a V descent, I'm not sure. Okay. All right. And so the proofs of these,

okay, so in the end, we see that it suffices to prove maybe just this statement or that excision. And so it's these properties of FPPF commutative that in the end give the key formula. And so how do we prove this based on the case of the key formula in positive characteristic that we assume there?

That's a step one. So the proofs reduce to the case when S and S prime are perfect FP algebras. Okay, I mean, initially these S and S prime

need not be even FP algebras. So certainly it has, I mean, the reduction is somewhat indirect. And more precisely, what is happening in here in the reduction is, well, if one just looks at the functor which computes cohomology after inverting P,

then this functor, and here the cohomology can be taken to be a tau cohomology because after inverting P, the group which is a P power order by decomposing it into primary parts, this functor satisfies descent for the arc topology, which is a slightly stronger topology than a P complete arc topology.

This is a result of Batt and Matthew, which, I mean, some of the consequences of this arc descent were already shown by Gaber and Fujiwara. One also uses deformation theory in a crucial way.

More precisely, the statement from deformation theory that one uses that if one has a square zero thickening, square zero thickening S of a ring S bar,

both simplicial actually, secretly with ideal, with ideal I, square zero ideal I, then the flat cohomology, the FPPF cohomology of S with coefficients in G is computed in terms of the FPPF cohomology of S bar with coefficients in G

in terms, I mean, the co-fiber of this map is the R-hom, R algebra R-hom from the pullback along the identity section of the cotangent complex of this five-file group scheme G over A, R-hom 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-hom, one wants this ideal I to be perhaps free over S bar in practice to be able to control this cohomology.

This is a perfect complex supported in degrees zero and minus one. But if the ideal I is some arbitrary ideal, this would be for a quantum coloring, then one couldn't control this deformation theory sufficiently, sufficiently, finally. Okay, and the third input. How long over R would you like?

Sorry? How long over R? 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 an object in the drive category of R modules because my group scheme is defined over my base ring R.

And in fact, I write in this way so that I don't have to discuss, I mean, S will be like simplicial ring in the end and so I don't really want to get into what is R-hom over simplicial ring. It's just, okay.

So these two ingredients give some of the reductions that go into the proof of these new properties of flat cohomology,

but really they by themselves don't allow us to pass the positive characteristic as we want to reduce the perfect FP algebras. And the third input is a certain p-adic continuity formula that I'm about to state, which is another property of FPP of cohomology

that we show. So the third in these inputs to the proofs is the following p-adic continuity formula. Here we assume that G is of p power order.

I'll just write it in formulae like so, that G is killed by a power of p. And then for a simplicial R-algebra S

for which the zeroed homotopy group is a Hanselian along p. So if R is just the usual algebra with no higher homotopy, then this assumption just means that this S is Hanselian along the ideal p.

So for instance, S could be a discrete and p-adically complete ring. We complete R-algebra. Okay, so under these assumptions,

the flat cohomology 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 a derived inverse limit of derived reductions of S modulo powers of p with 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 piece of characteristic zero 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 our 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 FPPF cohomology.

Okay, so first, I mean, certainly for simplicity, one could assume that S is discrete, let's say. But nonetheless, even in that case,

these derived reductions, they're not really discrete rings unless S is P torsion-free. 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 FPPF cohomology of a simplicial ring with coefficients in G actually is.

So first one defines FPPF site and map F of simplicial rings. So these are just simplicial objects in the category of rings, see? I mean, we can think of them like that for just for simplicity, see? 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 a homotopy in degree zero,

the natural base change map is an isomorphism. So the N homotopy group of S prime is just the base change of the N homotopy group of S along the map pi 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. Okay, so then we get the notion of a FPPF cover in particular, a faithfully flat map. And we get the site,

we get an FPPF site on simplicial rings. Right, that's right. I mean, I don't, I mean, I think it's not really a problem. Like, yes, so somehow really it's an infinity category

that's behind this. But everything in the end depends on the homotopy category. One takes this derived tensor product, but the property of the base change being flat or not, it really only depends on homotopy group. So it's not, I mean, I think it's not so abusive to say it's actually a site.

But one uses, for instance, quill and spectral sequence for a derived tensor product of simplicial rings to show that this notion is stable under base change, which perhaps is not immediately obvious. Okay, it's certainly stable under composition that is clear from the outset.

And so one defines the FPPF cohomology of S with coefficients in G by chiefifying, by, okay, by chiefifying sufficiently many times, transfinantly many times, just like in Lury's book on higher topos theory,

the functor which sends a simplicial ring S, simplicial R algebra S to the mapping space of, to the S point space of G, which can be thought of just as a simplicial abelian group.

And so one place as a simplicial abelian group in negative and non-positive cohomological degrees, and then one chiefifies that functor in the infinity categorical kind of sense. And so one gets the cohomology 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 periodic continuity. So firstly, in negative degrees, nothing happens. Just like if one forms a PPPF cohomology of usual ring and coefficients,

with coefficients in a group or in the shift and in degree zero, it shows global sections. And more generally, in the setting, the truncation in cohomological degrees, less than three point zero of this R gamma SG, it's just nothing else but the S point space of G itself. So in negative degrees, nothing happens.

And in positive degrees, one can show, I mean, it is not immediate from definitions and the fact that specific G being a finite locally free group scheme. Okay, so in positive degrees, this is actually nothing else than just the PPPF cohomology of the pi zero

with coefficients in G. So this PPPF cohomology of simplicial rings and how splices this simplicial abelian group with cohomology that happens in positive degrees, but splices in a way 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 map to simplicial R-algebras, Sn, Sn minus one is such a tower, for instance, given by Poznikov truncations

of the original S, the assumption is that the truncations in homological degrees less or equal and N are insensitive when passing to Sn, then the PPPF cohomology of S with coefficients in G

is just the inverse limit of the PPPF cohomologies of Sn with coefficients in G with respect to this 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.

And so, okay, so let me get back to this periodic continuity. Now, we assume that S is P-Henshelion and we want, and G is of P power order, and we want that the R gamma S G

is the same as our limb of R gamma modulo, modulo, well, right properly, the right tensor product. So in the first formula for H0 and lower, I suppose we are lower because ring is simplicial.

That's right. You wrote G of S. Yes. But this should depend on S. I suppose if you require isomorphism simplicial ring, this should not change your R gamma. Yes, that's right. But G of S, when G is not smooth, it's not so natural to evaluate G naively. I mean, there's a map.

There's a map from this to that, and it's a quasi isomorphism. It's just an isomorphism on homotopy. And like pi N H, H negative I of this is the same as H negative I of that, which is then really. The official resolution of something that,

so if you have coefficients like mu P or alpha P that are 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. Yes. And so then you have this simplicial ring coming with smooth things, and then you take G of this,

but when the group is like alpha P or mu, I mean, there are no non-trivial sections on. 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. Okay, not the naive ones.

Yes, it's not the naive morphisms. 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, I'm out of time.

Let me just give a basic step is to build, starting, well, first one reduces to S being just at pi zero by using this Posnicov-Tower argument.

Both sides behave well with respect to Posnicov-Towers because of this fact. And then the main somehow step is to replace, is to build an in-depth PPF hyper cover

of S by some SI such that each SI is a P-henzylian just as S was. And in addition, it has

no non-split, no non-split FPPF covers. Just like one builds local rings in the FPPF topology by taking limits overall, I mean, then one can also, for any scheme, for the ring S, one can build it in the FPPF cover, which has no non-split FPPF covers itself.

And if one alternates that with P-henzylization, one can make this such that the ring is P-henzylian and then one continues to do the hyper cover of this sort. This computes the R-gamma of this in terms of the R-gammas 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 R-gamma SI of G, because it has no non-split covers, it's just G of S. Then one wants to kind of show the same for S modulo P to the N, and then there's only, one uses a big array sequence, one uses a passage to derive P-hat decompletion, somehow derive Boor-Velles' law.

Anyway, this amount of time, so I won't enter into details there.

So I'm gonna go back to a question of Oakhurst. The G of S there, I would have thought of as just a simplicial group, but it's apparently not. I mean, it's in some infinity category, is that right? Yes, it is a simplicial abelian group, but it's to be understood as a simplicial abelian group in somehow the infinity category of simplicial abelian groups.

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 simpler approach by hand, so to speak, is perhaps to resolve your final 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 it out locally, there is no problem with lifting, so it should be possible to work more naively, like to define G of S as, at least in the, in using a tautopology as more naively.

I don't know, maybe the world theory needs you, but I believe that one can, in many cases, one can simplify, at least for the coefficient of the. 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 baguette resolution from the outset

and without entering this 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 its 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, so it's, I mean, it is true that our coefficients in the end are sufficiently simple, just a five-file group scheme, or a complex of small groups. I believe that, but also in your continuity thing,

of course, you're using this fancy kind of cohomology of, you even have to iterate infinitely many times because of this negative cohomological degree, so I'm not so sure about my question, but my question was, so you write Arlim,

but I suppose that in good cases, so the good cases, for example, you have an Inzylian, that's the usual Inzylian ring, where let's say that the phytogenicity is even bounded, then the situation is like a peak approximation, it is much better, you have not only the Arlim, but it is more or less essentially, not necessarily,

but I think that in every cohomology it will be the projective system will be essentially constant, so you have- Yes, in fact, it's this, I mean, we use this type of Elkic 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 in non-positive degrees, and on modulo p to the n, we then also show that it's concentrated in non-positive degrees, we take the Arlim, but we still need to show that the pi zero satisfies meta-Gluvla conditions, so that we don't jump off into H1.

And to do that, we use the Elkic 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, the Elkic I think is there in this step. But the Arlim in general is not essentially constant in every degree? Yeah, at least we don't know that.

Okay, I suppose if you start on the usual language, but okay. Yeah, I think it's reasonable to expect that it is.