2/6 On the local Langlands conjectures for reductive groups over p-adic fields
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00:00
Physical systemSpacetimeLimit (category theory)Network topology2 (number)InfinityRight angleCovering spaceAxiom of choiceCategory of beingArithmetic meanCounterexampleCohomologyMultiplication signPrice indexDegree (graph theory)Fiber (mathematics)ResultantProduct (business)Filter <Stochastik>Object (grammar)Vector spaceTheoremGrothendieck topologyMereologyPiMilitary basePoint (geometry)State of matterRhombusAreaSocial classFiber bundleSet theoryBasis <Mathematik>Group actionComputer animationLecture/Conference
09:02
Connected spaceSpacetimeLimit (category theory)Covering spaceSheaf (mathematics)ExistenceAlgebraic closureFiber bundleMathematicsCondition numberRational numberMultiplication signEinbettung <Mathematik>Spectrum (functional analysis)TopostheorieCategory of beingResultantKörper <Algebra>ModulformParameter (computer programming)Order (biology)Flow separationArithmetic meanRight anglePoint (geometry)Network topologyCompass (drafting)Position operatorShooting methodLecture/Conference
17:52
Element (mathematics)Right angleFilm editingMaxima and minimaAssociative propertyArithmetic meanPoint (geometry)Connectivity (graph theory)SpacetimeKörper <Algebra>SubsetPerfect groupProcess (computing)Product (business)ChainSet theoryConnected spaceMultiplication signEstimatorGroup actionOpen setLine (geometry)Limit (category theory)1 (number)Flow separationCondition numberSummierbarkeitEngineering physicsGeometryClosed setProendliche GruppeSpectrum (functional analysis)Variety (linguistics)Topologischer RaumFiber (mathematics)Bounded variationPower (physics)Ring (mathematics)RhombusModulformSparse matrixIntegerAlgebraic closureFinite setResidual (numerical analysis)Category of beingComplete metric spaceInverse elementRankingPanel painting
26:43
Hausdorff spaceRight angleArithmetic meanEquivalence relationNichtlineares GleichungssystemMultiplication signCovering spaceSpacetimeFiber (mathematics)Immersion (album)Uniqueness quantificationQuotientZariski topologyDiagonalNetwork topologyMorphismusEinbettung <Mathematik>RhombusSheaf (mathematics)Affine spacePositional notationGoodness of fitGeometryClosed setPoint (geometry)Flow separationMathematicsAlgebraAdditionParameter (computer programming)Compact spaceCategory of beingRankingCharacteristic polynomialSet theoryLocal ringDifferent (Kate Ryan album)HypothesisBasis <Mathematik>Linear subspaceDirected graphInsertion lossCausalityPrisoner's dilemmaMetric systemMetreTheory of relativityRing (mathematics)Total S.A.Lecture/Conference
35:33
SpacetimeEquivalence relationSheaf (mathematics)Multiplication signMultiplicationRhombusCovering spaceTorusLimit (category theory)Axiom of choiceComplete metric spaceAnalytic setFunktorCharacteristic polynomialProduct (business)Category of beingNatural numberFiber (mathematics)Goodness of fitStability theoryQuotientGlattheit <Mathematik>Differentiable manifoldPressurePower (physics)AlgebraPosition operatorGroup actionRight angleArithmetic meanMilitary baseExtension (kinesiology)Metric systemPerfect groupNoise (electronics)Analytic continuationNetwork topologyPoint (geometry)Theory of relativityLecture/Conference
44:24
Duality (mathematics)InfinityCovering spaceProof theoryPower (physics)Limit (category theory)Root3 (number)ApproximationObject (grammar)SpacetimeAutomorphismGroup actionParameter (computer programming)Module (mathematics)SequencePerturbation theoryFinitismusFiber bundleElliptic curveCategory of beingEquivalence relationAlgebraMorphismusRhombusSheaf (mathematics)TheoremAxiom of choiceRing (mathematics)IsomorphieklasseProcess (computing)Model theoryUniverse (mathematics)OpticsInsertion lossPhase transitionComplete metric spaceNichtlineares GleichungssystemSequelMultiplication signForcing (mathematics)Social classAffine spaceMathematicsDirected graphLecture/Conference
53:14
Topologischer RaumSheaf (mathematics)Abelsche ErweiterungProendliche GruppeGroup actionMultiplication signWell-formed formulaKörper <Algebra>Set theoryContinuous functionRhombusClassical physicsParameter (computer programming)Spectrum (functional analysis)TheoryCompactification (mathematics)Projective planeEquivalence relationSurjective functionSpacetimeCharacteristic polynomialClosed setLinear subspaceCompact spacePositional notationIdentical particlesGamma functionCategory of beingNatural numberElement (mathematics)Series (mathematics)Analytic continuationPower (physics)Quotient spaceRootProduct (business)Universe (mathematics)Thermal conductivityProfil (magazine)MathematicsExpressionPerfect groupPoint (geometry)Group representationExplosionLecture/Conference
01:02:04
TheoremAlgebraic structureNilpotente GruppeMorphismusCondition numberAnalytic setMereologyFunktorAnalogyFrobenius endomorphismSpacetimeRhombusSurjective functionCohomologyCovering spaceIsomorphieklasseRight angleGradient descentCharacteristic polynomialOpen setCurvatureHomomorphismusAnnihilator (ring theory)Group actionUniverse (mathematics)Normal (geometry)Multiplication signFocus (optics)SummierbarkeitNetwork topologyWeightPiCausalityBasis <Mathematik>Lecture/Conference
01:10:56
Maxima and minimaCategory of beingGradient descentBounded variationArithmetic meanFree groupFunktorParameter (computer programming)SpacetimeCondition numberFiber bundleConnected spaceCurvatureSet theoryAnalogyEquivalence relationGeometryRing (mathematics)ResultantTerm (mathematics)Network topologyPosition operatorLocal ringEnergy levelIsomorphieklasseMultiplication signCurveSocial classGlattheit <Mathematik>FamilyGroup actionKontraktion <Mathematik>Basis <Mathematik>Table (information)Connectivity (graph theory)SubsetOvalLink (knot theory)Nichtkommutative Jordan-AlgebraMathematicsCovering spacePiGame theoryFocus (optics)Lecture/Conference
01:19:47
Different (Kate Ryan album)Group actionOpen setRhombusAdditionFlow separationResultantSubstitute goodCondition numberFigurate numberMultiplication signCausalityRight angleProof theoryPrice indexEnergy levelFrequencySpacetimeProduct (business)Covering spaceGrothendieck topologyCategory of beingGradient descentFiber (mathematics)Variety (linguistics)FlagClassical physicsSocial classReduction of orderSubsetGroup representationTopostheorieMorphismusTheoremLecture/Conference
01:28:38
NumberCondition numberRhombusWater vaporPhase transitionPoint (geometry)SpacetimeCategory of beingCausalityMultiplication sign2 (number)Gradient descentPrisoner's dilemmaProjective planeMultilaterationCovering spaceFamilyGraph coloringMany-sorted logicEnergy levelProduct (business)ResultantEquivalence relationNetwork topologyFiber (mathematics)Affine spaceReduction of orderQuotientGoodness of fitMereologySurjective functionArithmetic meanTheoremTopologischer RaumLecture/Conference
01:37:29
Ring (mathematics)RhombusSpacetimeGroup actionDifferentiable manifoldPerfect groupMultiplication signWater vaporClosed setTotal S.A.Covering spaceLogicCategory of beingEnergy levelSpectrum (functional analysis)Affine spaceReduction of orderCondition numberProof theoryPropositional formulaCompact spaceAnalogyQuotientArithmetic progressionSet theoryLine (geometry)Game theoryConnected spacePoint (geometry)Algebraic structureTheoremAxiom of choiceGlattheit <Mathematik>Topologischer RaumBijectionTopostheorieIsomorphieklasseNetwork topologyLie groupKörper <Algebra>Bounded variationDivisorParameter (computer programming)AlgebraLecture/Conference
01:46:20
Point (geometry)Condition numberTopologischer RaumSpacetimeAbsolute valueRhombusLinear subspaceCovering spaceOpen setCompact spaceLogicBasis <Mathematik>Glattheit <Mathematik>Spectrum (functional analysis)1 (number)Category of beingTheoremAnalytic setPropositional formulaMaxima and minimaAnalogyCurvatureMultiplication signObject (grammar)Network topologyGroup actionMathematical singularityCausalityWater vaporAxiom of choiceSubsetFinitismusQuadrilateralShooting methodMilitary baseSign (mathematics)Line (geometry)Lecture/Conference
Transcript: English(auto-generated)
00:11
So, today's goal is to define diamonds and do some basic stuff with them. And so, let's start right away, defining proton morphisms, which play a important role.
00:25
So, a morphism, let's say from Y to X of perfectoid spaces, sorry, of a phenoid perfectoid spaces,
00:50
is, let's call it a phenoid proton, if it can be written as an inverse limit of a co-filtered
01:14
system of the tau maps f i from x i to x, or y i, to x, y i again. Just wait a second.
02:07
Okay, so you just take any inverse limits, like on the inverse limits that you have to say co-filtered of such a tau maps, where I should maybe remark that the category of
02:22
a phenoid perfectoid spaces has all inverse limits.
02:42
You mean filters only when… Well, I mean, you also have fiber… You have to define a log vector. Ah, ah, it's all connected, I have to say, right?
03:03
I mean, the indexing category has to be connected, because, sorry. So, in other words, it reduces to co-filtered guys and fiber products.
03:20
And, okay, then maybe you want a notion for all spaces. And let's just say that morphism from Y to X of general perfectoid spaces is proton,
03:45
if it is locally on source and target, a phenoid proton.
04:06
I should warn that these notions are really not so well behaved in general. So, say if you have such a map of a phenoid perfectoid spaces, and maybe locally on X, it is a phenoid proton, then it's not clear that it's globally a phenoid proton,
04:22
because it's not clear if you have local choices for these eta maps, how to build global choices. So, for schemes, there is a counter example to this, and I believe it adapts to this case. Similarly, this notion of proton for perfectoid spaces has also some caveats.
04:41
You can't really check it locally on X in the proton topology. There are some small caveats to this, but they will be addressed in a second. But let me first state, so you can define a proton site,
05:07
where covers are generated by, say, open covers. And subject to a phenoid proton map.
05:29
And although the site somehow may not be so well behaved, but the associated topology is actually a pretty canonical object.
05:44
Anyway, so you have the following theorem that this site is subcanonical. Also, if X is a phenoid perfectoid,
06:09
the proton cohomology of OX plus is almost zero for bigger than zero.
06:22
And in degree zero, it is what you think. I should say that OX is a sheath.
06:45
Let me very briefly sketch whether this is true. So, last time we had a similar result for the etaus site.
07:01
And so this implies that if you send some X with some spar R plus to R plus mod pi for some fixed pseudo-uniformizer pi, you really can't really choose this globally on the whole category of perfectoid spaces, but whenever you work over some fixed X,
07:23
you can somehow choose it on X and then take the pullback to everybody above. And if you want to check that something is a sheath, you can always work over some base space. So, this thing is almost an etaus sheath,
07:42
meaning that if you pass to the almost category, it becomes an etaus sheath. And it's almost a cyclic. But now if you want to pass through the pro settings,
08:06
and if you have some inverse limit of some spar Ri, Ri plus, this is some spar R infinity, R infinity plus,
08:23
then actually the, and this is co-filtered, then actually the R infinity plus mod pi is just the direct limit. Of the Ri plus one pi.
08:41
So, this is essentially how you compute the inverse limits. You just take the direct limits of the Ri pluses and then pi R complete to get the R infinity plus and then invert pi again to get R infinity. But this means that this sheath is somehow, on the proactive side, you get it by just somehow
09:01
evaluating it on, so assume that this is some kind of proactive thing over your base space of spar R plus. Then if you evaluate on this pro et al thing, you just get the direct limit of all the values on the etaus things. And then the pro et al sheath properties just falls by taking filtered direct limits from the etaus sheath.
09:34
What's the composition of a finite pro et al maps? They are still a finite pro et al.
09:41
But this is not clear how to, when you take the limit, it's not clear how to get to the center of the car map to, so if you have got z to y, which is etaus, how do you slightly descend it to something of a yi? You can certainly descend it to something et al. The question is whether it's still a finoid.
10:01
You can certainly, so there is a statement that if you look at quasi-compat quasi-separated et al maps to some limit, then this is a two categorical direct limit of the quasi-compat quasi-separated et al maps to some. Like what you said, like what you explained in the previous sense of, essentially you explained last time as you locally can write it, okay? But then you have to show it's a finoid.
10:21
Then I have, wait, that is, sorry, maybe I only, do I claim that in general these compositions are finoid pro et al?
10:41
No, sorry, I don't think, sorry, I take back that claim that in general finoid pro et al maps are composed to finoid pro et al maps, but you could assume that these are some kind of, well, it doesn't really matter. You can assume that these are composites of rational embeddings of finite et al maps in practice. In which case you can get through what you want.
11:10
I don't really care so much about this very notion of a finoid pro et al. I more care right now about the notion of the topos it generates, and for this it doesn't really matter.
11:25
All right, so there is this problem that notions are slightly tricky to check.
11:47
And I want to address this in some sense locally in the pro et al topology, so this can be resolved.
12:15
So this is based on the following definition.
12:28
I say that a perfectoid space X is totally disconnected,
12:45
or respectively strictly totally disconnected.
13:06
Well, I need that it's quasi-compact and quasi-separated, but more crucially, the condition is that any open cover splits in the totally disconnected case, or any et al cover has a splitting.
13:50
Meaning that if you have some UI to X, some et al open cover,
14:01
then I want that the map from the disjoint union to X has a splitting, a section.
14:34
And it follows from essentially arguing like the existence of algebraically closed fields,
14:43
or algebraic closure is that whenever you have, say QCQS perfectoid space to start with, you can build one which is pro et al over it and which is strictly totally disconnected. So if X is any, well, let me actually say this
15:00
with a phenoid, we can always find a subjective phenoid pro et al at X tilde to X
15:29
with X prime strictly totally disconnected.
15:46
So it just somehow takes the inverse limit of all possible et al maps to X in some sense. X tilde, yes, thank you.
16:03
What will be important later a little bit is that you can even arrange this map to be open and even open after any base change. But I think this will only be important
16:20
for some arguments that I'm not going to explain. So this means that locally in this pro et al topology, your spaces somehow are such strictly totally disconnected guys. So how do they look like?
16:43
So you can give a classification result for strictly totally disconnected and also just totally disconnected guys. They're pretty simple, if and only if.
17:14
First of all, X has to be a phenoid and every connected component of X is extremely simple
17:40
is of the form the edX spectrum of some KK plus where K is a perfectoid field and K plus and K is a bounded and open
18:05
variation subring. So these are the kind of spaces that takes the role of points in the world of edX spaces in some sense.
18:21
Whenever you have a point of an edX space, you always get a non-archeminian field K which is some of the residue field at this point plus such an open and bounded variation subring which corresponds to the variation on K corresponding to the point you're taking. And then you always get a map from Spark KK plus into your space.
18:42
But this is a totally ordered chain of points. But I should finish the statement first. So this is in the totally disconnected case and the strictly totally disconnected case you also need to assume that K is algebraically closed.
19:14
This is a totally ordered chain of points.
19:36
Okay, so you've essentially completely torn apart your space that it's just...
19:42
So let me make a remark that you always have a map from X to pi naught of X where pi naught of X is some pro-finite set that's true for any spectral space.
20:13
So you have this pro-finite set of connected components and all the connected components are really simple. And so you've essentially lost all geometry
20:22
after you did this. The underlying topological space. Here.
20:47
In defining it as an arctic space you have to restrict the structure shape and then do some completion probably in general for a connected component. That's right.
21:01
Yes, yes. So I mean by connected component I mean somehow the... They're closed, yes. But they're inverse limit of open and closed subsets. Each of these open and closed subsets is itself in a finite perfectoid space. And then you take the inverse limit in the category of a finite perfectoid space to see what this means.
21:37
In particular if it's strictly totally disconnected
21:40
then I would write these guys as sparse CC plus so it's just a lot of Cs. And then if you crush a lot of Cs together it's a diamond. Did anybody get this joke?
22:02
Okay. So locally you are of this form and if you are of this form then actually you can classify these proton maps more geometrically.
22:24
So if X is strictly totally disconnected and F from X prime to X
22:41
is a quasi-compact and separated map.
23:02
I should say what separated means. The easiest way to say it is separated just means that the map from X prime to the underlying topological space of the fiber product has closed image.
23:22
So for schemes that's one possible characterization. You can take it as a definition. Here there are several equivalent characterizations so you can also prove a variety of criterion for separatedness. Okay, we need to separate in this condition here. Okay, so then for such maps you can characterize when they are proton.
23:42
So then F is proton and only if the following happens that for all rank one points of X or let's say
24:01
for all generic points of X so this corresponds to some sparsi-posi mapping to X. So I said that
24:22
the connected components always had spare KK pluses which are a totally ordered chain of points and in every such connected component there's a unique generic point and also a unique special point. But it's considered a unique generic point. This corresponds to the maximal generalization of this variation which just is the one which forgets essentially about the K plus and replaces it just by the ring of integers
24:42
by the ring of power bonded elements inside of the field K. So you'll only look at those points. You want it for all these generic points. If I look at the fiber product and this is edX spectrum of COC
25:04
that is just some profinite set. Let's move to S times plus COC for some profinite set S.
25:23
What does this S underlined times mean? So if you have the profinite set you can write it as an inverse limit of finite sets and this product is the inverse limit of C products
25:48
with the finite sets. So if you have something which is a finoid pro-etal over just a geometric point like so then all the etal maps are just given by finite sets over it
26:00
and so the kind of a finoid pro-etal things should just be given in this way by profinite sets. And so that's the only thing you have to check. You only have to check that at the generic points you are such a profinite set. And in this case if you are pro-etal you're actually a finoid pro-etal.
26:34
Is it true also without strictly but taking the algebraic closure
26:42
of k, complete algebraic closure for every generic point? Is it true if I drop this strictly and then ask the same thing just for algebraic closures? Good question. I wasn't able to verify this
27:01
when I did this. One problem I had was that if you have such a SparkKK plus where k is not algebraically close and then you pass to finite etal cover then in general this is not somehow local anymore. You may have different close points. If you take such a local space
27:20
SparkKK plus which has a unique close point and then pass to finite etal cover you pass to finite etal cover of it then it may have different close points several different close points. And this was slightly complicating the arguments. So for this reason it was easier to really assume that you're in the strictly totally discrete.
27:42
Another question, a small one about the closed image and definition of separated. Is it like for schemes that for affine or finoid you have a kind of a close immersion that is a topological quotient of the... Yes, yes. So for a finoid it is always true and then this is a closed immersion in every possible sense.
28:02
So it's even defined by equations. So... Right. And so in general the diagonal map will be some kind of locally closed immersion as for schemes and then asking that it has closed image implies that it's a closed immersion in some sense of the word.
28:28
So in this secret manuscript there's a discussion about these notions and so. Okay, so this means that over such a strictly totally disconnected base
28:40
these notions of proton and finoid proton at least under these small hypotheses especially the separation hypotheses the equivalent to the minimal thing you would certainly ask for a proton map to be satisfied namely that over geometric points it's given by profinite sets. And so this means that in this case
29:00
you're satisfied with the notion and then in general maybe you like to redefine the notion. So a notion that becomes more important later than a proton is what I call quasi-proton.
29:21
So if after any base change
29:58
to a strictly totally disconnected X tilde
30:04
that map is proton. And then at least for morphisms which are in addition
30:21
quasi-compact and separated you can just check this by checking that for all such geometric rank bond points mapping to X the fiber is a profinite set. And then if the morphism is not separated
30:41
I mean there are again some subtleties to this definition but at least in the separated case which is one can often arrange things to be separated. This is a good notion.
31:01
Finally we can define diamonds.
31:27
A diamond is a proton sheaf from the category of perfectoid spaces of characteristic p
31:48
this is important such that Y can be written as a proton sheaf
32:02
Y so Y can be written as a quotient Y is X mod R. Okay so I'm here implicitly identifying the space for the sheaf it represents.
32:22
So I'm using no notation for the yonder embedding as a quotient Y X mod R where X is a perfectoid space
32:41
and R in X times X is a proton equivalence relation meaning that the source and target maps
33:00
from R to X are perfect. So also the R itself should be a perfectoid space. Maybe I should write representable somewhere.
33:27
Okay so you might fear that this definition is sensitive to these subtleties about the notion of a proton morphism but actually it's not sensitive to these issues
34:04
Y so whenever you can write Y in such a way you can without loss of generality also assume that X is just a disjoint union of strictly totally disconnected guys by passing to a further proton covering so disjoint union
34:27
and so then
34:42
R in X times X is a subspace and this is separated space and then by some general nonsense R must be separated again strictly totally disconnected
35:02
Well it might not be quasi-compact anymore. I mean so this disjoint union is only necessary because the space might not be quasi-compact and you need more and more spaces to fill out most of the space. Then the equivalence relation is automatically separated which also implies that the map is separated
35:26
and so now you're in the situation where you have the basis well the disjoint union of strictly totally disconnected spaces and the map is separated and then well to check that
35:41
something is pro-etal you can somewhat check it on quasi-compact open subsets so we're in the situation
36:00
where equivalence relation on a diamode represented
36:21
by a diamode where the maps are in the same sense pro-etal in the very weak sense then it's right so that Ophagava was asking whether if I now do this procedure again and try to take a quotient of a diament by some kind of pro-etal quasi-pro-etal equivalence relation is it still a diament and this is true
36:41
so as for algebraic spaces and so on so this notion of a diament has very good stability properties and so as I said so one justification for the name is that
37:01
how do you get a diament you take some strictly totally disconnected guy which is just a bunch of Cs totally disconnected and then you smash them together in particular this way of looking at a diament makes it not obvious that
37:20
they have a nice geometric structure I claim that you can still define smooth maps and so on that this will take some time and will only be done maybe at the end of the next lecture so let's do some examples
37:46
so the first thing I want to do is I want to relate this to the kinds of spaces we're usually interested in and so the other claim is the following that there is a natural
38:00
functor from analytic edit spaces over zp
38:24
to diaments extending the functor from all perfectoid spaces
38:40
to perfectoid spaces of characteristic p which are full subcategory here and these are full subcategory here
39:02
taking x to so in this sense you can define the tilt of any such analytic edit space now there will be such a diament
40:03
so let me first give you the intuitive definition so say x is
40:25
an analytic edit space of zp then you just choose
40:42
some proton cover x to x where x to is perfectoid let me not make completely precise by what I mean by this but
41:01
say an example would be that you just take the torus here in the example I did yesterday and then take the inverse limit of all multiplication by p maps or p's power maps then the inverse limit is such a perfectoid space so then if you look at the fiber product
41:23
this is proton over a perfectoid space and so there are some subtleties here what I mean by the fiber product because I need to do some uniform completion here really
41:40
but if you do it right this r will again be perfectoid the equivalence relation you get here and then define x diamond to be you take this perfectoid cover which you can tilt
42:01
and you can also tilt the equivalence relation and then the obvious problem with this approach is that even if you can make sense of all the words I said here it's not clear a priori that what you get is independent of the choice
42:41
and so for this reason the official definition goes differently so let's say x is analytic at x space over zp
43:04
then you define x is diamond as the following sheaf and let's say I only define it on a phenoid because it's supposed to be a sheaf you can then extend to all perfectoid spaces this is perfectoid
43:23
of characteristic p so the idea is that it should be all maps from this perfectoid space to x but this is of characteristic p and this might be of characteristic 0 so there are no such maps and so the way to correct
43:41
this is that you don't necessarily ask that you map this given perfectoid space to x but maybe instead some until which has some of the same information anyway in some sense so the definition is the following that it's centered to the following data
44:12
so this here is some perfectoid algebra which is an until of r in the sense that
44:21
if you tilt r sharp again it is r and then any such until automatically comes a distinguished choice for a plus sub ring and so then you just want a map from this bar r sharp
44:44
you take such data up to isomorphism with the remark that there are no automorphisms
45:20
and so the theorem is that this is really a diamond
45:41
and can be calculated as above let me summarize this but can be calculated intuitively
46:27
what do you mean that there are no automorphisms? well I mean if you define some sheaf on some category some modular problem then you and you do divide by well you usually want to divide by some morphisms
46:41
but usually this is not so well behaved to say automorphisms of objects right like some modular space of elliptic curves and so and so but this is not an issue for us because there are no automorphisms of these objects so it's really a well behaved thing to just divide by by isomorphisms
47:08
okay let me briefly say something about the proof so first you need to check that this is actually a proton sheaf
47:36
and that's actually a little bit of work in making
47:42
the tilting and un-tilting procedure because I mean it's even the sheaf if you forget about this map to x so just I mean adding this map to x back in is easy so the question is whether sending a perfectoid space to the
48:01
set of un-tilts is a proton sheaf and you can prove this but it's easy so then if x happens to be perfectoid
48:21
the tilting equivalent says that perfectoid spaces over x
48:43
are equivalent to perfectoid spaces over x-tilt via y-mixed to y-tilt
49:01
so if you have a base space then perfectoid spaces over that are equivalent without a base space the tilting is of course not an equivalent and if you think about what this means here it's exactly saying that this x diamond is equal to
49:22
x flat so if I'm looking for un-tilts which map to my given x then where this equivalence must be some other unique un-tilt under this equivalence so this means
49:40
that in general I mean for perfectoid spaces you know what happens and then in general you may assume it's a phenoid and then
50:06
there is an argument to Fontan and Gomez maybe that there exists some
50:26
sequence of finite atal gi tossers a to ai I'm gonna filter it
50:45
with a infinity which is a completed direct limit cais completed which then gets an action of g which is the inverse limit of the gis such that this is perfectoid
51:03
essentially just well one way to do it is just take an inverse limit of all possible finite atal covers passed to some universal cover at least if x is connected it's easy to do it's also enough to just take some
51:20
you know enough enough p power roots except if this is of mixed characteristic you have to be a bit careful because extracting p power roots is not a finite atal procedure so you have to slightly change the equation and do some art in triad covers but you can somehow extract some approximate piece roots by some finite atal covers
51:41
and then this means that in this algebra you will have enough approximate p power roots oh no these are finite and then in the inverse limit you get something okay you use
52:03
some perturbation of taking p so you the grammar group can be slightly okay you don't control yeah no actually so actually the argument I make is in the paper
52:20
if a is I mean I really passed to the universal cover and I make some sense of this if a is not connected and then you just need to check that after you pass this inverse limit that it always has approximate p power roots and for this you just write down any art in triad cover and you don't care about the Galois group anymore
52:43
okay so anyway you can do this and then you can define some x to infinity which is some spy infinity plus and then x diamond
53:01
will be isomorphic to the tilt of this x infinity module an action of this proof and group G well I mean the equivalence I mean this
53:26
notation is not exactly consistent with this notation because equivalence relation for a group action is really the group times the space with a projection into action method okay so
54:13
that's for example so if I take the edX spectrum of QP say and I
54:20
compute its diamond then this will be the edX spectrum of say the cyclotomic extension which you have to tilt by the action of the Galois group Cp cross and if you tilt this you get the edX spectrum of the field Fp around
54:40
series in T and all its p power roots which has a certain natural action of Cp cross explicitly if you have an element gamma in here then
55:32
gamma of is now over Fp
55:40
and why is it over Fp that's precisely because we insisted that so we only work with perfected space of characteristic p if we work with all perfected spaces and the base would still be Zp so this
56:14
is the main reason that
56:21
I want to work with this perfected space of characteristic p because then it's a more absolute theory okay so there's another funny example so
56:42
for this let me make the following remarks that if T is any topological space one can define a proton sheaf T
57:04
underlined which has implicitly occurred a couple of times already so particular you can make sense of this formula above there was this pro-finite set also using this definition can find a proton sheaf T underlined by saying
57:21
that the T underlined of any X is the continuous maps C0 so this means continuous maps from the underlying topological space of X into T so
57:54
you can this is a sheaf because all the covering maps are quotient maps
58:01
continuity can be checked after recovering and there's a following claim that if T is any compact host of space and K is
58:29
any perfect field say then if I take this sheaf corresponding to this topological space of compact host of space
58:41
and I base change it to bar K then this is a diamond so it's maybe slightly surprising that the sheaf compact host of space sneaks into this world
59:00
now so why is this true well there's this funny statement that whenever you have any compact host of space you can always find a surjective map from a profinite set onto it
59:22
there exists a surjection S to T where S is profinite it's some kind of classical fact I think the easiest argument for this is that you can take for S the Stone-Chesh
59:40
compactification of T as a discrete set
01:00:00
So the Stone-Chesh compactification of any discrete set is a profinite set. And it's universal for maps. So maps from the set into any compact host of space are the same as maps from the Stone-Chesh compactification to this compact host of space. And there's an obvious map from T into T, the identity. And by the universal property of the Stone-Chesh
01:00:21
compactification, this extends to a map from the Stone-Chesh compactification to T. And it's obviously surjective because always a map from T to T was surjective. And so you get what you want. But then the equivalence relation S times TS
01:00:42
is a closed subspace here. And this is a profinite set. So R, the equivalence relation is also profinite.
01:01:09
And so this means that then this funny sheaf here, you can actually write it as this profinite set times the point, which is a perfectoid space.
01:01:23
And you divide by the representable equivalence relation given by this equivalence relation here.
01:01:45
Yes, of characteristic P.
01:02:03
OK, maybe it's time for a break. All right. So the important technique for much of what will follow is what's called v-descent.
01:02:31
So the v-typology on perfectoid spaces is generated by, again, open covers
01:02:48
and all surjective maps of the phenoids.
01:03:09
So you basically allow everything to be a cover. And then the world is nice. So the v-typology is subcanonical.
01:03:25
The diamond functor, is it fully faceful? The diamond functor, the question was whether the diamond functor is fully faceful. It's not, just because some of them get the structure morphism. So somehow if two perfectoid spaces was the same tools they identified. Structure morphism to ZP.
01:03:43
What's more sensitive to us is whether if you, say, look at analytic spaces of a QP to diamonds over the spark QP diamond, whether this is fully faceful. And this is known to be true for semi-normal rigid spaces.
01:04:01
So under a finite mis-condition, you need semi-normality because the diamond functor, it will turn universal homomorphisms into isomorphisms. And so in particular, the semi-normalization map is always a universal homomorphism. So forget about this.
01:04:22
But then on semi-normal rigid spaces, this one can prove that it's fully faceful. Yeah, right. So maybe I should write this down somewhere. So there's a following theorem,
01:04:49
essentially due to K-diamond U, that if you look at semi-normal rigid spaces of a QP
01:05:03
to diamonds over the spark QP diamond, this is fully faceful.
01:05:22
And is it true of any non-alchemy field? Yes, you could take any non-alchemy field of characteristic zero here.
01:05:44
But it's critical that it's of characteristic zero because otherwise also the Frobenius map will be sent to an isomorphism. It's an interesting question whether it can prove something like this for some analytic edX spaces or some formal schemes which are flat over ZP and have some normality condition.
01:06:01
In this case, it also has a chance to be true.
01:06:21
It's sub-canonical. OX, OX plus a sheafs. And also if X is a phenoid,
01:06:43
the higher V cohomology is almost zero.
01:07:05
So in particular, any diamond defines a V-sheaf, sorry, any perfectoid space defines a V-sheaf. Part 2 says that actually also all diamonds are V-sheafs.
01:07:23
The cohomology of OX is zero and positive degrees, which is a consequence of systems, by just inverting pseudo-informizers. All diamonds are V-sheafs.
01:07:42
So 2 is an analog of a theorem of Gober from something like 2012 that all algebraic spaces
01:08:03
without any quasi-separatedness condition are FPQC sheets.
01:08:28
So it may be surprising that you can control all subjective maps here. I mean, for general edX spaces,
01:08:41
you can't even control open covers, but perfectoid spaces are so nice that they're all subjective names, so good. So why does this happen, that you can control everything? So by using pro-etal descent,
01:09:05
you can essentially reduce to considering only totally disconnected spaces. You might even consider it strictly, but I think for what I've been saying, it's enough to consider totally disconnected spaces.
01:09:28
But then there's a funny thing, so that if X is totally disconnected, and Y, so let's say this is some Spar RR+,
01:09:43
and Y is any affinoid perfectoid, Spar SS+, mapping to X, then there is some automatic flatness happening,
01:10:29
then the map from R plus one pi to S plus one pi. And it's faithful if that,
01:10:45
if Y to X is surjective. And so the rest of the V descent then follows from flat descent. Again, you work on this level mod pi first,
01:11:02
and then in some almost world, and then go back. Where does this flatness come from?
01:11:20
Well, you can check this on connected components, but then this is just some K plus mod pi, and the K plus is a variation ring.
01:11:46
And so K plus to S plus is automatically flat as it's torsion-free, over variation ring flatness can be checked
01:12:01
just by checking torsion-freeness. And so by base change, also the map mod pi is flat. Two connected components just closed,
01:12:20
so you check on it. I mean, these are both sheaves under the set of connected components, and to check for that flat, you can check at local rings. And then the local rings are somehow given by what you see on the connected component.
01:12:40
I mean, it takes a little bit of justification to make this precise, but that's the essential idea is that once you are such a totally disconnected base, you can essentially reduce, I mean, it's often you can reduce things actually to connected components by an extra argument. But this connected component is just a variation ring, so you get a lot of properties for free.
01:13:04
Okay, and then, okay, so that's the sketch for one. For two, you actually need some strong v descent properties.
01:13:31
So I want to state some of those.
01:13:41
Okay, so let's say we're in the following situation. Let F be a pre-stack, meaning just a functor to group words without any properties on the category of perfectoid spaces.
01:14:12
Then for a map from Y to X, let me denote by F of Y over X
01:14:21
the category of descent data, meaning a class in F of Y plus an isomorphism
01:14:43
in F of Y times X, Y, such that it was a co-cycle condition.
01:15:03
So there is a functor from F of X to F of Y over X. And F is a v-stack,
01:15:20
if and only if this is an equivalent for all v-covers, right? Well, you also need to do that this joint union is the form,
01:15:40
yeah, that the F of the joint union is the form, otherwise it doesn't follow from there. Okay, plus convertible with this joint unions.
01:16:19
Okay, and so
01:16:24
I've made this aggression here because the descent results will be slightly tricky to formulate in terms of that something is a v-sheet, if you can only say that if X happens to have some properties, then this kind of equivalence holds. Okay, so let me try to say this.
01:16:45
So if F is a thing which sends X to all perfectoid spaces over X,
01:17:03
then the functor from F of X to F of Y over X is at least fully faithful for all v-covers.
01:17:24
So that's an analog of the separated condition for a sheaf. But it might not be the case that whenever you have a given perfectoid space over a v-cover and the descent datum that you can descent to perfectoid space on X. I think that's even false for schemes, right?
01:17:46
For the total quality, I mean there is families of genus one curves which are not globally a scheme, but it's how locally they are a scheme, right? But I know. But you might hope that under extra conditions,
01:18:03
things are better. So now if instead I consider let's say F fenoid which takes which is only defined on the category of fenoid X and maps it to the category of a
01:18:20
fenoid perfectoid or a fenoid perfectoid. Somebody should resolve this question that one implies the other. Positively.
01:18:42
If I consider this functor, then while this functor is still fully faithful always but it's an equivalence if X is totally disconnected.
01:19:04
So if you're trying to descend something to a totally disconnected space and your total space is a fenoid, then you can do this. It fails more generally, but this is a failure.
01:19:25
Well, you don't see it for schemes and some usual flat topology say because a usual fpqc descent but it fails already in rigid geometry.
01:19:41
Smooth fenoid rigid spaces. Yeah, I mean also for the perfectoid case.
01:20:07
The examples are rather stupid. You can find some open subsets of rigid space which becomes an fenoid or even a rational subset after a covering of the space
01:20:21
but it's not globally a fenoid. It's for the two-dimensional ball. I just wanted to say that it's a failure which is already somehow in some kind of classical setup visible.
01:20:42
Didn't cause much trouble there. So then you can look at what else I want to say? Where am I here? If I look at separated and proital maps
01:21:01
which say you can again do on all perfectoid guys so it maps into this kind. Separated. Well, again fully faceless follows from one.
01:21:21
The question is whether you can in some sense prove descent effectivity of descent. And this holds true
01:21:41
if the base is strictly totally disconnected. This is another indication that the category of separated and proital maps for such a strictly totally disconnected base behaves well because
01:22:04
in this case these things glue even in the v-topology. And there's a final theorem for separated and proital maps
01:22:36
then it's actually a bona fide v-stack.
01:22:42
Sorry, I should include these. Then the formulation could be improved.
01:23:02
I just want to say that this guy is a v-stack. And also I think one should have that the category the atal topos of the perfectoid space is again be able to any atal sheave
01:23:20
probably you can descent. Any atal sheave you can also descent. Yes, yes, yes. So do you want to do something then? I think this should follow once I talk about atal sheaves and so on. Yes, yes, okay.
01:23:41
Let me defer this discussion to later. Okay. Okay, so you have some pretty nice descent results.
01:24:17
You can also prove that if x is analytic at its base
01:24:22
then the atal side of x agrees with the atal side of the diamond where maybe I should say what I mean.
01:24:46
So this means that if you want to construct
01:25:02
some atal maps to some rigid space say you can do it by constructing some atal map to a diamond and constructing a atal map to a diamond is equivalent to doing it v locally. So you can just cover your diamond by some huge strictly totally disconnected guy find an atal cover there
01:25:20
and then apply these descent procedures to in the end get some nice atal cover or an atal map to a usual rigid space.
01:25:45
as an aside, so this gets used to construct locus and more varieties which are, well these are again some gadgets which depend on some data of a reductive group, sigma-conjugacy class
01:26:00
and a core character and some level and then they should have some period map pi going to some flag variety which depends just on g and u. And this period map should be atal.
01:26:39
But I should
01:26:42
make a brief comment what I mean by this thing here because so far I didn't define what an atal map to a diamond is. So I need to define atal maps of diamonds or more generally of v-sheafs.
01:27:23
Atal if for all perfectoid spaces x is a map to y the fiber product is representable in the atal.
01:27:45
And secondly and here the relevant notion is the one quasi-proital
01:28:01
if again for all x but now I just ask it for strictly totally disconnected representable.
01:28:22
And then in the strictly totally disconnected base case quasi-proital and proital are the same. And then at least if the morphism is an addition separated then these descent results tell you that you can check these conditions locally.
01:28:49
I mean if you have a diamond you mean? I will answer this in one second.
01:29:03
So if f is separated these conditions can be checked locally by
01:29:22
the descent results. So there was a question whether if y is a diamond and you write y
01:29:41
as a portion of the perfectoid space by proital equivalence relation whether the map from x to y is quasi-proital I guess. And this is true.
01:30:01
It's not representable. It's not true or I'm not sure I have an example in mind but I don't know certainly I don't know how to prove that after pullback to any x prime mapping to y
01:30:20
the fiber product will be representable. Because I don't have the strong descent results except if the base is strictly totally disconnected. So what can be proved is that if you have a strictly totally disconnected x prime mapping to y then the fiber product is representable. Sorry and
01:30:49
I should have assumed that x is separated. Maybe to make this statement.
01:31:01
I mean you can always do this by just you can always replace x by disjoint union of phenoids or something like that. And what's the eta side of the diamond? I hoped I could come to this today
01:31:22
but I'm afraid I'm not. Well because of some issues with separatedness I don't actually allow all eta morphisms in this end. Let me defer the discussion of the eta side of a diamond actually a little bit later.
01:31:45
But covers are just given by families of maps which are jointly surjective as maps of sheaves. Which gives you the correct notion of covering. That is part of the question.
01:32:16
And so maybe another remark in the spirit
01:32:20
of y is a diamond and y to y prime is surjective and quasiproital and now I'm confused whether I need a small separate in this condition so let me check in the manuscript.
01:32:43
You can find it.
01:33:02
No, without any then y prime is also diamond. Is it any other
01:33:23
quasiproital sheave? You know it's enough
01:33:41
to assume it's a quasiproital sheave. Okay. Okay. Well if you want assume it's a v-sheave already but
01:34:01
so there's a way to set up definitions and statements in such a way that I could make this quasiproital sheave. Alright. Where am I going?
01:34:20
So So what are we doing? So we have this category of diamonds and we can define some good notions of italomorphisms and quasiproitalomorphisms of diamonds because we have good descent properties
01:34:43
like here and these are actually v descent properties and so this means this plus some other properties means that actually a few of the basic things don't just work for diamonds, they work for something much more general, namely any v-sheave.
01:35:02
So there's there's a following funny theorem
01:35:26
that let's say y is any v-sheave subject to a minor set-theoretic assumption, so such that there exists this surjective map, surjective meaning
01:35:45
surjective is a map of v-sheaves from a perfectoid space X.
01:36:02
Well then you can recover y is a quotient of x by r as v-sheaves where what is the equivalence relation of x times y x so I should say that
01:36:20
I refer to this condition by saying that y is small while the equivalence relation instead of x times x it's a sub-v-sheave it's always a diamond
01:36:48
so any small v-sheave can be written as a quotient of a perfectoid space by a diamond diamonds in turn can be written as a quotient of a perfectoid space by a perfectoid space and so on this kind of two-step procedure you can reduce
01:37:01
many statements for general v-sheaves to this case of perfectoid spaces which is quite useful sometimes. Yeah? So the word surjective, so here it was for epimorphism of sheaves I suppose. Yes. And in the original
01:37:20
definition you spoke of surjective maps of affinities it was on the level of topological spaces or? That's on the level of topological spaces, yes.
01:37:52
So this uses the following proposition that if y is a diamond
01:38:02
and y' and y is any sub-v-sheave then y' is again a diamond
01:38:24
which in turn one reduces to the following assertion that if x is a strictly totally disconnected space and x' and x is a sub-v-sheave
01:38:45
and the quasi-compact I should say there is some subtlety quasi-compacity that you have to take care of in the reduction then x' is actually automatically a strictly totally disconnected space.
01:39:14
That's the consequence, I mean I think what's happening here is that because being a v-sheave is so strong
01:39:21
it means that you automatically have some kind of geometric structure if you satisfy this v-descend. In particular what's happening in the proof of this proposition is that if somewhere you have any map from a finite perfectoid space into x that lies in the sub-sheave then
01:39:41
the map of perfectoid spaces has some image and the image by some general nonsense about the topology of spectral spaces will actually be represented by a strictly totally disconnected space. So there will be a strictly totally disconnected space over which this factors but
01:40:02
then this map must be a v-cover to the sub-space because it's surjective on topological spaces and then the v-sheave property tells you that actually the sub-space lies in the v-sheave itself and then Is the sub-space closed under generalization? Yes, all maps are generalizing
01:40:21
anyway, so yes they must be closed under generalization.
01:40:42
So this means that about general v-sheaves can be reduced
01:41:02
first to diamonds and then to perfectoid spaces.
01:41:20
So for example when it's a flowing lemma that if f from y' to y is a quasi-compact and quasi-separated map of v-sheaves so in any topology there is a notion of quasi-compact and quasi-separated maps and I just use this in the case of
01:41:40
the topos of v-sheaves then it's easy to check whether it's an isomorphism. So then f is an isomorphism if and only if for all algebraically closed non-arithmic field C
01:42:04
with an open and bounded variation ring sub-ring C plus in C it's a bijection on points.
01:42:51
So some kind of very general statement that there is no non-reduced structure of any kind anywhere here. It's all determined by points.
01:43:22
So let me try to finish this lecture by giving a criterion for a v-sheave to be a diamond.
01:43:42
It's some kind of analogue of Artin's theorem of giving criteria for algebraic spaces. That's enough to find a smooth cover or something like this. And this is actually used to prove that many of the examples that you care about
01:44:01
or at least in the original argument this was used to prove that many of the examples you care about are diamonds like the in this bungee setting and so on. Okay. So this is
01:44:20
so this uses the condition on the topological space which by the way is a condition that will also be important for ruling out the compact
01:44:41
host of spaces. So let me first define the
01:45:01
underlying topological space. Let's say you have a diamond then you define the underlying topological space
01:45:22
to be the quotient like so. And you check that it is independent of the choice.
01:45:46
And also by making the use of the theorem about small v-sheaves it also extends to small v-sheaves.
01:46:06
And for perfectoid spaces it just agrees with the usual underlying topological space. For example if T is compact host of the underlying topological
01:46:25
space of this funny sheave is T. for usual perfectoid spaces you get some
01:46:42
locally spectral spaces here you get some compact host of spaces and these are some kind of two extremes in the world of topological spaces. And the condition you're asking is that you are on the spectral side of things. So small v-sheave
01:47:07
y is spatial if essentially you would like
01:47:20
to say that the underlying topological space is quasi-compact and quasi-spectral. But you actually need to say a little bit more namely you need to say that the way the space is spectral somehow related to some properties of the sheave. And so the correct way to say this is that it has a basis
01:47:45
given by the topological spaces of v for via quasi-compact open subfunctors open subfunctors.
01:48:22
These give you quasi-compact open subsets but if you would just have a quasi-compact open subspace of the spaces it would not be clear that the associated sorry I should have said that y is QCGS and
01:48:44
so it's not clear that if the underlying topological space is quasi-compact then the sheave is quasi-compact.
01:49:06
The sheave is quasi-compact if it can be covered by well if any cover is a finite subcover so it's equivalent to asking that it's covered by finitely many affinoid perfectoids.
01:49:31
It's quasi-compact open so if you have a quasi-compact open subsheave it will automatically give rise to quasi-compact open subspace
01:49:41
of the topological space. So in general open subsheaves correspond projectively to open subspaces of the topological space but for quasi-compacities there's only an implication. So it is not clear that
01:50:01
every open subsheave is a union of quasi-compact? It's not clear that any I mean you might have the underlying topological space might be a compact host of space so then you don't have any quasi-compact open subspaces.
01:50:21
In the definition they require that the v absolute value is what? Well automatically if v is a quasi-compact open subsheave of y then the v absolute value is a quasi-compact open subspace of absolute value y.
01:50:40
You mean just the base of the topological space of y comes from something of the final object of y itself? Right, the quasi-compact. OK? And so then there is a proposition that if y is spatial
01:51:01
then the underlying topological space is spectral and maybe an important example is that if x is a QCQS analytic space
01:51:26
then the associated diamond is spatial. So the ones that come from usual things are spatial.
01:51:41
And right, the underlying topological space of x diamond is the underlying topological space of x. OK, so I'm already over time so let me just end by stating the theorem.
01:52:00
So if y is a spatial v-sheaf, so you've verified this topological condition and then you just verify a very minor statement about
01:52:22
points such that for all y is the underlying topological space of y there exists a quasi-croetal map
01:52:44
from some perfectoid space which you can assume to be some sparse CC plus to y with y in the image of logical spaces.
01:53:05
Then y is actually a space. You just need to check that the points
01:53:24
of the space are not too bad. But for example if you are in the situation of the F.N. Grassmannian there is already a stratification of the space for which you know that the strata are diamonds. And so any one of those points will lie in one of those strata and then you just
01:53:41
because it's a diamond there you know this condition. And so this means that you only have to verify the spatiality condition here. That's some points of topology which is fun or not, but anyway, let me stop here.
01:54:10
What is the R-theorem of which it's another one? It's roughly an analog of the theorem that if you want to prove that something is an algebraic space you don't actually have to find any tall atlas
01:54:21
but it's enough to find some smooth atlas or maybe flat atlas or something like that. Smooth atlas is lower and okay. And so here again for a small V-sheep you automatically have some V atlas. So the vague analogy is something that I consider the quasi-pro-toss side
01:54:41
as being vaguely analogous to etaussings in the scheme case and then any V-maps has been vaguely analogous to all flat maps in the in the scheme case.