6/6 Perfectoid Spaces and the Weight-Monodromy Conjecture
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00:00
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Transcript: English(auto-generated)
00:15
So this is now the final lecture of this lecture series, and I want to give some applications of the results.
00:27
So let me first summarize what you have proved so far. So again, let us fix a perfectoid field, k, as always. And so we had introduced its tilt.
00:44
And so let me recall again that an example would be something like, take a periodic field and then join all powers of the uniformizer at all square p power roots of the uniformizer, and then it's tilt. In this case, it's just some of the similar thing and equal characteristic.
01:06
And then we had introduced this notion of perfectoid k-algebras. And we had the theorem that the categories of perfectoid k-algebras was equivalent to the category
01:21
of perfectoid k-flat algebras, which was called the tilting equivalence, which sends any perfectoid k-algebra r via this Fontaine function to r-flat, which is the inverse limit over some of the piece power map of r, where this is always
01:42
in multiplicative. So the transition maps are multiplicative, but not additive, and so one has to do a little bit to define the addition here, and note in particular that there is a map from r-flat to r, by projection to the last coordinate, which again is multiplicative, but not additive, which I denoted by f maps to f sharp.
02:02
And so the deepest theorem that we have proved so far was that if r is a perfectoid k-algebra, and s over r is finite eta, then s is also perfectoid.
02:24
And so we can apply the tilting function to s, and this induces an equivalence between the finite eta of r algebras and the finite eta of that algebra, sending any s to s-flat.
02:45
And in fact, this implies a little bit more, so we know a version of the Fontaine's almost purity theorem on this side, and so we deduced it on the other side here. So in particular, we see that in this tilting equivalence, some of the finite eta covers behave in a nice way,
03:01
but we want to generalize this somewhat to an isomorphism of the whole eta of sites. And so what we did last time was that we introduced the category of perfectoid spaces, so we have the category of perfectoid spaces over k. So locally, these are of the form
03:24
the eddic spectrum associated to some perfectoid k-algebra and some ring of integers inside, so where r is perfectoid. And so we had introduced, so we have
03:46
the also somehow tilting extents to spaces, giving an equivalence of categories, x maps to x-flat. And what we did last time was that we
04:07
defined a notion of eta-amorphism of perfectoid spaces, which was not so obvious, because somehow perfectoid rings are always reduced, so we cannot just say that somehow eta-morphism is some
04:23
morphism which satisfies the infinitesimal lifting criterion. But there is some way around this, and so what we get from last time is the corollary that there exists a eta site, x eta,
04:42
and tilting induces equivalence of sites between the eta site and of x, and the eta site of its tilt.
05:00
And so somehow if x was just a point, this would just mean somehow that the absolute Gara group is isomorphic, and now we have extended this isomorphism to spaces. OK. But somehow we have to relate these eta-topoi of perfectoid spaces now to eta-topoi of more classical spaces
05:22
in order to make any use of this result. And so we have somehow to compare them with the eta-topoi of classical rigid analytic varieties. And so we have also somehow the proposition that if y is a locally noetherian eddic space over k,
05:47
so this means these are the eddic spaces that Huber usually works with, so there is some noetherianist condition imposed on the ring here. And x is a perfectoid space, and we
06:03
have a morphism from f from x to y. Then this induces a morphism of sites x eta to y eta,
06:24
which basically means that we can pull back an eta-map to y to an eta-map to x. And we basically have proved this last time. OK, so somehow these eta-topoi of perfectoid spaces
06:42
somehow have the expected property that somehow they are functorial even with respect to somehow objects in this larger category. And now we need the following result.
07:06
So let x be a perfectoid space, and xi ini somehow be a filtered inverse system
07:24
of noetherian eddic spaces. So noetherian is locally noetherian plus quasi-compacting quasi-separated eddic spaces
07:43
over k. Then we write that x is similar to the inverse limit of the xi. So no, first we have to give some maps.
08:04
Let x maps to xi be a map to the inverse limit, to the inverse system.
08:21
Then we write that x is similar to the direct inverse limit of the xi. If, first of all, on topological spaces it induces an isomorphism.
08:45
So homomorphism is a topological condition. And secondly, we need a condition somewhere on the rings. And it's enough to impose a rather weak condition, which is that for all x and x with images xi and xi, the map
09:10
from the direct limit of the residue fields, these map to the residue field of x, this has dense image.
09:30
So there's a similar definition where one somehow also requires the inverse limit to be still eddic, to be still
09:42
locally noetherian. And in this case, Hooper proves that the etaal topos of x can be written as the inverse limit of the etaal topoi of the xi. And some of the same theorems stays true here. So we have the following theorem that x is the inverse
10:09
limit of the xi. Then the etaal topos is the inverse limit of
10:25
the fiber topos. Taking the inverse limit of topos is a rather technical thing, and so makes use of this notion of a fiber topos, but don't worry about this too much.
10:40
I should say that this is the associated topos. The limit of the fiber topos somehow fibered over this
11:00
index category i. Yes, yes.
11:24
OK, and in fact, you need also a preposition about this notion of being this inverse limit, namely if you have such a situation, and you have to some xi some
11:43
etaal morphism of noetherian eddic spaces, then, as I said, you can take the fiber product, which again is a
12:03
perfectoid space. And this then is also just the inverse limit over the j which are bigger than i of y times xi. This will be important in a second.
12:30
And some of the corollary also is the theorem. Somehow what does it mean that something is an inverse limit of this fiber topos? If f i over xi eta is a sheaf with pullback fj to xj eta
12:58
and f to x eta, then you can just compute the cohomology
13:07
of f by taking the direct limit over the j which are at least i of the cohomologies on the xj's.
13:35
And another corollary of the theorem, someone
13:43
characteristic p, there are some cases. I mean, I recall that there somehow you can take any noetherian eddic space and pass to a perfectoid space just by taking the completed perfection of all the rings. And somehow this doesn't change the Thereski topology and the analytic topology and the eta topology because
14:00
Frobenius is clearly inseparable. And this is somehow incorporated in this theorem. So assume that x can be written as such an inverse limit. And all transition maps xi to xj induce isomorphism,
14:21
homeomorphisms xi, homeomorphic to xj, and purely inseparable extensions on completed residue fields.
14:48
Then the result of Hooper says that all of these topoi are equivalent. And hence, some are also the inverse limit is equivalent to all of these.
15:26
In my case, I think the reference I want to use is this condition, which is imposed. But I check. Sorry, the site, yes.
15:46
The topo, sorry. But it should be true for the site. And so we want to use all of this now in an example to
16:02
see what all of this really means, completely. And so we do it in the example of toric varieties. So let me shortly recall the definitions. Let K be some field.
16:23
And the toric variety over K is a normal separated scheme of finite type over K with an action of a split
16:52
torus g on x and a dense open subset such that there
17:05
exists somehow a dense torus orbit on this variety. So a dense open u in x and a point such that if you take
17:30
the torus orbit of x, then this is isomorphic to u.
17:48
Yes, sorry. So I should maybe say t isomorphic to u via acting on x. Yes, sorry. Yeah, I mean, it's probably better to take the
18:02
point as a part of the datum, but I don't want to.
18:36
And so the nice thing about toric varieties, there's, of course, an example, is that you can just take something
18:43
like a projective space. Or also, you can take a product of projective spaces and many more things. And the nice thing about them is that they can be classified completely combinatorially. And most things about them translate into combinatorics.
19:02
And so let me recall this classification of toric varieties and for this, let N be a free being group of finite rank, a strongly convex polyhedron cone in N is
19:43
a subset of the form. Or let's say an N tensor R is a subset of the form sigma, which is some of the cone generated by finite number of
20:02
elements, where the xi are all elements in this lattice N, and such that sigma contains no line, no line
20:23
through origin. And the second part is a fan in N tensor R is a collection of such cones.
20:50
It's a collection in big sigma of such cones. Sigma is such that it's stable under faces.
21:05
And if sigma and tau are in sigma, then also the intersection is a face of both of them.
21:26
And so for example, you can take N and z squared, and you can take a picture, and you can take the cones that come
21:41
from this subdivision. So you have three two-dimensional cones, three one-dimensional cones, and one zero-dimensional cone. And we can associate to sigma a toric variety x sigma.
22:08
And this goes as follows. So we glue it.
22:21
Yes, yes, yes, yes, yes. I guess I wanted to contain one maximal-dimensional cone to make it canonical. So some are not contained in a contains cone of dimension.
23:04
We can associate to sigma a toric variety x sigma, and this goes as follows. So do I want this, in fact?
23:22
Well, let's see what happens. So first for each sigma in each cone, we have its dual, which lives in M tensor R, where M is the dual of N.
23:42
And we set u sigma to be the spectrum. So I guess, yes, I just want sigma non-empty. The spectrum of the thing, so that's
24:02
some scheme of finite type.
24:30
And if tau is the phase of sigma, one gets an induced
24:41
open immersion from u sigma, from u tor into u sigma. And so somehow to get this functoriality, we really have to do this dualization procedure. And so somehow this allows us to glue everything together
25:06
and get this variety x sigma. And note that if you take the cone which is just 0,
25:26
then the sigma dual will be everything in tens. This is really just the spectrum of k of M. So this is a free abelian group. So this is really just a torus, so it's a split torus.
25:42
So let's call this t. And note that t naturally acts on x sigma. And there is a natural base point inside here. So 1 and t gives a point x in here.
26:04
And then this really becomes the open dense orbit that we wanted to have. And then the theorem is that any toric variety is of the
26:21
form x sigma. And if we fix this base point x and x, then sigma is a
26:45
unique fan in the co-characters of t tensor r.
27:02
OK, so this part is probably rather well known. I should say one more thing, somehow that if you have an element in U, then this gives rise to a function now on this open subset t and tends to a rational function on a variety U. And we let chi to the U be the
27:23
rational function, the associated rational function on x sigma. And but we will need a few statements about divisors on
27:43
toric varieties. And the nice thing about them that they as well are classified in a combinatorial manner. And so that's the following definition or proposition. So there are some divisors which you can describe
28:02
combinatorially. So let chi and sigma be the one-dimensional counts. Then we want to first associate to each such one-dimensional count some veiled divisor on the toric
28:24
varieties, namely the open subset corresponding to tori. After a change of coordinates, maybe it's isomorphic to the affine line times gm to something.
28:41
Or d minus 1 maybe is a better notation. And let the i in x sigma be the closure of 0 times gm.
29:03
Some of the open subsets were zeros, where the first coordinate is non-zeros. Exactly this open dense torus orbit. And then now we have something of code dimension one, a divisor.
29:30
Yes? Yes. Sorry.
29:41
Sigma is a unique fan, x tends to r, if x get a unique asymptotism. That's right. OK.
30:01
So then we define that a g veiled divisor is an element of direct sum g times di. And then it's true that any veiled divisor is
30:23
equivalent to a t veiled divisor. And if you fix the generator of this line tori, then when
31:01
you have the following expression for the global sections of and d is the sum of ai di, then the h naught of x sigma with values in o of d is the direct sum over all
31:28
u and m, such that the scalar product of u is ui is at least minus ai of k times the
31:40
expression of function k to the u. And so this is somehow what we need about toric varieties and now we want to define some of the eddic spaces and perfectoid spaces versions of these varieties.
32:02
And so let us fix a fan and assume now that k is a first and non-archimedian field. Then we define an eddic space, which I write curly
32:31
x sigma ed because it's not, in general, some of the
32:41
analogification of this scheme x sigma, which is glued out of u sigma, which is this thing.
33:08
So somehow if the toric variety was just affine space, then this would not be all of affine space but just the closed unit ball.
33:29
If x sigma is a n, then this curly x is maybe what you call bn, so it's the closed unit ball.
33:46
Meaning that if you, for example, consider its k valued points, then it's the set of triples x1 up to, as a set of triples x1 up to xn in k. So it sets the absolute value of all xi's at most 1.
34:08
But if x sigma is proper, which, by the way, is equivalent to the condition, it's pretty combinatorially again, that sigma covers n tensor r, then this curly x edd
34:32
is really just the space associated to x sigma.
34:53
And if moreover, k is a perfectoid field, we define
35:06
curly x perf as being glued out of u sigma perf, which is the space where now we somehow join all of the
35:23
p-power roots of the coordinates that we have. So we allow denominators in p.
35:43
And then we have the following comparison results.
36:03
By the way, I should have said in this example that I gave of this specific fan, the associated toric varieties would just be two-dimensional projective space.
36:24
So there's the following theorem. So we take a perfectoid field with tilt k flat.
36:43
And then we have some of this perfectoid version of this toric variety over k. And this tilts to the same thing over k flat.
37:08
Secondly, we have that this perfectoid version of this space is basically the inverse limit of the usual
37:25
versions, whereas this transition map phi isn't used by multiplication on p on m.
37:50
Then we get that, in fact, if one takes the attic thing over k flat and it's somehow underlying topological
38:02
space, then it's just the inverse limit over phi of the topological spaces associated to the thing over k. So in this sense, somehow the thing in characteristics, the toric variety in characteristic p can be seen as a pro cover of the same toric variety in
38:20
characteristic 0. And in fact, this is not only true on topological spaces, but it's also true on etal topoi. Take the etal topos, associate it. Then it's the inverse limit of this phi by topos with
38:48
the integers of this relation that this is a pro cover of this is true on topological spaces and etal topoi.
39:00
And we will need the following one more statement. So for any open u downstairs somehow, so over equal characteristic, mixed characteristic, somehow with
39:22
pre-image v on the other side, we have a commutative diagram of topoi, somehow that you have this
39:45
projection map somehow from here to inside.
40:01
You have the open subtopos u eta. And then there's a map here because, again, you can write v eta somehow as an inverse limit of the topoi in here.
40:43
So in particular somehow, if you take just the pn over this equal characteristic field, then in some sense it's equal to the inverse limit over phi of the
41:02
projective space over k, whereas this map on coordinates is just given by these powers. And so this induces a projection map somehow pi from
41:21
the pn over k flat to pn over k, which somehow does not really exist, but somehow at least morally exists. And for example, it exists on topological spaces in et al topoi. And some of this is on coordinates given by sending such a tuple to the sharp representatives.
41:51
Somehow that's the relation between the projective spaces over the two fields. Unfortunately, this is not completely functorial. So this identification somehow depends on the choice of
42:02
coordinates on pn. And so somehow in characteristic p, there's this canonical way of passing from this variety to a perfectoid space. But in order to pass in characteristic 0 from a variety to a perfectoid space, you have to make some choice.
42:26
So this basically follows directly from what we have proved. And we also need the following proposition that we can
42:41
compare the cohomologies of the toric varieties now. So assume that k is a separable average variably closed. Then the cohomology of the toric variety, and L is
43:05
invertible on p in k. No, no, no, no, in k flat, I mean.
43:20
So if you take the cohomology of this sheaf z mod L to the m, then this somehow via this projection map, pi, goes isomorphically to the cohomology.
43:52
And for the proof, just note that because of this
44:04
equivalence of 2 pi, that the one is the inverse limit of the others, you only have to see that the transition maps in this tower induce isomorphisms, which is now something purely in characteristic 0. Someone checks this by hand.
44:56
And for this application, the weight monotony conjecture, we will need an approximation lemma.
45:01
And this is the following. So fine proposition. Assume that the toric variety is proper.
45:23
And let y in x sigma over k be a hypersurface. And let us choose a small neighborhood.
45:41
So in some of the associated rigid analytic varieties, or analytic spaces, a small open neighborhood.
46:04
Then there exists the hypersurface z in some of the space on the other side, such that it's contained in the
46:26
inverse image of the small neighborhood of y. So some maybe short explanation for what this is supposed to mean.
46:48
It's a hypersurface, so it's of conimension one.
47:32
Let me give an example and continue, and then we will see what happens. So we can consider inside of P2 over k, we can consider
47:46
something like the set of all, this is a hyperplane where some of the coordinates are 0. And then we somehow have the P2 over k flat, which somehow is the inverse limit of the tower, where now
48:01
we here have these transition maps. And so we have pi inverse of y inside here. And so what is pi inverse of y? So we can first compute somehow it's the inverse limit of the pre-image into all of these P2 over k's. And so inside here we have pi inverse of y, which is now
48:25
given by the much more complicated equation where we raise to the P's powers here, and so on. At each stage, this equation gets more complicated. And so pi inverse of y is somehow the inverse limit of these.
48:44
The underlying topological space is the inverse limit of the underlying topological spaces. And so somehow, as a subset of this P2 over k flat, it has somehow a curve of infinite degrees, so some kind
49:03
of fractal, this pi inverse of y, and this maps to y, which is something very nice. And some of the lemma says that if it does something very bad, so this inverse image. But the lemma says that if you take somehow a small neighborhood of this, then we can't find a curve inside.
49:23
So we can approximate this by curves.
49:49
OK, and so let's give the proof.
50:04
So the divisor y is equivalent to the T-vale divisor d, which is the sum of ai di.
50:35
And we take a section on x sigma over k of O of d,
50:46
defining y. And so we want to approximate some of this section by section on the other side.
51:01
And for this, we use this approximation lemma that I had proved some or two lectures ago. And so in a slightly variant of it, so we consider the graded ring, which is the direct sum over all j, which
51:26
may have peace in the denominator of the H1, k, with values in O of j times d.
51:40
So somehow, on this effectoid signal, it makes sense to somehow take O of j times d, where j has peace in the denominator.
52:19
And so it can also be written as the direct sum over the
52:26
j's times the completed direct sum as a banner space of the u in M p inverse, such that someone's notation that I had used of k times chi 2, let R be its
52:53
completion. So somehow, for the obvious k0 sub-module, such that this
53:04
is open and bounded. And then if we form its tilde, then its tilde is somehow given by the similar construction over the tilde k
53:23
of that. So for this, we use somehow that this combinatorial data on d here, this is purely combinatorics somehow, and hence transfers to the tilde.
53:44
And so we may assume that somehow our neighborhood y tilde is given by the set of all x and x sigma k at
54:01
such that the absolute value of f is at most epsilon. So because somehow everything is defined by the toric construction, everything has an integral model intense, it makes sense to state this inequality here.
54:24
And so now we use this approximation lemma. So we approximate it by g and R flat, such that the
54:48
set of all points in x perf sigma k flat, such that the
55:01
absolute value of g of x at most epsilon is exactly somehow under tilting the set of all x here, such that
55:21
the absolute value of f of x is at most epsilon. It's curly x, yes. Mainly it's the same because it's toric somehow. It's proper.
55:41
I assumed in the proposition that my varieties, yes.
56:19
No, I mean, I still consider it, yes.
56:37
Yes.
56:47
Isn't it somehow true that some of this x, this toric variety has an integral model over k naught, and also some of this d has, O of d has an integral model. Yes. And so somehow you trivialize, oh, it's not a line bundle.
57:03
You keep it in there, but I need to trivialize this local integral model and define it.
57:35
OK. I guess I can make this assumption without destroying
57:42
any everything. I can just write smooth everywhere. And then it should be OK. OK, so then we have this.
58:02
And then take that to be the set where, so now, so g is also homogeneous of, so g lies in H naught of x perp
58:26
sigma k flat O of d. So because we have this homogeneity constraint, sum on this approximation lemma, so a priori is only the
58:41
completed direct sum of k flat times chi to the u over a certain u. But we can approximate g by element of
59:05
uncompleted direct sum.
59:39
And then if you take a high enough p power of g, then
59:52
this lies in fact n. Somehow we'll have no more p power roots of the coordinates.
01:00:00
in its expression. So it lies in sigma k flat at with various O of p to the n times d. And then we take that to be the local square g to the p to the n is equal to 0.
01:00:34
OK? And maybe let's have the break now for 15 minutes.
01:00:43
It's not so easy to write down. So the approximation algorithm is, I mean, there is sort of an explicit algorithm computing some of this approximation. But it's rather complicated to carry it out. OK, so by the way, it's, of course, enough to assume that the ambient variety is smooth.
01:01:01
And it doesn't matter whether the hypersurface is smooth. And so we get the following corollary that if, so assume again, x sigma is proper and smooth.
01:01:21
And that y and x sigma over k be a set-theoretic complete intersection. Well, set-theoretic is enough somehow.
01:01:42
And again, we take a small neighborhood.
01:02:08
Yes, that's all I want. Then there exists complete intersection.
01:02:27
Then there exists set on the other side. So you can assume it's irreducible and has the same
01:02:40
dimension as y, such that it's contained in the preimage of y tilde. And that's just by approximating each of the hypersurfaces. And it might even be that after this, the thing has two
01:03:02
large dimensions, and then we just cut down a little bit more and to take an irreducible component at the end. In the end, I will assume it's projective.
01:03:21
And then somehow because the intersection and the characteristic 0 is non-empty, you can cut by an ample line bundle a little bit more and see that the intersection of these dividers will be non-empty. And then somehow the same intersection will have to be
01:03:40
non-empty in characteristic p because everything is combinatorial. And hence, this intersection cannot be empty.
01:04:20
OK.
01:04:23
So generalizing from projective space to toric varieties is a little more subtle than I thought. OK. So finally, we will talk about the weight monodromy conjecture.
01:04:49
So let us first recall some things about analytic representation. So let K be a local field. And so GK is its absolute Galois group.
01:05:07
And we have the inertia subgroup. And let me also fix the geometric Frobenius.
01:05:21
And q is the residue field, which is finite. And so residue characteristic is p.
01:05:40
And I take a prime L, which is not p.
01:06:07
So recall that as a pro-L quotient, the inertia IK is
01:06:24
given by the homomorphism TL from IK to CL twisted by 1. So this is the inverse limit to the n-th roots of unity. So this TL is some of the inverse limit of certain
01:06:43
homomorphisms TLN. And how are these defined? Let me recall it. So if pi and k is a uniformizer, and you choose pi and L to the n-th root of pi, then if you apply
01:07:06
sigma to this thing, then it will be some root of unity depending on sigma time, which is just the TLN homomorphism times pi to the 1 over L to the n.
01:07:23
And now we have Grothendieck's quasi-unipotent theorem, which is the following proposition.
01:07:42
Let me write it on the right. Let V be a QL bar representation of GK.
01:08:07
So we have a homomorphism from rho from GK to GLV. Then there is an open subgroup, I1, subset of IK, the inertia,
01:08:28
such that for all sigma in I1, rho of sigma is unipotent.
01:08:40
Then there exists a unique neopotent operator N from V to V minus 1, say, such that for all sigma in this open
01:09:05
subgroup, rho of sigma is given by the exponential of N times TL of sigma.
01:09:23
So somehow, locally, this inertia will act through its pro-L quotient and do this via unipotent operators, which somehow give rise to a neopotent operator in turn by taking the logarithm.
01:09:42
And say, in order to ignore some twists, I will fix an isomorphism, QL1 with 1, QL. And then N is really a neopotent operator of V
01:10:06
itself, and it commutes with the geometric Frobenius up to the factor Q. So this follows from
01:10:21
uniqueness of this operator N, looking at how this commutes with the action of the Frobenius operator. And now we have the so-called monodromy filtration, which is the following. So it's in very great generality so that V be some
01:10:41
finite dimensional vector space. And N, a neopotent operator. Then there exists a unique, separated, and exhaustive
01:11:04
decreasing filtration, fill INV subset V, such that the following two properties are satisfied.
01:11:59
First of all, the operator maps the filtration step, the
01:12:06
i minus 2's, and decreasing. So this is larger than, no, it's too difficult for me.
01:12:41
And N to the i is an isomorphism from the grew iN of V to grew minus N.
01:13:08
And so, in fact, one can write down this filtration very explicitly. So it is a direct sum over i1 minus i2 equal to i of the
01:13:23
kernel of N to the i1 plus 1, intersected with the image of N to the i2.
01:13:43
And so the conjecture is now the following. So we draw the conjecture. So let's do it to the lean.
01:14:02
That if x is a proper smooth variety over k, and V is the talk homology of x over k bar with coefficients in k
01:14:25
over bar, then for all integers j, the jth graded
01:14:44
piece of V is pure of weight i plus j as a representation of the absolute group of this finite field, the
01:15:05
residue field. Meaning that all eigenvalues of Frobenius on the thing are real numbers of the correct weight.
01:15:36
And so I want to explain one possible way of looking at
01:15:40
this weight monodromy conjecture in terms of zeta functions.
01:16:01
Because it's maybe not so clear what some of the intuition behind this is in terms of r functions. So let x over q be some variety, or over any number field or global field, be some proper smooth.
01:16:26
And let us consider V, which is the i's et al cohomology group with coefficients of q over bar. And then we have the zeta function, or the l function,
01:16:43
associated to this cohomology group. And it's defined as a product of all places of local factors. So V runs through places of f.
01:17:01
And so let me recall the definition of V as a finite place of a residue characteristic not equal to l with local field k.
01:17:23
Then some of this local factor is just defined as the determinant of 1 minus q to the minus s times the luminous on the inertia invariance of this representation.
01:17:42
And if x has good reduction at P, at V, then we can apply the weight conjectures. And this says that all poles of this local factor have
01:18:06
real part i over 2. And in particular, except for finitely many factors, the
01:18:27
whole l function is absolutely convergent when the real part of s is greater than i over 2 plus 1. And some of the weight monodromy conjecture says that
01:18:41
all the other factors don't destroy this behavior. So weight monodromy implies that all other factors, q to
01:19:03
the minus s, yes? Ah, sorry, OK, sorry. All other factors contribute no poles of real part.
01:19:31
Well, not even bigger than i over 2. But in particular, not bigger than i over 2 plus 1. And in a sense, if one quantifies somehow over all
01:19:42
possible proper smooth x, then somehow the weight monodromy conjecture is somehow equivalent to this behavior that these other local factors do not contribute other poles. And in equal characteristic, Deling could turn this argument somehow into a proof of the weight monodromy conjecture.
01:20:07
Well, for the Riemann-Zeta function, I guess you have absolute convergence for real part greater than 1.
01:20:22
So there's a theorem of Deling that if k of equal characteristic and you have x over k, which is proper and smooth and defined over a global field, so function
01:20:46
field, then the conjecture is true.
01:21:05
And some of the argument that Deling gives in rail 2 is by somehow establishing this expected property of all local factors somehow by using what we know about L functions over function fields.
01:21:21
And in subsequent work by Ito and Terasoma independently, they showed that one can get rid of this assumption that it's defined over a function field by some approximation argument. But in mixed characteristics, the conjecture is basically
01:21:40
wide open. So it's known in dimension at most, too. And in context, this context, basically, which are related to some more varieties where one can apply some automorphic arguments. So I want to prove now the following theorem that I take
01:22:07
a local field of characteristics 0 and let y be a connected proper smooth variety over k such that y
01:22:30
is a complete set-theoretic complete intersection in a
01:22:40
projective smooth stoic variety x sigma over k.
01:23:01
Then the weight monodromy conjecture is true for x for y.
01:23:36
So for the proof, we will somehow use this
01:23:43
tilting machinery. And so we first have to produce a perfectoid field. And for this, we just join all of the p power roots of the uniformizer and complete. And let k flat be the tilt.
01:24:02
So k flat is the completed perfection fq and power series
01:24:21
over pi flat. And so this, let me call this k prime. And then we have that the absolute gamma group of k prime bar over k prime, or SEP, isomorphic to the
01:24:45
absolute gamma group of k bar over k, which is a subgroup of the absolute gamma group of k. And the weights and monodromy of gk representations can be
01:25:09
seen by restricting to gk prime. Because this extension is purely totally ramified.
01:25:30
And hence somehow the geometric Frobenius will survive. And it's pro-p, and hence the L-adic inertia will survive this tower.
01:25:42
And so we have the following picture. So somehow we have y inside of the stoic variety over k. So take its base change to k. And basically, what we have to do is we would like to take this projection at pi. So we have to first go to the x basis.
01:26:06
And take the pre-image of y. But again, we have seen that this is not something good. But so let us take first some small neighborhood here. Then take the pre-image here.
01:26:22
And then we will have something here. So what do we have there?
01:26:46
So it's a theorem of Uber that there exists an open neighborhood y tilde such that the cohomology with
01:27:12
coefficients in them, z mod L to the m, is the same as the cohomology of this.
01:27:22
Maybe you have to go to the completion. So first, there's this implicit theorem that the cohomology doesn't change when you go from the algebraic side somewhere to the rigid side. And then somehow you can take a small neighborhood.
01:27:40
And the cohomology still doesn't change. So imagine that there's just some small tube somewhere around y. And so you have this approximation result which
01:28:02
tells us that there exists z inside of this pre-image which is irreducible and such that the dimension is the dimension of y. And we let resolve singularities by an alteration.
01:28:28
So that prime should be projective and smooth.
01:29:09
And so what we get is a Gk, isomorphic to Gk flat
01:29:24
equivariant map from the cohomology of y with coefficients in z mod L to the m to the
01:29:41
cohomology of z prime, which, again, with coefficients in z mod L to the m. And I should say that we can assume that, by looking
01:30:00
back, the proof of this approximation lemma is defined over k prime. Or we could even assume, if you want to lean through that directly, that it's defined over a function field. Because we have this freedom to change
01:30:20
the equation slightly. OK. Yes? I don't know.
01:30:43
I haven't read the proof. And so it's in this paper on finiteness for direct image chiefs on the tau side of rigid analytic varieties.
01:31:01
And by taking the inverse limit over all m and tendering with QL bar, we also get a map with coefficients in QL bar. And the lemma is now that for in the top degree, this is
01:31:27
an isomorphism. So let's check this.
01:31:41
So we have the cohomology of y, which has a map from the cohomology of x sigma. So let me just write shorthand here. And this, on the other hand, is again the same as the cohomology of x sigma by this proposition that I
01:32:00
stated, that under tilting, the cohomology gets identified. And this maps to the cohomology of z prime. And we have this map. And so these are both one-dimensional QL bar vector spaces. So if the map is not an isomorphism, it's a zero map.
01:32:26
So we have to exclude the case that it's a zero map. So we would have a zero here. And by this diagram, we would see that there's also a zero here. So we have to exclude the case that the restriction map from the cohomology of historic variety to the
01:32:43
cohomology of z prime is not a zero map. And just note that if you take the first term class of an ample line bundle to the dimension of y's power, so
01:33:05
x sigma k flat has non-zero image.
01:33:28
So somehow what you should think is that somehow morally there's a retraction somehow back to y. And so this z prime can be seen somehow as an alteration of y along this map somehow.
01:33:42
And what this lemma says is that somehow the degree of this alteration is non-zero. So somehow it does not go to somehow a closed sub-scheme or something like this. So somehow we have covered our mixed characteristic
01:34:01
variety by somehow a variety in equal characteristic now. And now the Poincare radiality pairing shows that this becomes a direct summand of this. So h1 of y is a direct summand of the
01:34:29
cohomology of z prime. And here we have Deline's theorem.
01:34:45
And it's clear that if you have a direct summand of something which satisfies the weight monotony conjectures, then so does this direct summand, and hence we are done. OK, and that's the end of this lecture series.