Fibrant Resolutions of Motivic Thom Spectra
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01:01:43
Numerical digitMeeting/Interview
Transcript: English(auto-generated)
00:17
Okay, thank you very much. So I'm gonna talk about the joint work
00:23
with Grigorik Garkusha. And the main idea of this work is to construct an explicit fiber replacement of certain spectra in the stable material model category. So in this talk, we're gonna work over the
00:41
base field key, which is infinite and perfect field of characteristic different from two. And we'll be interested in the stable
01:01
material model category. Right, we're gonna need two descriptions of this category. Either we can think about this as a category of P1 spectra or the category of GM S1 by spectra.
01:21
And the main target is to construct explicit fiber replacements of certain spectra
01:49
in this stable particular model category. So the main example, the main motivating example is the algebraic keyboardism spectrum, MGL.
02:17
That's the spectrum that consists of the spaces, which are the term spaces of the tautological
02:25
vector bundles of rank I. So tau I here is the vector bundle over the Grassmannian I, where Grassmannian I is the co-limit of Grassmannian of subspaces
02:45
of dimension I in the space of dimension N and N goes to infinity. So then there is a tautological vector bundle over this Grassmannian and then the term space of this tautological vector bundle is the, is the space in the algebraic keyboardism spectrum, MGL.
03:06
And let me first formulate our main result for the spectrum MGL and then I'm gonna talk a bit about similar spectra that look similar to MGL that also consists of the term spaces of certain vector bundles. That's what we are calling the material spectrum.
03:23
So let me start with the construction, with explicit construction of the fiber replacement of the spectra in the stable material category. So for this construction, I need several steps.
03:42
The first step, I need to introduce the notion of omega correspondences. So if U and X are smooth varieties over K, I'm gonna introduce omega correspondences from U to X as a groupoid of the certain objects.
04:08
So the objects look like this, these are the heads, U, Z, X, where the map P from Z to U is finite,
04:23
A-flat, and locally complete intersection morphism. And Z here is some scheme over K.
04:41
It not need to be reduced, it's just a certain scheme over K that's with a locally complete intersection map to U, which is finite and flat. So these are the objects and isomorphisms in this groupoid are just the usual isomorphisms
05:00
of these heads. If we have two such heads, Z and Z prime, an isomorphism between them is just a map that commutes with both projections in both maps from the top of the heads to the variety X.
05:20
So that's the groupoid, and then for every smooth variety U, we can associate the nerve of the groupoid core omega from U to X, and this is gonna be appreciative of simplicial sets.
05:47
So that's the first object I need to introduce. Then we can make, we can construct a five inch replacement from this object just in two more steps. So the second step I need to do,
06:00
I need to apply the simplicial, the singular simplex construction by C star. So C star n core omega of U X. That's a simplicial object that looks like this. So it's zero simplices are just curve and core omega U X.
06:23
It's one simplex, one simplices is n core omega of U cross delta one over K into X. Two simplices are n core omega of U cross delta one plus delta two K X and so on.
06:50
So then that's again, it's gonna be a simplicial presheaf and delta one and delta two here are just algebraic simplices. So delta one K is just a spectrum.
07:02
K X zero X one where zero plus X one is equal to one. Delta two K is the algebraic two dimensional simplex
07:20
and so on. So we apply this simplicial construction to this nerve of the group void of the omega correspondences. Excuse me, Sasha, there's a question for you. What does the omega stand for? Omega. Omega here is just a reference to the keyboardism.
07:42
It's really, I mean, it's just for some historical reasons, there's no real reason to call it core omega. That's just how we call it. Okay. Okay, thank you. So yeah, so the next step,
08:01
we need to make it an S one spectrum. So we're gonna construct a fabric replacement in the category of GM S one by spectrum rises naturally as the GM S one by spectrum rather than P one spectrum. So here we constructed just the spaces that simplicial presheafs.
08:21
So now we need to make an S one GM by spectrum. So first of all, we construct the S one spectrum and it's constructed also in a standard way. So to every U, we just associate the spectrum C star and core omega U X mesh S.
08:44
And that's a spectrum that consists of the following things. C star and core omega U X mesh S one C star and core omega U X mesh S two and so on.
09:05
Instead of X, we can put a simplicial scheme as X plus meshes one X meshes two and so on. And then we get this construction that makes it C star and core omega X of U into simplicial scheme
09:20
is also simplicial presheaf. So then the sequence of simplicial presheafs naturally becomes an S one spectrum. And then the last step, we need to make it into S one GM by spectrum.
09:43
So it's a GM spectrum of S one spectra. So it consists of the following things. So the first element is the S spectrum C star and core omega U X mesh S.
10:00
The next, the C star and core omega U X smash GM's mesh one, smash S and so on. The next is gonna be C star and core omega U X
10:22
smash GM, smash two, smash S and so on. Where GM's mesh one here is a simplicial scheme. Also, that's just a cone, simplicial cone of the inclusion of the point into the pointed variety G.
10:41
Sasha, a question for you again? Would you mind describing the structure maps in step three? Oh, the structure maps in step three. Okay, so we have C star and core omega X, sorry,
11:05
U X smash S I, smash S one, goes to C star and core omega from U to X
11:21
plus smash S I plus one. So basically this map is constructed in as the following sequence. So first of all, we have a map into C star
11:41
and core omega U X plus smash S I, smash S one and then it goes over here. Where, yeah, the idea here is that the C star and core omega X, it's covariant with respect to the second argument.
12:01
So when I have any simplicial scheme and I put it on the second argument, I automatically get a simplicial ratio here. Okay, great, thanks. Okay, right, so here I did the first step. So here I got the S one GM by spectrum
12:22
and this is everything we need to know. I can formulate the following theorem. Yes, sir, I didn't give a name to this. Let me call this by spectrum C star N or omega U X smash G smash S.
12:43
So then our main theorem tells that this S one GM by spectrum C star N core omega X smash G smash S is locally equivalent
13:05
to a motivic fibrin replacement of the spectrum X smash MGL.
13:25
So it's not a fibrin replacement itself, but it's locally equivalent. In particular, it has the same homotopy groups as the fibrin. So it has the same local homotopy groups as the onus motivic fibrin replacement
13:41
of the spectrum X smash MGL. Or more precise, to make it a fibrin replacement, what I need to do, I just need to,
14:00
in every step of my GM spectrum, of my S one GM by spectrum,
14:20
I just need to apply the fibrin replacement in the local model structure without any A one coolances.
14:45
So if I do this on every step,
15:03
then this S one GM by spectrum will be motivic fibrin and will be weakly coolant to the spectrum X smash MGL. Right, so in particular, if we wanna compute, say, sheafs of homotopy groups of the spectrum MGL,
15:22
the sheafs of homotopy groups will be the same as, so A one homotopy groups of the spectrum X smash MGL, computed on local Hinsl and U,
15:41
will be just homotopy groups of this S one spectrum MGL. Spectrum C star and, or omega of U X smash S.
16:17
Okay, so that's the main application.
16:21
And similarly, we can do a similar result for other spectra that are similar to MGL that, so other examples where we can state similar results are MSL and MSP. So these are the spectra for special linear
16:42
cabortives that are subletic cabortisms. Let me talk a bit about how this result is related with the theory of frame correspondence. Basically, this result is just an application of the techniques that are available to us
17:00
thanks to the theory of frame correspondence. So the frame correspondences were introduced
17:24
by Vojvotsky and developed by Cush and Pani.
17:45
So here, let me look at a bit more general situation. Suppose E is a Tom spectrum, and by Tom spectrum, I mean the spectrum
18:00
that consists of Tom spaces. So E has spaces E zero, E one, and so on, where E n is a Tom space of some vector bundle V n, where V n is a vector bundle of rank n over X n.
18:28
So the main examples of this Tom spectrum are suspension spectrum, X plus mesh T,
18:41
X plus mesh T square, and so on. The second example that we've already seen is just a X plus mesh MGL.
19:00
Similarly, we can think about X plus mesh MSL, and X plus mesh MSP. But we need to be a bit careful here because naturally there rises T square spectrum, but again, similar things can be done for these two spectra. But let me mostly concentrate on these examples
19:22
on the suspension spectrum of a smooth variety and on the MGL spectrum. Then we can, if we have such a Tom spectrum, then we can introduce the E frame correspondences from U to X as just the home set
19:48
in the category of pointed misnavid shifts from U, smash P, smash n, into X plus smash EN.
20:02
And by Lemma-Voyvotsky, this set of homes in the pointed category of misnavid shifts has the following geometric description. So all these homes can be described using the following geometric data. So first of all, that's a closed subset
20:22
into affine space A n over U, which is finite over U. Second ingredient here is an etal neighborhood of Z in the affine space. And the next is the map from the etal neighborhood W
20:48
into the vector bundle Vn such that Z is the pre-image of the zero section Q. And also the second part of the data
21:01
is just a map from etal neighborhood to the variety X. So this map phi here that defines Z inside V, we call them a framing, and that's the frame, the E frame correspondence that rise with respect to the Tom spectrum E.
21:23
So yes, so these sets of data describe this morphisms up to a certain equivalence where the data Z W phi F is equivalent to Z W prime, phi prime, F prime if W and W prime
21:45
are two etal neighborhoods of the same closed subset Z and all phi and phi prime F and F prime
22:00
coincide on their common refinement on W cross A new W prime. So in this data, we can just shrink our etal neighborhood W and the correspondence stays the same. So then that's the description of this homestead.
22:23
Excuse me, just a quick question, Sasha. Right. Does this also holds without A1 invariance? Yes. That's here, we don't discuss any A1 invariance. So these are just, so here we just talk about pointed Niesznaevic sheaves.
22:52
So there is no A1 invariance here so far. Okay, thanks. Okay, and then if E is just a suspension spectrum,
23:06
S T is just point T, T square and so on. Then this frame, the E frame correspondences from U to X are just usual frame correspondences
23:21
from U to X. When we consider just a suspension spectrum here instead of vector bundle VN, we just have a trivial vector bundle over a point.
23:44
So then we can use their theory of frame correspondences to use this E frame correspondences to construct a vibrant replacement for the material spectrum E. So again, we can do this in two steps.
24:01
First of all, we construct an S1 spectrum and EX that again, we need to apply the C star construction to this pre-sheaf frame E from U to X
24:21
and then make it into an S1 spectrum. Right, and I didn't say what is frame E here. Frame E from U to X is just a co-limit when N goes to infinity of frame N E from U to X
24:45
where I can embed frame N E from U to X into frame N plus one E from U to X as just, I just need to embed A and U inside A and plus one U and then instead of W, I take W across A one and so on.
25:02
So this is a natural stabilization process that makes the level of this frame correspondence bigger. Right, so then I take the co-limit with respect to N, apply the simplicial construction C star and make it an S1 spectrum. So then what I get, I get this S1 spectrum MEX
25:25
and then we can make it into G spectrum using the same construction as we saw before. It's a S1 GM spectrum consisting of MEX,
25:41
MEX smash GM smash one and so on. So that's gonna be an S1 GM by spectrum and then the general theory of frame correspondences tell us that MEGX is locally equivalent to a motivic fabric replacement of X smash E.
26:03
So MEGX is locally equivalent to the motivic fabric replacement of E smash X
26:22
and this motivic fabric replacement looks like this. Again, I just need to apply a local fabric replacement. So MEX smash GM smash I.
26:46
F here is a fabric replacement of MEX smash GM smash I
27:03
in local stable, S1 stable Nissenovitch model structure.
27:23
So in particular, again, if we're interested in the, yes, it's locally equivalent to this one, sorry, and this one, this S1 GM spectrum is motivically vibrant
27:42
and is equivalent to X smash, sorry, smash E. So in particular, if we're interested in the homotopy groups, then the homotopy groups
28:03
can be computed explicitly as the homotopy groups of the corresponding S1 spectra. So that's the general result. It works for every Tom spectrum.
28:21
So it works for the suspension spectrum, it works for MGL and so on. And next, for MGL and some others, Tom spectrum, we can further make a reduction and make this construction a bit simpler.
28:41
So let me show you the steps of the reduction we needed to do in order to obtain the result for MGL.
29:10
So the first step, that is also a general one that works for every Tom spectrum is the following. Instead of considering the frame correspondence,
29:22
we can consider an object, which we call normal frame correspondences. And that's the following thing, denoted as frame and E tilde from U to X.
29:45
So that's the set of the following geometric data. So first of all, again, we have override U, we have affine space A and U, and we have a closed sub-scheme of A and U.
30:05
So Z inside A and U is a closed locally complete intersection sub-scheme.
30:20
And the map from Z to U is finite and flat. Then again, we consider a tall neighborhood of Z
30:42
inside A and U. But now instead of the whole map from W to the vector bundle VN, we just construct a map. We just need to consider a map from W to the base XN
31:01
of the vector bundle. So here VN is the vector bundle over XN from the definition of the Tom spectrum. Here we consider the map from the tall neighborhood psi, the inclusion I, and then instead of framing here,
31:22
we have the isomorphism between the normal bundle of Z inside A and U and the induced vector bundle by this map psi I from Z to XN.
31:50
And then the second piece of information is the map from Z into X. And then we have a similar equivalence relation
32:02
where we can shrink a bit a tall neighborhood W when we shrink a tall neighborhood W the normal bundle will not change. And we just need for this isomorphism phi to stay the same when we shrink the tall neighborhood. So here, what we did instead of having the,
32:23
having a whole set of equations that define Z inside W, we just remember one piece of information, we just remember how these equations give us an isomorphism between the normal bundle of Z inside W,
32:40
which is the same as normal bundle of Z inside A and U, and the actual induced bundle on Z that's induced from XN, right? So that's the normal frame correspondence. And as a side remark, it looks especially nice
33:01
where XN is just a point, right? So when VN is just a trivial vector bundle of rank N, XN is just a point, then what we have here, this data is just equivalent to the following data. We have Z, which is finite flat over U.
33:27
It's a locally complete intersection sub-scheme of A and U. And then we have a trivialization of a normal bundle
33:41
of Z inside A and U. Let's say X is also a point here. So then we have just locally complete intersection sub-schemes of A and U with a trivialization of their normal bundle. And that, it's really very similar to what's known
34:05
in topology as frame cavortism introduced by Panchragin.
34:27
So that's the first reduction step we need that instead of the whole framing, we just keep track of the isomorphism on the normal bundle of their supports. We're gonna call Z here, the support of the framed correspondence.
34:41
So it turns out that instead of the whole frame correspondence, we can just consider the isomorphism of the normal bundle of the support with the bundle induced from via. So then the theorem says that if I construct
35:10
in the same way the normal frame motif E of X, which is just C star frame E tilde of X,
35:23
C star frame E tilde of X meshes one, and so on, then M E tilde, so there's a natural map from M E of X to M E tilde, right,
35:43
on the level of frame correspondence. So when we have a frame correspondence like this, then it defines Z, if I consider not just a closed subset defined by the equations phi by the whole closed sub-scheme Z defined by phi, then it's gonna be a locally complete intersection,
36:02
it's gonna be phi net flat, automatically gonna be flat over U, and then this phi gonna induce the isomorphism between normal bundle of Z inside A and U with the bundle induced from VN. So we have naturally a map from a usual framed E frame correspondence to the normal E frame correspondence
36:23
just by forgetting some structure. Then it induces a map on the level of S1 spectra, and it turns out that this map is locally an equivalence.
36:41
So it turns out that for any E frame spectrum, we can simplify our construction like this. So that's the first step, it works for every Tom spectrum. And now let me show you other steps that work specifically for the spectrum MGL.
37:01
So for E equals MGL, we can simplify the construction.
37:21
Further, so when E is MGL, then VN we have here is just a total logical vector bundle of rank N, right? So this normal frame correspondence,
37:41
we're gonna give us that isomorphism of the normal bundle of Z with a pullback of the total logical vector bundle. And then it turns out that we can get rid of this information as well. So we can introduce another object,
38:02
which I call embedding N from U to X that consists of the following data that consists of embeddings of Z inside A and U,
38:21
which is locally complete intersection, so that the projection onto U is finite and flat. And then just a map from the support Z into X, right?
38:49
So here, then we have a natural map from a normal frame correspondence from U to X into this embeddings,
39:03
and UX that just takes all this data, Z, WZ, psi, phi, and F,
39:20
and just forgets everything, keeps only Z and F, right? So then we can show that
39:40
this thing, after we apply the C star construction, is an equivalence. So C star frame and E from U to X into C star embedding N from U to X is an equivalence
40:06
of simplicial sets for any smooth affine scheme U, right?
40:32
So for the case of MGL, we can forget about the framing and the only thing we need to keep track of is the support, actually.
40:41
And then the last step, for MGL, we can also forget about the embeddings of Z inside the affine space, right? So there is a natural map from embedding N,
41:02
UX, into the set of core omega, UX, where we take this embedding data, and just forget the embedding. So we have, we just keep Z, U, and X.
41:25
So we need this map P here to be finite, flat LCI map from Z to U. So that's what I started with,
41:41
the set of omega correspondences, right? And then it turns out that when I apply the simplicial construction again, so here I map to the set of, to the zero simplices of the nerve.
42:02
So when I consider a map from embedding Z into nerve, of core omega of UX, then it turns out that this is also, is a weak equivalence of simplicial sets
42:31
when U is smooth and affine.
42:44
And then this is how we arrive at the result I started with. So we start with the general construction of a five inch replacement for every term spectrum,
43:00
MEG of X, then we reduce it to the spectrum that we get using the normal framed correspondences. And then for MGL, we first of all reduce this normal frame correspondence to the spectrum that can be constructed using these embeddings,
43:20
and then to the spectrum that we construct using this omega correspondences. All right, and another remark here is that embeddings and UX is representable
43:48
by a smooth scheme. So that's gonna be the smooth part. It's gonna be LCI part of the Hilbert scheme
44:02
of A and U, and then C star, then the spectrum C star embedding X Smash G Smash S that we can construct from this data will be a local model for motivic
44:30
five inch replacement of X Smash MGL,
44:41
which is representable just by co-limits, directed co-limits of smooth simplicial schemes.
45:14
So I guess it's a good point to stop here. That's what I wanted to tell them.
45:26
Okay, thanks. Thanks a lot, Sascha. Thank you. Wonderful, very impressive result. Let's see, are there some questions?
45:48
A little question I ask. To what extent does this tell you about the various P1 suspensions, the maps on the P1 suspensions of things like MGL or MSL,
46:03
like negative or positive suspensions? So naturally these things they are, they live in the GM S1 spectrum. So it's not clear,
46:23
but does it tell you about the P1 suspensions? Mark, positively suspense, down spectra, everything works. Yeah, but on the negative ones, it's unclear. Negative ones aren't clear, exactly.
46:42
Okay, good. Okay, great. Thanks. Any other questions? Sascha, maybe this is a question from one of them.
47:00
So in this remark, you described the say take X to be point. And so take local model for the Markievic pregnant replacement of MGA.
47:23
And it is representable, as you described by smooth-synthesial, tiny units of smooth-synthesial schemes. Can we just work in with this model
47:40
proof cancellation theorem for this bi-spare pair? The GM cancellation? Yeah, means not taken back to original kind of omega MJ correspondence, yes.
48:06
I'm not sure if you can, so your question is, can you get the GM cancellation directly from this description? Yes, yeah. I don't know, it's hard to say.
48:23
It's just not clear to me. Okay, so that was the question. I got another one, thank you. Thank you. Thank you, Vayner.
48:44
Can wait a little bit longer for possible other questions. And maybe I can ask one more question. Okay, could you take that a little bit up?
49:08
Say, okay, say for the step three. In the step three, you require U is to be smooth and fine.
49:23
Okay, could you comment why it is enough? Means that typically one should expect that you should be taken to be local in the area.
49:42
And in your case, it is smooth and fine. The point is why it works for smooth and fine because actually we are able to construct an explicit homotopy in between. So if you have the same Z over U,
50:01
but you have different embeddings of Z inside an affine space, then you can construct a homotopy that drags one embedding into the other embedding. And for this thing to work, it's just enough to consider a smooth and fine U. Okay, very good.
50:21
And is it kind of the same principle which works for the previous step, means for the step number two. Yeah, for the step number two, you mean the step for a normal frame correspondences.
50:46
This you have shown to me the step number one, but go a little bit downstairs. Yes, step number two. Yeah, here is the theorem. The step number two goes from the normal frame
51:01
correspondences to the embeddings. Yes, again, the reason why it works over this smooth, affine U is that we're able to construct an explicit homotopy that works over smooth and affine U. So the point here is that when we talk about
51:24
this normal frame correspondence for MGL, then what we have here, we have just a tautological vector bundle, VN. And so the framing here is an isomorphism of a normal bundle with a pullback
51:41
of the tautological vector bundle. And every vector bundle is isomorphic to the pullback of the tautological vector bundle over the fine variety. So actually it turns out that we can just, if we have two isomorphisms to the pullback
52:01
of the tautological vector bundle, then again, what we can do, we can produce a homotopy that sort of transforms one into another in a consistent way so that it makes this equivalent possible. Okay, thank you. Okay, we have some more questions for you Sasha.
52:22
I can read them. So Tony Anala asks, what part of the geometric part of algebraic co-bodism can be represented by these methods? What part of? Of the geometric part of algebraic co-bodism can be represented with these methods?
52:41
So geometric part, you mean two and diagonal? I guess. Okay, so actually, locally we can compute everything using these methods because if you have a A, B homotopy group of A one,
53:02
say MGL, smash X on U, U is local, zillion. Then using these methods, this is gonna be equal to just by A of C star
53:29
and core omega of, sorry, I guess I need to write A minus B here, right? Because we have simplicial sphere of dimension A minus B
53:50
and here we have GM, smash B, smash U into X. Yeah, the point here is that, so GM, smash B, smash U is not local,
54:02
not local and zillion anymore but because everything here, all the sheaves we have here, they are framed sheaves and for the framed pre-sheave, which is A one invariant and with some additional properties that hold here, we know that it coincides with its sheafification
54:21
on the open subsets of affine space. So actually, this computation here of the A one homotopy groups, it works not only for local and zillion schemes but also works for all the open subsets of affine space.
54:40
So every A one homotopy group can be explicitly, can be computed as the usual simplicial homotopy groups of this. So sorry, I need to write down as one spectrum.
55:07
Okay, so when U is not local, we don't have this result but when U is local and zillion, we can get this description.
55:20
So maybe we'll specify a little bit. The previous question, not of mine but you try to answer it. They substitute U to be a step of the field going on.
55:45
When U is what? U is step of the field. Okay.
56:02
And say, X is a point.
56:30
So you answer on question of the person before mine.
56:40
As I understood your answer, it's likely that you take the pi A minus B of this simplicial set and so this is the answer, yeah? Yeah, this simplicial spectrum but it's omega S one spectrum starting from the first. So we can actually rewrite it like this.
57:02
Can say it's pi A minus B plus one of this simplicial set C star N or omega of GM, smash B, T plus, smash S one. So then that's an explicit simplicial set
57:23
that computes the mod A one homotopy of MGL. Okay, good. Then there's two more questions. At least two more questions, Sasha. So one of the questions says,
57:41
is this the non Borel-Mortisim of X? Is non... Non Borel-Mortisim of X. Non Borel-Mortisim of X. I'm not sure what's exactly the question.
58:03
So maybe that's because the map to X is not proper. What's the difference with the algebraic keyboardism, or more algebraic keyboardism of this construction?
58:21
Yeah, the point is that Z here, first of all, Z is not a variety, Z can be a scheme. And also, yes, the map from Z to X is not proper. It's just any map. So the only, so Z here is a scheme. And the only thing we need from Z
58:42
is that it's the map from Z to U is a locally complete intersection map and it's finite and flat. Okay, sounds good. Then another question from Ora Sander. Could you explain why you need to take the nerve
59:02
of a core omega? Oh, that's because that's because when we forget about them, when we forget about the embeddings here, the point is that we need to keep track
59:22
of the isomorphism between the tops of the heads, because it's not gonna work if I just take, say, an isomorphism classes of the heads of core omega. Just will not get an, will not get a weak equivalent here.
59:43
So because when we forget about the embeddings here, we still have to keep track of the isomorphism on the top of the heads. Okay, great. Then Kirsten asks if you can say something more about the relation with Hilbert.
01:00:00
schemes oh yeah so the the relation of Hilbert scheme is is is pretty straightforward so when when X is a point say then this embedding and a few into the point that's just by definition that's the sub sub scheme of a Hilbert schemes that consists of LCI of LCI schemes and it's known
01:00:26
that this so that this part of the Hilbert scheme is smooth so this embedding so this embedding so u goes to embedding n from u to k is just
01:00:45
represented by a sub scheme of Hilbert scheme of a n that that then the sub
01:01:03
scheme of Hilbert scheme of a n is exactly this sub schemes of Hilbert sub sub schemes that form the Hilbert schemes that are locally complete intersection sub schemes okay seems that wraps up the question round so
01:01:25
thanks a lot Sasha and also thanks to all the attendees thank you how to say
01:01:57
our pleasure in French our pleasure was simply wasn't really something like that
01:02:06
yeah yeah that's it do you see I'm improving Sasha during during my stay in digital Paris very good at least I promise to learn more French words
01:02:23
during the two weeks that's it you did okay