Yeah, thank you very much for the invitation to speak here. I should make two preliminary remarks.

First of all, the results I want to speak about today are not as complete as I would like. And second, I mean, I should say that most of them somehow came around discussions with Matthew Morrow. OK, so in this talk, by a scheme X, I always mean a scheme which is quasi-projective

over a Noetherian ring of finite cool dimension, for simplicity.

OK, and first of all, we want to study k groups. So let's start with k0 of the scheme X. And of course, classically, you can just define this as the free abelian group generated by the vector bundles V on X modulo the exact sequences.

So modulo the subgroup generated by elements of the form V prime plus V double prime minus V,

where you have a short exact sequence of this form. OK, and then Beth tried to define not only k0, but also define negative k groups. And his idea is as follows.

So he made some algebraic delooping process. So he defined k minus 1 of X to be some sort of deloop version lambda of k0. And this lambda is a construction which works as follows. Namely, it's defined as the co-kernel of the map from k0

of X times a1. Let me just write X times a1 as xt, so t is the parameter for a1, plus k0 of X times a1, but now the parameter t to the minus 1.

And then this goes to X times gm with a natural map. And then you take the co-kernel of this and call this k minus 1. And now you can go on and define k minus 2 of X similar,

just by the same construction, lambda construction, applied to k minus 1. It's the same co-kernel, but just replacing k0 with k minus 1. And so now Beth suggested to study these groups.

And this is motivated by the following fundamental theorem, due to Beth and in this form Quillen and also, I think,

Houghton-Dieck. And this says the following. Namely, first of all, that if you apply this lambda construction to positive k-fury as defined by Quillen, then you just get ki minus 1. So for i bigger than 0, this lambda construction

applied to ki of X, this is just ki minus 1 of X. And the second statement that is also somehow part of this fundamental theorem is that if you apply this, if you make look at this construction of negative k groups

applied to regular scheme, then this vanishes. So for X regular, it is true that ki of X vanishes for i less than 0.

So some of these negative k groups are measure for the singularities of the scheme. But it's a very complicated environment to calculate. And the most important conjecture about these negative k groups of schemes is the conjecture of Weibull that I

want to discuss in this talk. This was originally formulated as a question by Weibull in 1980. And it says the following two things. Namely, even for singular schemes of finite Quill dimension, these negative k groups

finally vanish. More precisely, ki of X is 0 if i is less than minus the Quill dimension of X. And there's also a second part of this conjecture,

namely that ki when i is equal to the dimension minus the dimension of X, then this is homotopy invariant. So ki of X mapping to ki of X times aj, the affine line,

is an isomorphism. Or this is true, actually, for all i less than or equal to minus the dimension of X. So this is the conjecture. There's also a definition of homotopy invariant k theory.

So they define a version of k theory, which you like. Yeah, Weibull defines it, yeah. Yes, and so there is the same conjecture for this. Yeah, let me come to this. We just proved this conjecture, but let me come to this in a minute. So then let's call this conjecture WcX for this scheme

X. And just one more small notation. So let us call WcD, the statement that you know this conjecture for all such schemes of dimension at most d.

So for all x of dimension at most d. And then classically, or in the last years, the following theorem has been shown. First of all, the first major progress is that Weibull showed this conjecture up to dimension 2.

So we have the statement Wc2, which is due to Weibull. Then we have the important result that it holds for varieties of characteristic 0.

So for x over some field k of characteristic 0, a finite type. And this is a result due to Kotinnias, Heisenmeier, Schlichting, and Weibull. And then also if we assume very strong form

of resolution of singularities and positive characteristic over a perfect field, we also get the Weibull conjecture there for such x. So k is a perfect field. And we assume that exactly the same statement

as proved by Hironaka in category 0 holds for varieties over k. Let's call it strong resolution of singularities. So this is what is in the literature, more or less.

Now we can come to Gaba's question. Namely, you can also ask the same conjecture for homotopy k field. Let's just mention this shortly.

So homotopy k-fury is the following k-fury defined by Weibull. First of all, consider the standard algebraic j simplex over the integers, which is just back the z.

Then you take polynomial ring and variables t0 to tj, modulo ideal. And then homotopy k-fury is defined as follows. Namely, the i-th homotopy k-group of the scheme x

is defined as the i-th homotopy group of the homotopy co-limit. Now when I write k of a scheme, by this I mean the non-connective k-fury spectrum.

And now you just take the homotopy co-limit over this simplicial spectrum, and then you take the homotopy groups. And then the new theorem I want to present today, let's call it star, is the following.

Namely, first of all, if you replace in the formulation of Weibull's conjecture the k-group by the homotopy k-group, then this is true. So Weibull conjecture on k-h.

So I think this was Gabba's question, yeah? And second, so what can we say for k-fury? So the results are not as complete.

So we can prove the Weibull conjecture for the scheme x and the following assumptions. First of all, you need some sort of fineness assumption. Namely, if all the residue fields of positive characteristics are Frobenius finite. Let's say x is a point of capital X of positive

characteristics, then the Frobenius is a finite morphism on this. And if this is only a very small assumption, which depends on some technical problems with topological

cyclic homology, and if a very weak form of resolution of singularities holds, which I want to call Res, and I will explain later. So this is just an extremely weak form, and I'm quite optimistic that we are able to prove this. Or you could also assume just that the dimension of x

is less than or equal to 3. And in this situation, we can prove this kind of resolution problem. So that's the new result. And moreover, third statement. But you don't need quasi-excellence for this resolution assumption, or? No, no.

I come to this. And there is some way to reduce it to quasi-excellence schemes, in particular, so to say, a special point of this part two is part three. Namely, that the viral conjecture holds for all schemes of characteristic 0, so for any

scheme containing q. So in particular, this is mostly due to Matthew Moore, also with some contribution by myself. So now, Gabo already mentioned, you have to reduce somehow this kind of viral conjecture to a

statement with better schemes, like quasi-excellence schemes and so on. So let me now explain these reductions. And although these reductions are really elementary, I couldn't find them anywhere in the literature. So it's quite interesting.

But at first, your x was a very nice scheme. Yes, it's always a very nice. No, no. It's always a scheme of this form. At the top of the board, it's quasi-projective over something. When I say schema, it always means such a good scheme. You can generalize it, of course, to more general schemes, but then you must not work with this kind of

K theory, but K theory of perfect complex or something like this. So even for these schemes, you need the reductions? Yeah, the reductions, I'm coming now. So let me explain what I mean by reductions. So first of all, we have the following kind of reduction.

Namely, if I know, let's say x is any scheme, if I know the viral conjecture for all the stocks of the structure sheaf, then I know the

viral conjecture for x. So the idea is very simple. Let me explain it quickly. So for this kind of K theory, we have a Tzelski descent spectral sequence of the following form.

The E2 term is just the Tzelski cohomology of the Tzelski sheafification of this non-connective K theory. Oops. This converges then to K minus P minus Q of x.

And now if you assume that you have this viral conjecture, you get some vanishing of this sheaf at certain stocks, depending on the dimension of this point. And then you have here Tzelski cohomology, and then there's this sort of vanishing result of Croutendieck, which tells you that Tzelski

cohomology vanishes for low-dimensional schemes, but you can somehow generalize it or give a version of this, which works also for such any sheafs which have low dimensional support, roughly speaking. So then you see that for minus P minus Q smaller than

the dimension of x, assuming this here, this actually vanishes. And then you use Croutendieck vanishing for this. So this is really quite simple.

The next reduction allows us to pass to excellent rings or excellent schemes in particular. So let's say D is some non-negative fixed integer. Then we have the following reduction.

If we know the viral conjecture, now for certain rings R, which are complete local rings of dimension at most D.

For all such rings, then we actually know the viral conjecture for all schemes of dimension up to D.

So to prove the viral conjecture, in other words, you just have to prove it for complete local rings. In particular, they are excellent. And we have now for them, we later have the whole strength of Hironaka's resolution of singularities.

OK, the idea again, it's not very complicated. So we proved this by induction on the dimension of x, of course from 0 up to D. So first of all, by

the first proposition, I can assume that x is a local scheme. Then I denote by x hat its completion with respect to

small x, the closed point of x. Now, we have the following commutative diagram with exact rows.

The rows are just the localization sequences of k3 over support of Thomas and Torbeau. Take k i plus 1 of x without the closed point. Then we take k i of x support in the closed point.

Then we take k i of x. And then we take k i of x without the closed point. And then we can write the same for the completed ring

like this. And then we have, of course, vertical maps like this.

So then the rows are exact. And what else do we know? We know that this here is an isomorphism. This is the excision theorem of Thomas and Torbeau for k over support because this map is flat. We also know now if we assume that i is less than

minus the dimension of x, then these schemes here, they have dimension one less than the dimension of x. So by an induction assumption, we know that actually these groups vanish as well as these groups.

So we know that in effect here we have also isomorphism. So we see that here we have an isomorphism. And we assume that here this group vanishes. This was our assumption. So that's the simple proof.

So now we have reduced to some simpler situation, particularly in characteristic 0. We now have resolution of singularities because Hironaka proved this for quasi-excellent local rings. So the basic idea now in all proofs that I know of Weibull's

conjecture is somehow, first of all, you need some sort of vanishing result. Vanishing like, for example, if x is regular, then all negative k groups vanish. And you need some sort of descent result. So you somehow resolve the singularities of x. Then you apply this vanishing, and then you apply some sort of descent.

This is always more or less the same structure in the proof. So one definitely needs some sort of descent under blowups, let's say. So this is something I want to explain now.

And this is motivated especially, again, by some work of Matthew Morrow. So let's consider what kind of descent do we want to consider. Let's consider a pullback square of schemes of the following form.

x prime maps to x via map p. Then we have a closed sub-scheme here, y of x. And then we take the pullback and call it y prime. So then this is a pullback square, call it plus. And we assume that p is projective.

And we assume that p is an isomorphism outside y. So in other words, p induces an isomorphism from x prime or y prime to x without y.

We call such a thing an abstract blowup square. Of course, if you blow up x and y, then this would be a special case.

Now, to formulate the descent result, so let's say y is defined by the coherent ideal sheaf i. Then let us set yn to be equal to the closed sub-scheme defined by the coherent ideal sheaf i to the n.

And let us define the k-fury spectrum of the formal scheme, associated formal scheme y hat, to be the homotopy limit over all n of the k-fury spectrum of yn.

And then the k groups itself are just, of course, defined as the homotopy groups of this spectrum. OK, now I can consider the following.

Just apply a k-fury to my square, more or less. I consider the following square. Let me apply the k-fury spectrum to x. Then I consider the k-fury spectrum of this formal scheme y hat. Then I consider the k-fury spectrum of the analogous formal scheme y prime hat.

And then, oh, sorry, this goes in this direction. Here, consider a k-fury of x prime. And I ask that this square is homotopy Cartesian. So this is some sort of descent property. Let's call this property dx. I say that dx holds if this is homotopy Cartesian.

For all abstract log squares where x sits in this corner, then I say that dx holds.

OK, so why should we consider this? So some examples where this kind of descent under abstract logs holds are the following, which are known. For example, if the morphism p from x prime to x is finite,

then this follows from the work of Suslin-Wachitsky on the excision for k-fury and some work of Matthew Morrow. Another example is where we know this. So in this case, we know dx. Another case is if x prime is the blowup of x

in the closed-subscheme y, and if y to x is a regular closed-immersion.

So then we also know this. For this special case, we know, well, I should, OK, this is not right. I should say like this, OK. Then we know it for this dx, x prime, and this y.

And this was shown by Tomasson. Tomasson didn't consider the completion. He just do it immediately. And so you have to do the same work for the completion compact. Yeah, yeah, but in the completion, it's proof. I mean, of course, the powers are not regularly immersed. But if it's affine, then you can set up the powers.

You can just take a different powers of, I mean, just a different system of ideas, which are, I mean, in the equivalent, yeah? OK, and these are regularly immersed. Yeah, yeah, exactly.

OK, so but this is just a special case. So what we really need is some more general statement. And this is the second main theorem. Let's call it double star. And it says that we know this descent conjecture for k

field for the scheme x for x quasi-projective over ring R, where R is complete local ring.

And we, again, have to impose this Frobenius finiteness. So let's say m is the maximum ideal. And we want that this field is Frobenius finite.

And we also have to impose this certain very weak form of resolution of sine lattice that I want to explain now, just for quasi-projective schemes over this ring R, to be precise. So now I really have to explain this unpleasant assumption here about, so to say, resolution.

It's rather than resolution of sine lattice. It's some sort of principalization of ideal sheaves. So let's start. Let's recall the following definition of Hironaka. A closed immersion of schemes y into x.

We say that this immersion is normally pseudo-flat

by definition. Let's say x is an integral scheme. Let's restrict to this case. If, first of all, y is not dense,

and secondly, if I consider the blowup of x and y, then I ask that all the fibers of this blowup over y have the same dimension.

And the dimension necessarily has to be the co-dimension of y and x. You mean all fibers over y? But do you allow lower dimensional components in the fiber, or you should be purely a good dimension?

I don't care. I don't care. So this is Hironaka's definition. Maybe this can happen. I don't have an example here. So there are at least of these dimensions, and then you require a lot more? No, no. This is a path found from above. So the dimension could, the fibers always has to be at least this co-dimension.

But I don't want that the main dimension can go up anywhere. I don't say anything about whether all components have the same dimension or so. So this is, of course, a weaker version than normal flatness. Normal flatness would mean that somehow

the exceptional divisor is flat more or less over y, roughly speaking. And now I can formulate this resolution assumption. So this is a statement for all quasi-projective integral schemes over this fixed, let's say, complete local ring R.

And for any non-dense closed sub-scheme z of x,

there exists some composition of blow-ups in normally pseudo-flat centers, which I want to call P from x tilde to x.

NPF means normally pseudo-flat, and such that the pullback of z to x tilde

is defined by a locally principled ideal chief. You don't require that the blow-ups are above z.

They can be answered. Can be everywhere in any non-dense center. So the point is that this is exactly the way that Hironaka approaches resolution of singularities. He states that he can principalize some closed sub-scheme by a certain sequence of blow-ups,

namely in regular, normally flat centers. That's the important result of Hironaka. But here, it's much weaker. So I'm quite optimistic that actually this kind of resolution process can be achieved.

So unfortunately, I have no final result yet. So I'll just state some partial results. And what's the conclusion of TRM in the rest of it? The conclusion is that this descent property for k-theory

holds for any abstract law of square where x sits in one of the corners. If x is quasi-projective over this complete local ring, which is Frobenius finite, and you have this resolution assumption.

So this kind of resolution problem is actually much closer to a process called

Kolmec ollification, which was developed by Faltings. So I cannot say very much about this. Just one remark. First of all, if R contains the rational numbers,

then this resolution problem follows from the work of Hironaka. So if the residue characteristic is 0, R is always a complete local ring, or could be any quasi-excellent local ring, if you like.

This is Hironaka. But the following thing I can prove more generally, namely this kind of resolution problem. Now for some fixed scheme x, let me write it like this. And any closed non-linear sub-scheme z

can be proved up to dimension 3. It can also be proved for the closed sub-scheme z and x if the dimension of z is less than or equal to 1.

And actually, I think I learned this from a letter of Gaber to Eleninou, which deals with a similar question. Now this was just preliminary, so we

want to come back to the proof of Weibull's conjectures under a certain assumption. So now what we want to do is we want to deduce theorem star, or to be more precise, I just want to restrict to the second part, namely this here,

where you have this resolution assumption and this Frobenius fineness using this theorem double star.

So this is quite nice. So as I said, we need some vanishing, something like vanishing for regular schemes. Then we need some descent. OK. So let me first explain. We cannot really resolve singulatives, so we cannot

stick to this fundamental theorem about the vanishing of K-fear of regular schemes. So here's a replacement that is quite useful. Let's call it the key proposition. So consider a reduced scheme, and let's consider a K-fear

class psi in K minus J of x, where J is any positive integer. Then I claim that I can, after some modification,

I can make this psi vanish. Namely, there exists a modification p from x prime to x, so a proper bi-rational, a projective bi-rational map,

such that the pullback p upper star of psi in K minus J of x prime, this vanishes. So if you have a resolution of singularities of this x,

then you can just take x prime as this resolution of singularity, because it will kill, of course, all the classes, because then this group vanishes. So this is already one replacement for resolution of singularities. And the key thing that we will use is the Renault-Cresson platification by Éclatement.

So one preliminary remark is the following. Namely, there exists a natural surjection to the group K minus J of x. So I explained to you that this comes from this delooping, multiplying x by some gm. Or if I repeat this, I multiply x by J copies of gm,

and then I mod out something of this K0 group of this scheme. So at least there exists now a surjection from K0 of x times gm to the J. What you know, the image in this K0 group of K0 of x times aj.

At least this is mod out, maybe more, but we don't care for the moment. We just consider this surjection. So let us come now to the proof of this proposition, key proposition.

So first of all, we can assume, so psi is now represented by an element in this group. So without loss of generality, we can assume that it's represented by a vector bundle on x times gmj.

Then, as is well known, we can extend this vector bundle on this scheme to this bigger scheme as a coherent sheaf, at least.

So we can extend this to something I want to call V bar, which is a coherent sheaf on x times aj. Now, because x is reduced,

I can find an open dense sub-scheme of x such that V bar is flat over this. So such that V bar restricted to u times aj,

mapping to u, is flat. I mean, what I mean is, of course, that this sheaf is flat over this, OK? Now, Renaud-Croissant's Platification par eclatement

tells you the following. Namely, I can find a blowup in x without u,

which makes somehow this V bar flat. More precisely, they tell us the following.

Namely, there exists some projective bi-Russian map,

p from x prime to x, which is an isomorphism over u, and such that if I consider the so-called strict transform

of V bar, and let's call this V bar prime, which is now coherent sheaf on x prime times aj,

then this is flat over x prime. Now, what is the strict transform? It just means to take the usual pullback of coherent sheaves, and then you kill the torsion, kill all sections which vanish over the preimage of u, OK?

So this is just a quotient of the usual pullback. So they give you this. But then it follows that automatically, this coherent sheaf has finer tau dimension, as you can easily check.

The reason is just that it's flat over this x prime. And of course, this scheme is smooth over this x prime. So then it's not difficult to check. But any coherent sheaf with finite tau dimension

gives rise to some class in k0, as shown by Krotnik, by just resolving it by vector abundance, finite resolution by vector abundance. So then this element, this coherent sheaf, gives rise to a class in k0 of x times aj.

And of course, if I restrict this coherent sheaf here to x times gm to the j, I get the class of the pullback of my original element xi. So then this restricts to p upper star of xi.

This induces this element. And then, of course, any such element vanishes in negative k groups, as I explained on this blackboard. So then it follows that p upper star of xi vanishes.

That's an important observation. So now I can come to the actual proof of theorem star. So first of all, by my reductions I just explained, I can now restrict to schemes which

are quasi-projective over complete local rings. So from now on in this proof, by scheme I always mean a scheme which is quasi-projective over some fixed complete local ring R.

And for such schemes, we want to prove this vanishing conjecture of y by induction on the dimension of x.

So let's consider some element xi in ki of x, where i is less than minus the dimension of x. So we want to show that xi vanishes. So I want to stick now to the first part

of Weibull's conjecture, not the homotopy invariance, for simplicity. Now, the key proposition tells us, I mean, first of all, we can easily reduce to x reduced. I skipped this. We can then find some modification which kills this xi.

We'll choose a modification p prime from x prime to x,

such that the pullback of this xi under p vanishes. Now, because this is projective by version, I definitely find a closed, non-dense sub-scheme y of x,

such that this p is an isomorphism outside y.

Now, we can consider the following diagram of exact rows.

Sorry, not the diagram of exact rows, but we just use our descent property.

Namely, we use this theorem double star, which gives us some sort of descent relation. Namely, we get an exact sequence of the following form, ki of x receives ki plus 1 of the formal scheme y prime hat.

And this goes to ki of x prime plus ki of the formal scheme y hat. So this is exact. And what can we say? Now, we have our element xi in here.

Well, y and y prime all have these formal schemes have dimension less than the dimension of x. So by our induction assumption, we can actually apply. We can deduce that ki minus i plus 1 of y itself vanishes.

This is the induction assumption. But then it's not difficult to see that somehow this non-reduced structure, which is sitting in this formal scheme, doesn't change k-theory. So this is some easy object. It's just based on the observation at the end that k0 of a ring does not depend on the important elements.

So I can just mode out the important elements without changing k0 of a ring. You can use this. Well, you get the vanishing here, and by the same argument, this is just the k-theory of y prime. And then also by our induction assumption, this vanishes.

But this means, of course, that I can forget about this map here and say this is injective. But I already know that xi goes to 0 here. So I deduced that xi is 0. So this is the basic idea.

So we have assumptions you needed just for the unit of this property, just for the blue, which is given by the Green's law. Maybe you are asking that too much. So you will apply it for this law. Yes, to find this x prime, I have to use this. So classically, the resolution of single lattice was used twice in all these approaches,

if k-theory, let's say. Yeah, but this d x is just here. It means you will apply it just for the square, which is given by the Green's law. No, no. You need this condition, this resolution, in order to prove, first of all, descent for k-theory.

And then classically, instead of Renaud-Croissant, people used resolution of single lattice here. So classically, resolution of single lattice is used in characteristic 0, for example, in the work of Faber-Cortinius and so on, in two different ways. One, to obtain descent, and one to prove this kind

of key proposition that I have here. Yeah? OK, so let me now say just a few words about how we can obtain this descent for algebraic k-theory, so this theorem double star.

So there are two ingredients.

This is an idea, going back to Cortinius, his Maya Bible, and so on, this group of people. Namely, we split up k-theory into some linear part and some part which doesn't see the non-reduced structure. So this works as follows.

So instead of the k-theory of x, we consider the so-called infinitesimal k-theory of x, which is just defined as the homotopy fiber of the k-theory of x mapping via the trace map to topological cyclic homology

spectrum of x. And for this topological cyclic homology,

the analog of this kind of descent property for k-theory was proved by Dandas and Mauro. So here, we know this kind of descent. Here, we want to prove it. So in the end, it's enough to prove this descent for this so-called infinitesimal k-theory. And the important property of infinitesimal k-theory

is that k-inf of x is the same thing as k-inf or is weakly equivalent to the k-inf of the associated reduced scheme. This is the result of Goodwillie and McCarthy. So this is the weak equivalence. So this is the most important thing we want to use.

Now, for this k-inf, I can apply some very general kind of descent result, which generalizes work of Heisenmayer. So I want to explain now some axioms about some cohomology theory of schemes, which guarantee that this cohomology theory satisfies something like descent along abstract Blau op squares.

The original version was in Heisenmayer's thesis. And he really used the whole strength of Hironaka's resolution of singularities. And the new thing is that I make this work, so to say, with this very weak form of resolution of singularities, or principleization of ideals.

So now we work over some fixed complete local ring. This is fixed. And we consider a cohomology theory of schemes.

Namely, we consider schemes which are quasi-projective over R to the category of spectra, topological spectra. And let's call it H. And then we also have a relative H. Namely, if we have a closed immersion, y, sitting inside x,

we consider the relative H theory, H of x relative to y, which is just equal to the homotopy fiber of H of x mapping to H of y.

Now I can state some properties, which are usually easier to check, which

will guarantee that this H theory satisfies descent under abstract Blau ops. Namely, theorem. So if, first of all, we assume our extremely weak form of resolution of singularities over this ring R,

and if H satisfies the following properties, namely, first of all, excision for affine schemes,

which means that H of, let's say, we consider a finitely generated algebra over R, which we call A. So H of spec A, relative to the closed sub-scheme defined by some ideal I of J, maps as a weak equivalence

to H of spec A prime relative to the closed sub-scheme defined by I, where A to A prime is some ring extension

with a property, while I is some, first of all, some ideal in A, but we assume that I is also an ideal in A prime. So then we assume that this is a weak equivalence.

This is excision. And it holds for K inf in particular. The next thing we need is Meyer-Weitouris for finite Zariski coverings.

And then we need this kind of descent for regular Blau ops in the sense of Thomason. I will come, I can explain it in a minute.

We also need this invariance under infinitesimal extension. So H of any scheme X maps as a weak equivalence to H of the reduced scheme. And then these properties imply that for an abstract Blau op

square, X prime mapping to X, and containing here this closed sub-scheme Y and Y prime, the following square is homotopy Cartesian,

H of X mapping to H of Y, mapping to H of Y prime, mapping to H of X prime, X. Here we don't need anything about the formal scheme because this theory doesn't see the non-reduced structure.

So this should be homotopy pullback. So where does the assumption of the residue field and characteristic P having a Frobenius finite enter? Does it enter here or? Yeah, here. I mean, I explained to you we know

this resolution in Cartesian zero. No, I'm saying you have the special assumption in theorem star star about R mod N Frobenius finite. And when you discuss now the general formalism for obtaining this, you did not prove this assumption. It's hidden in topological cyclic homology.

This is in order to prove this kind of descent in certain situations for topological cyclic homology, you need this Frobenius finiteness. I don't want to say anything about this. I'm not really an expert on this, so I cannot say much about it. So the point is that now descent

under abstract Blau ops for this H field means that this is a homotopy pullback square. And by descent for regular Blau ops, I of course mean that if x prime is a Blau op of x inside this y, where y is regularly immersed, then you have this kind of Blau ops square. And while his and Meyer's approach to result like this

is very involved, he makes really full use of the strength of resolution of singularities, whereas I find a simplification of his approach. Somehow you just, if you have any abstract Blau op, x prime to x, then some...

how you just dominate it by this resolution assumption by blow-ups in normally pseudo-flat centers. So then by some simple trick, you are reduced to the case where x prime is a blow-up in a normally pseudo-flat center. So you just have to prove descent for this.

And the idea is to reduce descent for this to descent in regular blow-ups. Because if you have some normally pseudo-flat blow-up, let's say your scheme x is local, then you can just consider this ideal defining this y. You find a so-called reduction of this ideal defined, constructed by Northcott-Reece,

which is generated just by co-dimension of y and x many elements. And if you apply some additional tricks from commutative algebra, you can make sure that actually this ideal is generated by a regular sequence. Okay? Yeah, I will stop here.

Are there any questions? What about homotopic K-theory? So I think people, I remember in some talks in 2004, people considered the, I don't remember, I think it was maybe the result of Wilson method,

but I remember they considered homotopic K-theory. And you claimed that for homotopic K-theory, you don't need the... This resolution assumption. Okay, so how does this... Okay, so the point about homotopic K-theory is that, in principle, the proof works very similar.

First of all, I should say this is published. You can find it on the archive. The result on homotopic K-theory is published. The rest of it is not published. The idea is, I mean, again, you need some sort of way to kill negative K-groups. This is made in more or less the same way as for the key proposition, yeah?

Key proposition can easily prove a variant for homotopic K-theory. The next thing is you need some sort of descent result for homotopic K-theory. But for homotopic K-theory, we know exactly descent along abstract lob squares in this sense, because it was proved by Ciesiński

using completely different techniques. So for homotopic K-theory, we have descent along abstract lobs, and we don't need to do all these complicated things that sits inside this proposition, yeah? I have a general question.

So what kind of information does negative K-theory give on the classical singularity set of surfaces? Okay, yeah, so for surfaces, you have two possible non-managing groups, so K minus two and K minus one. So for example, if you take K minus two of a normal surface, let's say you have,

then it's just an isolated singularity, and then let's say you take just of a local ring, and then it just counts the number of loops in a resolution. And well, K minus one is more complicated. But the thing one can say, what it means, if you have a dimension D scheme, and you consider K minus D of the scheme,

then somehow, more or less, it just depends on the combinatorics of some resolution of singularities of this scheme. So combinatorics of exceptional divisor, very roughly speaking. So it is finite type of B-theory? Yes, yeah. Yeah, but this is conjecture, I guess, but.

But? No, the K minus one is not finite type. So the K minus dimension. K minus the dimension. Minus dimension, yes. K minus one contains the three groups of H2O star, which is very, very deep. Yeah, so there exists some description of a normal surface,

singularity by Weibel, of this K minus one in terms of some Peacock groups. But can you prove that K minus the dimension is finite degenerate in some new cases? I mean, in the old, in the old methods, that's what makes it drop down for free, if I remember correctly. You mean now in Kato 6-0, whether one can prove it, for example, or assuming resolution of singularities, or?

Yeah, I can't remember what the states of the old results were if I just wanted to be got in any new cases. I'm not sure, I didn't think about it, but I suppose you could prove it if you assume a solution of singularities, so. Which are the questions? Then we meet again in 15 minutes.