 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 Moro. OK, so in this talk by a scheme x, I always mean a scheme which is quasi-projective over in the favorite ring of finite cold 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 Bess tried to define not only k0, but also define negative k groups. And his ideas as follows. So he made some algebraic deloping 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 gn 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 Bess suggested to study these groups. And this is motivated by the following fundamental theorem about YouTube, Bess, and in this form, Quillen, and also, I think, Rotendieg. And this is the following. Namely, first of all, if you apply this lambda construction to positive k theory as defined by Quillen, then you just get k i minus 1. So for i bigger than 0 lambda, this lambda construction applied to k i of x, this is just k i 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 k i of x vanishes for i less than 0. OK? So somehow these negative k groups are measured for the singularities of the scheme. But it's a very complicated invariant to calculate. And the most important conjecture about these negative k groups of schemes is the conjecture of Weibel that I want to discuss in this talk. This was originally formulated as a question by Weibel 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, k i 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 k i, when i is equal to the dimension minus the dimension of x, then this is homotopy invariant. So k i of x mapping to k i of x times a j, the f i in 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. We also have the definition of homotopy invariant k theory. So they define a version of k theory, which you like. Yeah, Weibel defines it, yeah? Yes. And so they have 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 W c x for this scheme x, and just one more small notation. So let us call W c d. 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 Weibel showed this conjecture up to dimension 2. So we have the statement W c 2, which is due to Weibel. Then we have the important result that it holds for varieties of characteristics 0. So for x over some field k of characteristics 0, a finite type, and this is a result due to Coutini's Heismeyer-Schlichting and Weibel. And then also, if we assume very strong form of resolution of singularities in positive characteristic of a perfect field, we also get the Weibel conjecture there for such x. So k is a perfect field. And we assume that exactly the same statement as proved by Hironaka in vertex 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 it can come to Gabba's question. Namely, you can also ask the same conjecture for homotopy k theory. Let's just mention this shortly. So homotopy k theory is the following k theory defined by Weibel. First of all, consider the standard algebraic J-simplex over the integers, which is just back z. Then you take polynomial ring and variables t0 to tj, modulo ideal. And then homotopy k theory 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 core limit. Now when I write k of a scheme by this, I mean the non-connective k theory spectrum. And now you just take the homotopy core limit over this simplisher 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 Weibel's conjecture the k group by the homotopy k group, then this is true. So I have a conjecture kh. So I think this was Gavas, your question. And second, so what can we say for k theory? So the results are not as complete. So we can prove the Weibel conjecture for the scheme x and the following assumption. First of all, you need some sort of fineness assumption. Namely, if all the residue fields of positive characteristic are Frobenius finite, let's say x is a point of capital x of positive characteristic, 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 single lattice holes, which I want to call res, and I which 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. So 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-excellent smoothies for the resolution by some channel. No, no. I'll come to this. And there is some way to reduce it to quasi-excellent schemes, in particular, so to say, a special point of this part 2 is part 3. Namely that the Weibel conjecture holds for all schemes of characteristic 0. For any scheme containing q. So in particular, this is mostly due to Matthew Moore, so with some contribution by myself. OK, so now, Gaba already mentioned, you have to reduce somehow this kind of Weibel conjecture to a statement with better schemes, like quasi-excellent 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 first yours, your x was a very nice scheme. Yes, it's always a very nice scheme. No, no. It's always a scheme of this form. Yeah. The top of the board, it's a very projective over some. Yeah, this is, when I say a schema always means such a good schema. You can generalize it, of course, to more general schemes, but then you must not work with this kind of k-fear, but k-fear of perfect complex or something like this. So even for these schemes, you need reductions. Yeah, the reductions, I'm coming now. So yeah, let me explain what I mean by reductions. So first of all, you have the following kind of reduction. Namely, if I know, let's say x is any scheme, if I know the Weibel conjecture for all the stocks of the structure sheaf, then I know the Weibel conjecture for x. So the idea is very simple. Let me explain it quickly. So for this kind of k-fear, we have a Tzelski descent spectral sequence of the following form. The E2 term is just the Tzerski homology of the Tzerski sheafification of this non-connective k-free rubes. This converges then to k minus p minus q of x. And now, if you assume that you have this Weibel conjecture, you get some vanishing of this sheaf at certain stocks, depending on the dimension of this point. And then you have here Tzerski homology, and then there's this sort of vanishing result of Grotendieg, which tells you that Tzerski homology vanishes for low-dimensional schemes. But you can somehow generalize it, or give a version of this, which works also for such any sheaves, which have, so to say, low-dimensional support, roughly speaking. So then you'll see that for minus p minus q smaller than the dimension of x, assuming this here, this actually vanishes. And then you use Grotendieg vanishing for this. So this is really quite simple. The next reduction allows us to pass to excellent schemes in particular. So let's say D is some non-negative fixed integer. Then you have the following reduction. If we know the viable 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 viable conjecture for all schemes of dimension up to D. So to prove the viable 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, is not very complicated. So we prove 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 the maximal ideal. And I denote by 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 k-thever support of Thomas and Troubault. Take k-a 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 Troubault for k-thever 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 our 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 a solution of singularities because here on Narkaproof, this for quasi-excellent local rings. So the basic idea now in all proofs so that I know of Vibals 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 blow-ups, let's say. So this is something I want to explain now. And this is motivated, especially again, by some work of Matthew Moro. 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 out y prime to x without y. We call such a thing an abstract blow-up square. Because 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-feural spectrum of the formless scheme associated formless scheme y had to be the homotopy limit over all n of the k-feural spectrum of yn. And then the k groups itself are just, of course, defined as the homotopy groups of this spectrum. Now I can just apply a k-fury to my square, more or less. Consider the following square. Let me apply the k-fury spectrum to x. Then I consider the k-fury spectrum of this formless scheme y had. Then I consider the k-fury spectrum of the analogous formless scheme y prime had. And then, oh, sorry, this goes in this direction. Here consider 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 blob 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 lobs 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 Suslind Wojcicki on the excision for k-fury and some work of Matthew Moro. Another example is where we know this. So in this case, we know dx. Another case is if x prime is the blow up of x in the closed sub-scheme y. And if y to x is a regular closed immersion. So then we also know for this special case, we know, OK, this is not right. I should say like this, OK. Then we know it for this dx prime and this y. And this was shown by Thomason. Thomason didn't consider the completion. He just did it. And so you have to do the same work for the completion. But in the completion, it's proof. I mean, of course, the powers are not regularly immersed. But I mean, if it's f-hine, then you can set up the powers. You can just take a different powers off. I mean, you just a different system of ideals, which are, I mean, in the equivalent, yeah? OK. And these are regularly met. Yeah, yeah, exactly, yeah. 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. It's called a double star. And it says that we know this descent conjecture for k through 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 singletties 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 singletties. It's some sort of principalization of ideal sheets. So let's start, let's recall the following definition of hieronaca, a closed immersion of schemes y into x. We say that this immersion is normally pseudo-flat by definition. Let's say, OK, 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 blow-off of x and y, then I ask that all the fibers of this blob of y have the same dimension. And the dimension necessarily has to be the co-dimension of y and x. You mean all fibers over one? Ah. But do you allow lower dimensional components in the fiber or there should be fewer of these dimensions? I don't care. I don't care. So this is hieronaca definition. Maybe this can happen. I don't have an example. So there is least of this dimension and then you require that or not more? No, no. This is a buff bound from a buff. So the dimension could, the fibers always has to be at least this co-dimension. But I don't want that 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 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 a factorization, sorry, and such that the pullback of Z to X tilde is defined by a locally principal ideal sheaf. You don't require that the blow-ups are above Z. They can be elsewhere. They 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 just state some partial results. What's the conclusion of TRN in double-stop? The conclusion is that this descent property for k-fury holds for any abstract blob 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 Kuhl-McOllification, which was developed by Fultings. 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 zero. 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 generality, namely this kind of a solution problem. Now for some fixed scheme X that we write it like this and any closed non-land subscheme Z can be proved up to dimension three. It can also be proved for the closed subscheme Z and X if the dimension of Z is less than or equal to one. And actually I somehow I think I learned this from a letter of Gabber to Elenino, which deals with a similar question. Okay, so now this was just preliminary, so we want to come back to the proof of Weibels' 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 it to the second part, namely this here, where you have this resolution assumption and this four-binius fineness using this theorem double star. Actually, so this is quite nice. As I said, we need some vanishing, something like vanishing for regular schemes, then we need some descent. Okay, so let me first explain, we cannot really resolve singularities, so we cannot stick to this fundamental theorem about the vanishing of k-phere 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-phere 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- or projective biational map, such that the pullback p up a star of psi in k minus j of x prime, this vanishes. Okay, 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. But so this is a replacement for, already one replacement for resolution of singularities. And the key thing that we will use is Renaud-Crescent Platification par Eclatmore. So one preliminary remark is the following, namely there exists a natural surjection to the group k minus j of x. So explain to you that this comes from this deloping, 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 k zero group of this scheme. So at least there exists now a surjection from k zero of x times gm to the j. What you do the image in this k zero group of k zero of x times aj. At least this is mod out, maybe more, but we don't care for the moment. We just consider the 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 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. Now Renault-Crescent's Platification par Éclatement tells you the following. Namely I can find a blow-up in x without u, which makes somehow this v bar flat. More precisely, they tell us the following. Namely, there exists some projective compression 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 a 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 pre-image of u. 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 torque 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. Okay, but any coherent sheaf with finite torque dimension gives rise to some class in K0, as shown by Kotnik, but just resolving it by vector buttons, finite resolution by vector buttons. 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 pullback of my original element psi. So then this restricts to p upper star of psi. 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 psi 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-protective over complete local rings. So from now on in this proof by a scheme I always mean a scheme which is quasi-protective 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 psi in Ki of x where i is less than minus the dimension of x. So we want to show that psi vanishes. So I want to stick now to the first part of y-ables conjecture not the homotopy environments for simplicity. Now the key proposition tells us I mean first of all we can easily reduce to x I skip this we can then find some modification which kills this psi. Choose a modification p prime from x prime to x such that the pullback of this psi under p vanishes. Now because this is projective byrush I definitely find a closed non-dance sub-scheme y of x such that this p is an isomorphism outside y. Now we can consider the following diagram of exact rows or 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 psi 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 this somehow this non-reduced structure which is sitting in this formal scheme doesn't change k-fury so this is some easy it's just based on the observation at the end that k0 of a ring does not depend on new potent elements so I can just mode out new potent 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-fury of y prime and also by our induction assumption this vanishes okay and this means of course that I can forget about this map here and say this is injective but I already know that psi goes to 0 here so I deduce that psi is 0 so this is the basic idea so we have you need this property just for the blow-up which is given by Renaud Grison maybe you are asking him too much so you will apply it for this blow-up yes to find this x prime I have to use this so classically the resolution of singlet is was used twice in all these approaches in k0 say this dx is even just here it means you will apply it just for the square given by Renaud Grison no no you need this condition this resolution in order to prove first of all descent for k-fury and then classically instead of Renaud Grison people used resolution of singlet is here so classically resolution of singlet is is proved used in k0 for example in the work of Fibre, 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 ok so let me now say just a few words about this how we can obtain this descent for algebraic k-fury so this theorem double star so there are two ingredients this is an idea going back to Cortinius here's my arrival and so on this group of people namely we split up k-fury into some linear part and some part which doesn't see the non-reduced structure ok so this works as follows so instead of the k-fury of x we consider the so called infinitesimal k-fury of x which is just defined as the homotopy fiber of the k-fury of x mapping via the trace map to topological cyclic homology spectrum of x and for this topological cyclic homology this kind of the analog of this kind of descent property for k-fury was proved by Dundas and Moran 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-fury and the important property of infinitesimal k-fury is that k-inf of x is the same thing as k-inf or as weakly equivalent to the k-inf of the associated reduced scheme this is the result of Goodwill and McAfee so this is the weak equivalence so this is the most important thing we want to use ok now for this k-inf I can apply some very general kind of descent result which generalizes work of Hezemeier so I want to explain now some exumps about some homology theory of schemes which guarantee that this homology theory satisfies something like descent along abstract blow-up squares the original version was in Hezemeier'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 Resolution, principalization of ideals so now we work over some fixed complete local ring this is fixed and we consider a homology 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 blobs 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 transition 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 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 in in particular the next thing we need is Meijer-Vetourez for finite surface coverings and then we need this kind of descent for regular blow-ups in the sense of Thomason I can expand 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 blow-up square X prime mapping to X and containing here this closed sub-scheme Y and Y prime the following square is homotopic Cartesian H of X mapping to H of Y 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 on the residue-filling characteristic pre-aving of a various finite enter does it enter here or here I explain to you we know this resolution in Cartesian 0 I'm saying you had a special assumption in theorem star-star about arm-o-den for benus-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 in order to prove this kind of descent in certain situations for topological cyclic homology this is for benus-finite 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 blobs for this H theory means that this is a homotopy pullback square and by descent for regular blobs of course mean that if X prime is a blob of X inside this Y where Y is regularly immersed you have this kind of blow-up square and while his a Meier's approach to a result like this is I mean very involved I mean he makes really full use of the strength of resolution of single lattice whereas I find a simplification of his approach somehow you just if you have any abstract blow-up X prime to X then somehow you just dominated by this resolution assumption by blow-ups and 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 and then you can just consider this ideal defining this Y you find a so-called reduction of this ideal defined, constructed by Northcott Ries which is generated just by co-dimension of Y and X many elements and if you make some 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 any questions? what about Homotopy K theory? so I think people I remember in some talks in 2004, people considered I don't remember, I think it was maybe the result of Wussamer but I remember they considered Homotopy K theory and you claim that for Homotopy K theory you don't need the resolution assumption okay so how does it work? okay so the point about Homotopy 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 Homotopy 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 key proposition can also easily prove a variant for Homotopy K theory the next thing is you need some sort of descent result for Homotopy K theory, but for Homotopy K theory we know exactly descent along abstract blobs squares in this sense because it was proved by Chisinski using completely different techniques so for Homotopy K theory we have descent along abstract blobs and we don't need to do all this complicated things that sits inside this proposition what kind of information is negative for the K theory views on the classical singularity for surfaces for surfaces you have two possible non vanishing groups, so K-2 and K-1 for example if you take K-2 of a normal surface, let's say you have 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 K-1 is more complicated but the thing one can say what it means if you have a dimension D scheme and you consider K-D of this 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 billion loop. Yes, well that is conjecture, I guess. But not the K-1, it's not finite type. K minus the dimension. Minus the dimension, yes. K-1 contains the three loops of H2 of star, which is very big. So there exists some description of this singularity by viable of this K-1 in terms of some p-car groups. But can you prove that K minus the dimension is finite in general in some new cases? I mean in the old method that's a little short term for free if I remember correctly. You mean now in characteristic zero whether one can prove it for example or assuming resolution of singularities? I'm not sure I didn't think about it, but I suppose you could prove it if you assume resolution of singularities. Questions? Then we meet again in 15 minutes.