On the vanishing of negative K-theory
Kerz, Moritz
Weibel's conjecture predicts that negative algebraic K-theory vanishes in degrees less than minus the dimension of the ring. The conjecture is known in characteristic zero. In the talk I will explain an approach which reduces the general conjecture to a very weak form of resolution of singularities.
 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 Moro. OK, so in this talk by a scheme X, I always mean a scheme which is quasi-projective over a netherian ring of finite-grille 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 idea is 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 pock 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 minus 1. And then this goes to X times gm with the natural map. And then you take the co-coil 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. This is the same co-coil, 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, while due to Bess and in this form Quillen and also I think Hohten-Diek. And this is the following. Namely, first of all, that if you apply this lambda construction to positive k theory defined as defined by Quillen, then you get, I mean, you could just get k i minus 1. So for i bigger than 0, 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 you just vanishes. So for X regular, it is true that k i of X vanishes for i less than 0. 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 cool dimension, these negative k groups finally vanish. And also precisely, k i of X is 0 if i is less than minus the cool dimension of X. And there's also a second part of this conjecture, namely that k i when i is equal to the dimension of minus the dimension of X, then this is homotopy invariant. So k i of X mapping to k i of X times aj, 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 leave the same conjecture for this. Yeah, let me come to this. This is, 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 Weibel showed this conjecture up to dimension two. So we have the statement Wc2, which is due to Weibel. Then we have the important result that it holds for varieties of characteristic zero. So for X over some field k of characteristic zero, a finite type. And this is a result due to Kortini's Hezemeier, Schlichting and Weibel. And then also if we assume very strong form of resolution of singularities and positive characteristic over 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 Heronaka in characteristic zero 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, sorry, to Gabba's question. Finally you can also ask the same conjecture for homotopy k3. 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-symplex over the integers, which is just back the z, then you take polynomial ring and variables t0 to tj, modulo, this 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 homotopic 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 Weibel's conjecture kh. So I think this was Gabbas' 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 the point of capital X of positive characteristic, then Frobenius is the 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 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 solution for this resolution, right? No, no, I'll come to this. And there is some way to reduce it to quasi-excellent schemes, in particular, sort of a special point of this part two is part three, namely that the Vibre conjecture holds for all schemes of characteristic zero. So for any scheme containing Q. So in particular, this is mostly due to Matthew Moore. So with some contribution to myself, by myself. OK, so now, Gabba already mentioned, you have to reduce somehow this kind of Vibre conjecture to a statement for 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 mean, I couldn't find them anywhere in the literature. It's quite interesting. But first yours, your X was a very nice scheme. Yes, it's always a very nice. No, no, it's always a scheme of this form. The top of the board. So it has a projective over some. Yeah, this is when I say 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 the reductions. Yeah, the reductions, I'm coming now. So let me explain what I mean by reductions. So first of all, you have the following kind of reduction. Namely, if I now, let's say X is any scheme, if I now the Vibre conjecture for all the stalks of the structure sheath, then I know the Vibre conjecture for X. So the idea is very simple. Let me explain it quickly. So for this kind of K-fear, we have a Telski descent spectral sequence of the following form. The E2 term is just the Telski homology of the Telski sheafification of this non-connective K-free loops. This converges then to K minus P minus Q of X. And now if you assume that you have this Vibre conjecture, you get some vanishing of this sheath at certain stalks, depending on the dimension of this point. And then you have here Telski homology, and then there's this sort of vanishing result of Grotenlieg, which tells you that Telski homology vanishes for low-dimensional schemes. But you can somehow generalize it or give a version of this, which works also for such any sheaths which have, so to say, 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 Grotenlieg 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 Vibre conjecture, now for certain rings, R, which are complete local rings. R is the ring of dimension at most D. For all such rings, then we actually know the Vibre conjecture for all schemes of dimension up to D. So to prove the Vibre conjecture, in other words, you just have to prove it for complete local rings. In particular, they are excellent. And 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 idea. 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-thiever support of Thomas and Toboe. Take kI plus 1 of X without the closed point. Then we take kI of X support in the closed point. Then we take kI of X and then we take kI 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 Toboe for k-thiever 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 1 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 single lattice because here on our proof this for quasi-excellent local rings. So the basic idea now in all proofs so that I know of Weibels 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 single lattice 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. Of course if you blow up X and Y then this would be a special case. Now to formulate this descent result, so let's say Y is defined by the coherent ideal shift I, then let us set Yn to be equal to the closed sub-scheme defined by the coherent ideal shift I to the n. And let us define the k-fury spectrum of the formless scheme associated formless 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. Now I can just apply a k-fury to my square more or less, consider the following square, namely apply the k-fury spectrum to X. Then I consider the k-fury spectrum of this formless scheme Y hat. Then I consider the k-fury spectrum of the analogous formless scheme Y prime hat. 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 notice, 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 this, for this special case we know, I should say like this, OK. Then we know it for this dx. X prime and this Y. And this was shown by Thomas. Thomas didn't consider the completion. He just did it. So you have to do the same work for the completion. Yeah, yeah, 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 power, I mean, just a different system of ideals which are, I mean, in the equivalent, yeah. OK, and it's regularly. 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. It's called a double star and it says that we know this descent conjecture for k-phil 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 singlet is 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. And rather than resolution of singlet is it some sort of principalization of ideal sheets. So let's start, let's recall the following definition of Hieronaca, the 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 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. I don't care. I don't care. So this is Hieronaca's definition. Maybe this can happen. I don't have an example. No, no, this is a buff found 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 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 subscheme 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 a locally principal, is defined by a locally principal ideal sheaf. You don't require that the blocks are above Z. They can be elsewhere. And 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 subscheme 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 the first term? 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 for binius finite, and you have this resolution assumption. So this kind of resolution problem is actually much closer to a process called a Kuhl-Mech olification, 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 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, mod generality, namely this kind of resolution problem. Now for some fixed scheme X, let me write it like this, and any closed non-land subscheme Z can be proved up to dimension 3. It can also be proved for the closed subscheme Z and X if the dimension of Z is less than or equal to 1. Actually, I think I learned this from a letter of Gabba to Elenino, which deals with a similar question. So now this was just preliminary, so we want to come back to the proof of Fibre's conjectures under certain assumptions. 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 Frobenius finalness using this theorem double star. Actually, so this is quite nice. So as I said, we need some vanishing, something like vanishing for regular schemes, then we need some descent. So let me first explain, we cannot really resolve singularities, so we cannot stick to this fundamental theorem about the vanishing of k-fierre of regular schemes. So here's a replacement that's quite useful, let's call it the key proposition. So consider a reduced scheme, and let's consider a k-fierre 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 bi-ational map, such that the pullback p-apa 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. But so this is a replacement for already one replacement for resolution of singularities. And the key thing that we will use is renault-crescent, platification by 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 is 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 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 is represented by a vector bundle on x times gmj. Then as it 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 subscheme 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 that, I mean what I mean is of course that this sheaf is flat over this. Now Renault-Crescent's Platification by Eclatement tells you the following. Namely, I find a blow up in x without u, which makes somehow this v bar flat. More precisely, they tell us the fact that namely there exists some projective by 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 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 preimage 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 finite tor 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 tor dimension gives rise to some class in k0, as shown by Kotnik, but just resolving it by vector, but it's finite resolution by vector, but it's 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 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-protective over complete local rings. So from now on in this proof, by 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 xi in k i 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 Weibels conjecture, not the homotopy advance for simplicity. Now the key proposition tells us, I mean first of all we can easily reduce to x reduced, I skip this, we can then find some modification which kills this xi. So 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 byrational I definitely find a closed non-dent variance sub-scheme y of x such that this p is an isomorphism outside y. Now we can consider the following diagram with exact rows. Sorry not the diagram, 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 k i of x receives k i plus 1 of the formal scheme y prime hat and this goes to k i of x prime plus k i 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 k i 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 theory. So this is some easy, it's just based on the observation at the end that k 0 of a ring does not depend on the important elements. So I can just mod out the important elements without changing k 0 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. And 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 deduce that xi is 0. So this is the basic idea. So we have… Sometimes you need this property just for the blow up which is given by reno grisso. Sometimes 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 k 0. This dx is even just here. It means you will apply it just for the square which is given by reno grisso. No, no. You need this condition, this resolution in order to prove first of all descent for k theory. And then classically instead of reno croissant people used resolution of singlet is here. So classically resolution of singlet is proved… used in k 0 for example in the work of Fierbeck or Kortiniessenssohn in two different ways. One to obtain descent and one to prove this kind of key proposition that I have here. Okay, so let me now say just a few words about this how we can obtain this descent for algebra k theory. So this theorem double it up. So there are two ingredients. This is an idea going back to Kortiniessenssohn here is my 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. Okay, so this works as follows. So instead of the k-pho of x, we consider the so called infinitesimal k theory of x which is just defined as the homotopy fiber of the k-pho 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 theory was proved by Dundas and Moral. 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 Goodwill and MacAfee. So this is the weak equivalent. So this is the most important thing we want to use. Okay, 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 axioms 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 the 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 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 subscheme defined by some ideal I of J maps as a weak equivalence to H of spec A prime relative to the closed subscheme 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 in particular. The next thing we need is Meier-Vetouris for finite Zyrzki 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 subscheme 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. Where does the assumption on the residue field characteristic P of a radius finite enter? Does it enter here or? Here, I mean, I explained to you we know this resolution in characteristic zero. No, I'm saying you have this special assumption in theorem star-star about arm-o-den for P of a radius finite. And when you discuss now the general formalism for obtaining this, you did not pull 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 for P of a radius 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. But okay, so the point is that now descent under abstract blobs for this H field 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, then you have this kind of blob square. And while his amias approach to result like this is, I mean, very involved, I mean, 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 blow up X prime to X, then somehow you just dominate it by this resolution assumption by blow ups and normally pseudo flat centers. So then by some simple trick, you're 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 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 Wusser Merz, but I remember they considered homotopy K theory. And you claim that for homotopy K theory, you don't need the solution. 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. The key proposition can also, you can 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 lobs squares in this sense, because it was proved by Chisinski using completely different techniques. So for homotopy K theory, we have descent along abstract lobs, and we don't need to do all these complicated things that sits inside this proposition. Okay, yeah, so for surfaces you have two possible non-vanishing groups, so K minus 2 and K minus 1. So for example, if you take K minus 2 of a normal surface, let's say you have, then it's just an isolated singularity, and then in general, 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 1 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 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 if this finite type of real loop. Yes, yeah, well that's conjecture I guess. But not the K minus 1 is not finite. K minus dimension. K minus dimension. Yes. K minus 1 contains the three groups of H2O stuff, which is very, very big. Yeah, so there exists some description of a normal surface singularity by Weibel, of this K minus 1 in terms of some peak HAR groups. But can you prove that K minus the dimension is finite in general, if there's anything you can say? I mean, in the whole matter, that's sort of a short term for free, if I can't prove it correctly. You mean now in cathode 6.0, whether one can prove it, for example, or assuming resolution of singularities? Yeah, come and work the states of the oldest also, if I just wanted to get in many cases. I'm not sure, I didn't think about it, but I suppose you could prove it if you assume resolution of singularities. Which are the questions? Then we meet again in 15 minutes.
 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 Moro.  OK, so in this talk by a scheme X, I always mean a scheme which is quasi-projective over  a netherian ring of finite-grille 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 idea is 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 pock 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 minus 1.  And then this goes to X times gm with the natural map.  And then you take the co-coil 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.  This is the same co-coil, 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, while due to Bess and in this form  Quillen and also I think Hohten-Diek.  And this is the following. Namely, first of all, that if you apply this lambda construction  to positive k theory defined as defined by Quillen, then you get, I mean, you could just  get k i minus 1.  So for i bigger than 0, 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 you just vanishes.  So for X regular, it is true that k i of X vanishes for i less than 0.  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 cool dimension, these negative k groups finally  vanish.  And also precisely, k i of X is 0 if i is less than minus the cool dimension of X.  And there's also a second part of this conjecture, namely that k i when i is equal to the dimension  of minus the dimension of X, then this is homotopy invariant.  So k i of X mapping to k i of X times aj, 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 leave the same conjecture for this.  Yeah, let me come to this.  This is, 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 Weibel showed this conjecture up to dimension  two.  So we have the statement Wc2, which is due to Weibel.  Then we have the important result that it holds for varieties of characteristic zero.  So for X over some field k of characteristic zero, a finite type.  And this is a result due to Kortini's Hezemeier, Schlichting and Weibel.  And then also if we assume very strong form of resolution of singularities and positive  characteristic over 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 Heronaka in characteristic zero  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, sorry, to Gabba's question.  Finally you can also ask the same conjecture for homotopy k3.  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-symplex over the integers, which is just  back the z, then you take polynomial ring and variables t0 to tj, modulo, this 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 homotopic 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 Weibel's conjecture kh.  So I think this was Gabbas' 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 the point of capital X of positive characteristic, then Frobenius is  the 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 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 solution for this resolution, right?  No, no, I'll come to this.  And there is some way to reduce it to quasi-excellent schemes, in particular, sort of a special  point of this part two is part three, namely that the Vibre conjecture holds for all schemes  of characteristic zero.  So for any scheme containing Q. So in particular, this is mostly due to Matthew Moore.  So with some contribution to myself, by myself.  OK, so now, Gabba already mentioned, you have to reduce somehow this kind of Vibre conjecture  to a statement for 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 mean, I couldn't find them anywhere in the  literature.  It's quite interesting.  But first yours, your X was a very nice scheme.  Yes, it's always a very nice.  No, no, it's always a scheme of this form.  The top of the board.  So it has a projective over some.  Yeah, this is when I say 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 the reductions.  Yeah, the reductions, I'm coming now.  So let me explain what I mean by reductions.  So first of all, you have the following kind of reduction.  Namely, if I now, let's say X is any scheme, if I now the Vibre conjecture for all the  stalks of the structure sheath, then I know the Vibre conjecture for X.  So the idea is very simple.  Let me explain it quickly.  So for this kind of K-fear, we have a Telski descent spectral sequence of the following  form.  The E2 term is just the Telski homology of the Telski sheafification of this non-connective  K-free loops.  This converges then to K minus P minus Q of X.  And now if you assume that you have this Vibre conjecture, you get some vanishing of this  sheath at certain stalks, depending on the dimension of this point.  And then you have here Telski homology, and then there's this sort of vanishing result  of Grotenlieg, which tells you that Telski homology vanishes for low-dimensional schemes.  But you can somehow generalize it or give a version of this, which works also for such  any sheaths which have, so to say, 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 Grotenlieg 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 Vibre conjecture, now for certain rings, R, which are complete local rings.  R is the ring of dimension at most D. For all such rings, then we actually know the  Vibre conjecture for all schemes of dimension up to D.  So to prove the Vibre conjecture, in other words, you just have to prove it for complete  local rings.  In particular, they are excellent.  And 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 idea.  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-thiever support of Thomas and Toboe.  Take kI plus 1 of X without the closed point.  Then we take kI of X support in the closed point.  Then we take kI of X and then we take kI 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 Toboe for k-thiever 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 1 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 single lattice because here on our proof this for quasi-excellent  local rings.  So the basic idea now in all proofs so that I know of Weibels 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 single lattice 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.  Of course if you blow up X and Y then this would be a special case.  Now to formulate this descent result, so let's say Y is defined by the coherent ideal  shift I, then let us set Yn to be equal to the closed sub-scheme defined by the coherent  ideal shift I to the n.  And let us define the k-fury spectrum of the formless scheme associated formless 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.  Now I can just apply a k-fury to my square more or less, consider the following square,  namely apply the k-fury spectrum to X. Then I consider the k-fury spectrum of this formless  scheme Y hat.  Then I consider the k-fury spectrum of the analogous formless scheme Y prime hat.  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 notice, 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 this, for this special  case we know, I should say like this, OK. Then we know it for this dx.  X prime and this Y. And this was shown by Thomas.  Thomas didn't consider the completion. He just did it. So you have to do the same work  for the completion.  Yeah, yeah, 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 power, I mean, just a different system of ideals which are, I mean, in the  equivalent, yeah.  OK, and it's regularly.  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. It's called a double  star and it says that we know this descent conjecture for k-phil 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 singlet is 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.  And rather than resolution of singlet is it some sort of principalization of ideal sheets.  So let's start, let's recall the following definition of Hieronaca, the 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 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. I don't care. I don't care. So this is Hieronaca's  definition. Maybe this can happen. I don't have an example.  No, no, this is a buff found 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 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 subscheme 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 a locally principal, is  defined by a locally principal ideal sheaf.  You don't require that the blocks are above Z. They can be elsewhere.  And 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 subscheme 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 the first term?  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 for binius finite, and you have this resolution assumption.  So this kind of resolution problem is actually much closer to a process called a Kuhl-Mech  olification, 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 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, mod generality, namely this kind of resolution problem.  Now for some fixed scheme X, let me write it like this, and any closed non-land subscheme  Z can be proved up to dimension 3. It can also be proved for the closed subscheme Z  and X if the dimension of Z is less than or equal to 1. Actually, I think I learned this  from a letter of Gabba to Elenino, which deals with a similar question.  So now this was just preliminary, so we want to come back to the proof of Fibre's conjectures  under certain assumptions. 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 Frobenius finalness using this theorem double  star. Actually, so this is quite nice. So as I said, we need some vanishing, something  like vanishing for regular schemes, then we need some descent. So let me first explain,  we cannot really resolve singularities, so we cannot stick to this fundamental theorem  about the vanishing of k-fierre of regular schemes. So here's a replacement that's quite  useful, let's call it the key proposition. So consider a reduced scheme, and let's consider  a k-fierre 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 bi-ational map, such that  the pullback p-apa 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.  But so this is a replacement for already one replacement for resolution of singularities.  And the key thing that we will use is renault-crescent, platification by 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 is 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 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 is represented by a vector bundle on x times gmj. Then as  it 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 subscheme  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 that, I mean what I mean is of course that this sheaf is flat over  this. Now Renault-Crescent's Platification by Eclatement tells you the following. Namely,  I find a blow up  in x without u, which makes somehow this v bar flat. More precisely, they tell us the  fact that namely there exists some projective by 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 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 preimage 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 finite tor 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 tor dimension gives rise  to some class in k0, as shown by Kotnik, but just resolving it by vector, but it's finite  resolution by vector, but it's 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 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-protective over complete local rings.  So from now on in this proof, by 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 xi in  k i 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 Weibels conjecture, not the homotopy advance  for simplicity. Now the key proposition tells us, I mean first of all we can easily reduce  to x reduced, I skip this, we can then find some modification which kills this xi. So  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 byrational I definitely find a closed non-dent  variance sub-scheme y of x such that this p is an isomorphism outside y.  Now we can consider the following diagram with exact rows.  Sorry not the diagram, 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 k i of x receives k i plus 1 of the formal scheme y prime hat  and this goes to k i of x prime plus k i 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 k i 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 theory.  So this is some easy, it's just based on the observation at the end that k 0 of a ring  does not depend on the important elements. So I can just mod out the important elements  without changing k 0 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. And 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 deduce that xi  is 0. So this is the basic idea. So we have…  Sometimes you need this property just for the blow up which is given by reno grisso.  Sometimes 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 k 0.  This dx is even just here. It means you will apply it just for the square which is given  by reno grisso. No, no. You need this condition, this resolution  in order to prove first of all descent for k theory. And then classically instead of reno  croissant people used resolution of singlet is here. So classically resolution of singlet  is proved… used in k 0 for example in the work of Fierbeck or Kortiniessenssohn in two  different ways. One to obtain descent and one to prove this kind of key proposition that  I have here. Okay, so let me now say just a few words  about this how we can obtain this descent for algebra k theory. So this theorem double  it up. So there are two ingredients. This is an idea going back to Kortiniessenssohn  here is my 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. Okay, so this  works as follows. So instead of the k-pho of x, we consider the so called infinitesimal  k theory of x which is just defined as the homotopy fiber of the k-pho 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 theory was proved by Dundas and Moral. 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 Goodwill and MacAfee. So this is the weak equivalent.  So this is the most important thing we want to use. Okay, 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 axioms 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 the 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 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 subscheme defined by some ideal I of  J maps as a weak equivalence to H of spec A prime relative to the closed subscheme 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 in particular. The next  thing we need is Meier-Vetouris for finite Zyrzki 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 subscheme 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.  Where does the assumption on the residue field characteristic P of a radius finite enter?  Does it enter here or?  Here, I mean, I explained to you we know this resolution in characteristic zero.  No, I'm saying you have this special assumption in theorem star-star about arm-o-den for  P of a radius finite. And when you discuss now the general formalism for obtaining this,  you did not pull 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 for P of a radius  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.  But okay, so the point is that now descent under abstract blobs for this H field 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, then  you have this kind of blob square. And while his amias approach to result like this is,  I mean, very involved, I mean, 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 blow up X prime to X, then somehow you just dominate it by this resolution  assumption by blow ups and normally pseudo flat centers. So then by some simple trick,  you're 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 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 Wusser  Merz, but I remember they considered homotopy K theory. And you claim that for homotopy K  theory, you don't need the solution. 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. The key  proposition can also, you can 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 lobs squares in this sense, because it was  proved by Chisinski using completely different techniques. So for homotopy K theory, we have  descent along abstract lobs, and we don't need to do all these complicated things that  sits inside this proposition. Okay, yeah, so for surfaces you have two possible non-vanishing  groups, so K minus 2 and K minus 1. So for example, if you take K minus 2 of a normal  surface, let's say you have, then it's just an isolated singularity, and then in general,  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 1 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 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 if this finite type of real loop. Yes, yeah, well that's conjecture I guess. But  not the K minus 1 is not finite. K minus dimension. K minus dimension.  Yes. K minus 1 contains the three groups of H2O stuff, which is very, very big.  Yeah, so there exists some description of a normal surface singularity by Weibel, of  this K minus 1 in terms of some peak HAR groups.  But can you prove that K minus the dimension is finite in general, if there's anything  you can say? I mean, in the whole matter, that's sort of a short term for free, if I  can't prove it correctly. You mean now in cathode 6.0, whether one can prove it, for  example, or assuming resolution of singularities? Yeah, come and work the states of the oldest  also, if I just wanted to get in many cases. I'm not sure, I didn't think about it, but  I suppose you could prove it if you assume resolution of singularities.  Which are the questions? Then we meet again in 15 minutes.
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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 Moro. OK, so in this talk by a scheme X, I always mean a scheme which is quasi-projective over a netherian ring of finite-grille 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 idea is 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 pock 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 minus 1. And then this goes to X times gm with the natural map. And then you take the co-coil 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. This is the same co-coil, 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, while due to Bess and in this form Quillen and also I think Hohten-Diek. And this is the following. Namely, first of all, that if you apply this lambda construction to positive k theory defined as defined by Quillen, then you get, I mean, you could just get k i minus 1. So for i bigger than 0, 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 you just vanishes. So for X regular, it is true that k i of X vanishes for i less than 0. 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 cool dimension, these negative k groups finally vanish. And also precisely, k i of X is 0 if i is less than minus the cool dimension of X. And there's also a second part of this conjecture, namely that k i when i is equal to the dimension of minus the dimension of X, then this is homotopy invariant. So k i of X mapping to k i of X times aj, 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 leave the same conjecture for this. Yeah, let me come to this. This is, 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 Weibel showed this conjecture up to dimension two. So we have the statement Wc2, which is due to Weibel. Then we have the important result that it holds for varieties of characteristic zero. So for X over some field k of characteristic zero, a finite type. And this is a result due to Kortini's Hezemeier, Schlichting and Weibel. And then also if we assume very strong form of resolution of singularities and positive characteristic over 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 Heronaka in characteristic zero 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, sorry, to Gabba's question. Finally you can also ask the same conjecture for homotopy k3. 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-symplex over the integers, which is just back the z, then you take polynomial ring and variables t0 to tj, modulo, this 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 homotopic 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 Weibel's conjecture kh. So I think this was Gabbas' 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 the point of capital X of positive characteristic, then Frobenius is the 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 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 solution for this resolution, right? No, no, I'll come to this. And there is some way to reduce it to quasi-excellent schemes, in particular, sort of a special point of this part two is part three, namely that the Vibre conjecture holds for all schemes of characteristic zero. So for any scheme containing Q. So in particular, this is mostly due to Matthew Moore. So with some contribution to myself, by myself. OK, so now, Gabba already mentioned, you have to reduce somehow this kind of Vibre conjecture to a statement for 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 mean, I couldn't find them anywhere in the literature. It's quite interesting. But first yours, your X was a very nice scheme. Yes, it's always a very nice. No, no, it's always a scheme of this form. The top of the board. So it has a projective over some. Yeah, this is when I say 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 the reductions. Yeah, the reductions, I'm coming now. So let me explain what I mean by reductions. So first of all, you have the following kind of reduction. Namely, if I now, let's say X is any scheme, if I now the Vibre conjecture for all the stalks of the structure sheath, then I know the Vibre conjecture for X. So the idea is very simple. Let me explain it quickly. So for this kind of K-fear, we have a Telski descent spectral sequence of the following form. The E2 term is just the Telski homology of the Telski sheafification of this non-connective K-free loops. This converges then to K minus P minus Q of X. And now if you assume that you have this Vibre conjecture, you get some vanishing of this sheath at certain stalks, depending on the dimension of this point. And then you have here Telski homology, and then there's this sort of vanishing result of Grotenlieg, which tells you that Telski homology vanishes for low-dimensional schemes. But you can somehow generalize it or give a version of this, which works also for such any sheaths which have, so to say, 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 Grotenlieg 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 Vibre conjecture, now for certain rings, R, which are complete local rings. R is the ring of dimension at most D. For all such rings, then we actually know the Vibre conjecture for all schemes of dimension up to D. So to prove the Vibre conjecture, in other words, you just have to prove it for complete local rings. In particular, they are excellent. And 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 idea. 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-thiever support of Thomas and Toboe. Take kI plus 1 of X without the closed point. Then we take kI of X support in the closed point. Then we take kI of X and then we take kI 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 Toboe for k-thiever 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 1 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 single lattice because here on our proof this for quasi-excellent local rings. So the basic idea now in all proofs so that I know of Weibels 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 single lattice 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. Of course if you blow up X and Y then this would be a special case. Now to formulate this descent result, so let's say Y is defined by the coherent ideal shift I, then let us set Yn to be equal to the closed sub-scheme defined by the coherent ideal shift I to the n. And let us define the k-fury spectrum of the formless scheme associated formless 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. Now I can just apply a k-fury to my square more or less, consider the following square, namely apply the k-fury spectrum to X. Then I consider the k-fury spectrum of this formless scheme Y hat. Then I consider the k-fury spectrum of the analogous formless scheme Y prime hat. 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 notice, 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 this, for this special case we know, I should say like this, OK. Then we know it for this dx. X prime and this Y. And this was shown by Thomas. Thomas didn't consider the completion. He just did it. So you have to do the same work for the completion. Yeah, yeah, 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 power, I mean, just a different system of ideals which are, I mean, in the equivalent, yeah. OK, and it's regularly. 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. It's called a double star and it says that we know this descent conjecture for k-phil 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 singlet is 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. And rather than resolution of singlet is it some sort of principalization of ideal sheets. So let's start, let's recall the following definition of Hieronaca, the 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 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. I don't care. I don't care. So this is Hieronaca's definition. Maybe this can happen. I don't have an example. No, no, this is a buff found 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 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 subscheme 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 a locally principal, is defined by a locally principal ideal sheaf. You don't require that the blocks are above Z. They can be elsewhere. And 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 subscheme 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 the first term? 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 for binius finite, and you have this resolution assumption. So this kind of resolution problem is actually much closer to a process called a Kuhl-Mech olification, 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 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, mod generality, namely this kind of resolution problem. Now for some fixed scheme X, let me write it like this, and any closed non-land subscheme Z can be proved up to dimension 3. It can also be proved for the closed subscheme Z and X if the dimension of Z is less than or equal to 1. Actually, I think I learned this from a letter of Gabba to Elenino, which deals with a similar question. So now this was just preliminary, so we want to come back to the proof of Fibre's conjectures under certain assumptions. 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 Frobenius finalness using this theorem double star. Actually, so this is quite nice. So as I said, we need some vanishing, something like vanishing for regular schemes, then we need some descent. So let me first explain, we cannot really resolve singularities, so we cannot stick to this fundamental theorem about the vanishing of k-fierre of regular schemes. So here's a replacement that's quite useful, let's call it the key proposition. So consider a reduced scheme, and let's consider a k-fierre 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 bi-ational map, such that the pullback p-apa 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. But so this is a replacement for already one replacement for resolution of singularities. And the key thing that we will use is renault-crescent, platification by 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 is 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 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 is represented by a vector bundle on x times gmj. Then as it 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 subscheme 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 that, I mean what I mean is of course that this sheaf is flat over this. Now Renault-Crescent's Platification by Eclatement tells you the following. Namely, I find a blow up in x without u, which makes somehow this v bar flat. More precisely, they tell us the fact that namely there exists some projective by 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 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 preimage 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 finite tor 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 tor dimension gives rise to some class in k0, as shown by Kotnik, but just resolving it by vector, but it's finite resolution by vector, but it's 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 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-protective over complete local rings. So from now on in this proof, by 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 xi in k i 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 Weibels conjecture, not the homotopy advance for simplicity. Now the key proposition tells us, I mean first of all we can easily reduce to x reduced, I skip this, we can then find some modification which kills this xi. So 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 byrational I definitely find a closed non-dent variance sub-scheme y of x such that this p is an isomorphism outside y. Now we can consider the following diagram with exact rows. Sorry not the diagram, 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 k i of x receives k i plus 1 of the formal scheme y prime hat and this goes to k i of x prime plus k i 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 k i 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 theory. So this is some easy, it's just based on the observation at the end that k 0 of a ring does not depend on the important elements. So I can just mod out the important elements without changing k 0 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. And 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 deduce that xi is 0. So this is the basic idea. So we have… Sometimes you need this property just for the blow up which is given by reno grisso. Sometimes 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 k 0. This dx is even just here. It means you will apply it just for the square which is given by reno grisso. No, no. You need this condition, this resolution in order to prove first of all descent for k theory. And then classically instead of reno croissant people used resolution of singlet is here. So classically resolution of singlet is proved… used in k 0 for example in the work of Fierbeck or Kortiniessenssohn in two different ways. One to obtain descent and one to prove this kind of key proposition that I have here. Okay, so let me now say just a few words about this how we can obtain this descent for algebra k theory. So this theorem double it up. So there are two ingredients. This is an idea going back to Kortiniessenssohn here is my 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. Okay, so this works as follows. So instead of the k-pho of x, we consider the so called infinitesimal k theory of x which is just defined as the homotopy fiber of the k-pho 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 theory was proved by Dundas and Moral. 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 Goodwill and MacAfee. So this is the weak equivalent. So this is the most important thing we want to use. Okay, 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 axioms 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 the 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 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 subscheme defined by some ideal I of J maps as a weak equivalence to H of spec A prime relative to the closed subscheme 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 in particular. The next thing we need is Meier-Vetouris for finite Zyrzki 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 subscheme 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. Where does the assumption on the residue field characteristic P of a radius finite enter? Does it enter here or? Here, I mean, I explained to you we know this resolution in characteristic zero. No, I'm saying you have this special assumption in theorem star-star about arm-o-den for P of a radius finite. And when you discuss now the general formalism for obtaining this, you did not pull 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 for P of a radius 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. But okay, so the point is that now descent under abstract blobs for this H field 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, then you have this kind of blob square. And while his amias approach to result like this is, I mean, very involved, I mean, 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 blow up X prime to X, then somehow you just dominate it by this resolution assumption by blow ups and normally pseudo flat centers. So then by some simple trick, you're 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 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 Wusser Merz, but I remember they considered homotopy K theory. And you claim that for homotopy K theory, you don't need the solution. 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. The key proposition can also, you can 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 lobs squares in this sense, because it was proved by Chisinski using completely different techniques. So for homotopy K theory, we have descent along abstract lobs, and we don't need to do all these complicated things that sits inside this proposition. Okay, yeah, so for surfaces you have two possible non-vanishing groups, so K minus 2 and K minus 1. So for example, if you take K minus 2 of a normal surface, let's say you have, then it's just an isolated singularity, and then in general, 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 1 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 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 if this finite type of real loop. Yes, yeah, well that's conjecture I guess. But not the K minus 1 is not finite. K minus dimension. K minus dimension. Yes. K minus 1 contains the three groups of H2O stuff, which is very, very big. Yeah, so there exists some description of a normal surface singularity by Weibel, of this K minus 1 in terms of some peak HAR groups. But can you prove that K minus the dimension is finite in general, if there's anything you can say? I mean, in the whole matter, that's sort of a short term for free, if I can't prove it correctly. You mean now in cathode 6.0, whether one can prove it, for example, or assuming resolution of singularities? Yeah, come and work the states of the oldest also, if I just wanted to get in many cases. I'm not sure, I didn't think about it, but I suppose you could prove it if you assume resolution of singularities. Which are the questions? Then we meet again in 15 minutes.
by Heronaka in characteristic zero holds for I want to call res and which I will explain of Vibre conjecture to a statement for with that I is less than minus the dimension of X, of Weibels conjecture is somehow, first of all with this very weak form of resolution of of resolution of singularities over this ring R,