 It's a great pleasure to speak with this conference. And of course, I was also interested a lot by Maxim's results and ideas. And so I will speak about different versions of draft categories which are related to singularities and so on. And I will start, well, let's, for simplicity, fix a basic field to be complex numbers. And I recall that, yeah, and the schemes which we will consider will be of finite type, separated over k. And well, you may just assume it to be quite projective. Then I'll recall that we have the bounded draft category of coherent shifts, which I will just denote by db of x. And inside it, we have a category of perfect complexes, which by definition objects which locally have finite free resolutions. Well, if it's quite projective, you may, of course, consider global resolutions. Yeah, and it's basically known that this category of perfect complexes is, well, it's also a dg category. And it is smooth if and only if x is smooth. And it is proper if and only if x is proper. And I recall that if t is some dg category, then t is smooth if the diagonal by model is perfect. And t is proper if for any objects, the space of x from x to y is fine dimensional. Well, another way to say is that this diagonal by model should be sort of perfect. Yeah, this is classical. And also, a couple of years ago, I proved that for this category db of x, that then it is always quite nice. Namely, db of x is homotopically finely presented. Double the size? Well, yeah, I'll try. Homotopically finely presented. Well, I wouldn't like to define what it means in full generality. I'll just say that actually what I proved is that it can be represented as a quotient of some smooth and proper dg category by a subcategory generated by one object. So e is just some objects. And yeah, and this c is glued from draft categories of some smooth and proper varieties. And these are actually smooth centers which appear when you construct a resolution of a complication of x using blow ups with smooth centers. So well, maybe I should write that by Hiranaka, if you choose some complication of x, they could reduce parts. And then you have some sequence of blow ups such that xn is smooth. And each x i plus 1 is blow up of the previous one with some smooth centers. And these guys actually appear here, but maybe in different order and maybe with multiplicities. So this category in general has very nice properties. Now another general result, which is even more classical, is that for this category of perfect complexes, we have a localization theorem by Thomason, namely if z side x is closed, then we have a short exact sequence of triangulated categories. So perfect complexes supported on z. I mean, homology supported on z. Perfect complexes on x. And perfect complexes on x minus z. And well, yeah, this is exact sequence up to summons. And similarly, which is even easier, we have a similar sequence for draft categories of coefficients. And here it is just exact without even adding direct summons. So we have this localization, which in particular applies to key theory and to all kinds of homology, cyclic homology. And here we also have, it's known that periodic cyclic homology of category of perfect complexes is identified, well, up to parity just with the usual single homology of x with complex coefficients. Well, if instead of c, we have some field of characteristic 0, then here there will be crystalline homology. And so exact up to summons means? It means that the function from the quotient of this by this to here is fully faithful. And each object here is a direct summons of the image of something. Yeah, and here is just literally the image of something. Yeah, and so this is approved by Weibel. Well, for smooth axis, just an exercise. And for non-smooth, it requires some technique with this. Well, for a fine x, it was proved by Fagan and Sigan. And then there's some technique to globalize the result. And periodic cyclic homology of db of x is actually more homology of x, well, which is due to the homology with complex support. And this, I think it should be written by Preigl, maybe not yet published. So there's such a general computation. And these exact sequences are compatible with the usual exact sequence for homology, and more homology. OK, so yeah, another object which is also quite known and which appears naturally in mirror symmetry is the category of matrix factorizations. Namely, suppose that we have a regular function on x. First, assume that x is smooth. So xw is a Landau-Gisburg model, well, b part of Landau-Gisburg model, and then the associated category, the triangulate category associated to it is the category of matrix factorizations. And which is defined as follows. We take the objects are pairs of locally free shifts and maps from 0 to 1 and twice versa, such that both compositions are multiplication by our function, by w. And then if, yeah, so first, you have naive category, which obtained just as a homotopic category of such factorizations. And if x is not known and fine, you should take the quotient by convolutions of exact triples of factorizations. So here you have. So this factorization form also an abelian category, and we can take any exact, short exact sequence. Yeah, naive is just a homotopic category of such a case. And here, after quotient, you get the right category, which should be, actually. Yeah, it can be also defined in del bohr resolutions, basically, yeah. Well, yes, yes. But it's actually, yeah, it's like for perfect complex, you have different enhancements. Yeah, because it's quotient, and you have to calculate it. Actually, in every definition, it's maybe not so easy to calculate here. Yeah, you can calculate using del bohr or check enhancement. Yeah, it's as usual. Yeah, so we take quotient by convolutions of such exact triples, and so this is the right category of matrix factorizations. Yeah, and here there is a result of Orlov, which says that this category is equivalent to the quotient of, yeah, we take 0 fiber of our function, and this is the derived category of the 0 fiber quotient by perfect complexes. And well, yeah, here it's, let's assume that w is non-constant, otherwise, if w is 0, then you should just take dg fiber, it holds in this case. OK, and yeah, and just by definition for general scheme, why the category of singularities is defined to be this quotient? Yes, it's not correct. Yes, in general, it's not correctly closed. OK, and here we actually also have a notion, first we have a notion of support, which is maybe not so evident from the definition, but it's quite natural. Namely, if you have some object here, then support is just by definition smallest. It's smallest closed subset z in x such that f restricted to x minus z is a zero object in the category of maximizations of x minus z and w. And also one can show that there is a localization for such categories. So we have a short sequence of factorizations with support contained in z, all factorizations and factorizations on the complement. And here, of course, yeah, all these guys are not carbacomplete, but one can show that, well, it's actually a short exact, even without adding direct summons. So any factorization from here can be obtained as an image. OK, and yeah, and also for such categories, we have a computation that predicts that the homology is actually, well, under even Hilbert correspondence. Well, this is using Gessler-Gausmanian connection. You can make this guy into a local system on the form of punctured disk. And this will under even Hilbert correspondence will correspond to the homology of the shift of vanishing cycles, well, on x analytic with the usual monodrome, well, twisted by sine. Yeah, so yeah, and again, we have the same version with supports, and this short sequence is compatible with standard exact sequences in homology. OK, so now what I want to consider is to consider the case when x is singular itself. So we have not just a singular fiber over zero, but x is itself singular. This much more complicated situation. Before the assumption was that there is only one singular fiber. Well, not only one, but there are just finally many fibers, so you can consider them just separately. I'm sorry, can you spell out the form of the word before the human Hilbert affords again? It's under Riemann Hilbert, it's, well, it's, ah, yeah, it's here, yes, under Riemann Hilbert, it corresponds to, it should correspond to a vector space, well, the integrated vector space with an automorphism. And this is the integrated vector space and just even and odd parts of the homology of the perverse shift of vanishing cycles with the standard, with the usual monodrama on it. Well, twisted by sine, you just multiply by minus one for the odd part. It's just technical. Well, you always have a monodrama or automorphism. Yeah, OK, so now consider the case when x is, x has singularities. And then, let's consider, let's denote by the absolute of x and w, the category which is obtained by, well, category of coherent matrix iterations, which is obtained in the same way as for locally free. Then, again, this category is always actually nice. And in particular, to illustrate it, I also just mentioned my result that it's, well, this is, all these matrix iterations are, of course, Z2 graded DG categories. So both the matrix iterations defined above and this absolute derived category. That's because we consider a function. Of course, we could consider a section of line bundle, then shift by two will be twisting by this line bundle. Yeah, so statement is this category is also homotopically finely presented. And, well, again, we have an analogous result that it's a quotient of something smooth and proper, also Z2 graded. And this category, again, has a similar to the composition into categories of matrix iterations on some yi and wi. And here, you have additional assumptions that yi is smooth and the critical loss of wi in the fiber over zero are compact, well, proper. And this guarantees that this category is smooth and proper. So because of smoothness, I can replace this V absolute by... Yeah, yeah, yeah, you're right. I should mention it, yes. If X is smooth, then these categories are equivalent. And I don't need add importance to make this happen. You don't need add importance. You just take resolution by factorizations in abandoned categories and then take the convolution. Yeah. So how are the morphisms defined in Dx, Xw? How the morphisms are defined. Again, you take the homotopic category defined in the same way, and then you take the quotient by... Yeah, it's general. Yeah, so it turns out that this category is always nice. And also, again, you have... If this is closed, then you have short exact sections. This category is not necessarily closed, yeah? Again, yes, yes. And what does smooth and proper... Well, it doesn't matter. You always... Yeah, yeah, yeah, yeah. Yeah, so... It's some subgroup in K0, yeah. Yeah, it corresponds to some subgroup in K0, yeah. Yeah, and the quotient is related to K-1, as usual, of this... Well, I'll explain in a minute. Okay, so here we also have this localization short exact sequence. Well, here it's an analogous result, so our eyes are also smooth centers of the blow-ups in the Kharanaka process for the... But here you take relative complication over A1, and then... Yeah, okay, so this is short exact. And also, I'm not sure that someone has written this, but, however, this pre-cyclic homologer is, again, dual to the homologer with compact support. Well, again, under Riemann-Hilbert correspondence, it's dual to the homologer with compact support of the shift of vanishing cycles. Okay, and... So, yeah, and, of course, I should mention that if the potential is zero, if W is zero, then this absolute drive category is just two-periodic drive category of coherences. So you just consider two-periodic complexes. And metricizations are just two-periodic perfect complexes. And, yeah, and it's quite a natural question. What is the analog of perfect complexes for this situation? And the naive answer would be that this category defined in the same way for a singular X should be category of perfect complexes. But in terms of how this category is, in general, it can be completely horrible and doesn't satisfy the properties which you expect. Well, so let me still concentrate on, first on the coherent case. So here there's a generalization of theorem of Orlov, which is by Posicelsky, that this category of coherent metricizations is equivalent to the quotient of db of the fiber over zero by the subcategory generated by direct images of db of X, or by inverse images of objects in db of X. So here J is close to embedding of X zero into X. So this is a morphism of third-dimension one. So you have a well-defined direct inverse image on db. Yeah, and you have some subcategory which is generated by this inverse images, and it contains, of course, perfect complexes, because perfect complexes on X zero are generated by the restrictions of perfect complexes on X. So yeah, so this is kind of a relative version of the category of singularities. And I'd like to generalize this picture, namely, let... Sorry, in which sense is this a relative version? Relative, I'll explain in a minute. So suppose that instead of X zero, we have some morphisms from Y to X, morphism F, such that the third dimension of F is finite. In particular, the direct inverse image is well-defined. Then let's define the category... Well, I'll denote it db of Y over X. This will be... So this will be a full subcategory in db of Y, and it consists of objects F such that for any point of small y in Y, we have that... So you have algebra of jumps of functions in the neighborhood of Y, and you have a shift fiber, I mean, just localization of F, which is... So you have Fy, which is an object of db of this local ring, the requirement is that this Fy should be generated by inverse images of db of O of the local ring of F of Y. Well, maybe in this... Well, to simplify a bit the definition, let's consider the special case. Well, first I claim that in this special case, this subcategory is precisely described by this property. Well, the proof is quite technical, but not hard. It just requires some machinery with co-derived categories and so on. But just to clarify a little bit, let's consider the special cases. First, if X is smooth, then this db of Y over X is just the category of perfect complexes on Y. That's because restrictions of perfect complexes are just perfect complexes, and so the requirement is that in each point, the object is a perfect complex, but this is the same as to say that the object itself is a perfect complex on Y. And also if F is smooth, so if F is a smooth morphine, then db of Y over X is just db of Y. And in general, this is some intermediate category. And one can show that actually... Well, again, it's not too hard that actually this relative derived category of Y over X is actually generated by object of the form. So you take some perfect complex over Y and take its derived tensor product with inverse image of some object on db of X. So in particular, if it's just a closed embedding, then this is just the category generated by inverse image of db of X. And also, well, it turns out that, again, it's quite a reasonable class of categories, and one can show that, in particular, if Z inside Y is some closed subset, then you have a short exact sequence of categories. Also it's short exact, and it's quite natural to conjecture that its periodic cyclic homology can be described in the following way. I think it should be not hard, but actually I didn't prove it that periodic cyclic homology of this category db of Y over X should be homology of the inverse image of the dualizing complex on X. So, yeah, this is an object of constructible derived category and it should be just its homology. Yeah, again, it agrees with special cases when X is most or F is most. Okay, so this is the case of coherent factorizations, and now what we have for locally free matrix factorizations, well, maybe let's denote them in different way. Let's write the absolute locally free. Then also, Przestewski proved that, in this case, this category is equivalent to, in this case, you should take relatively perfect complexes. I'll explain in a second what it means. Of X0 over X and take the quotient by perfect complexes on X0, and again, by definition, well, in this case, definition is even simpler if F is a morphine that has finite torque dimension, then this category is just, it consists of objects in Db of Y such that, well, let's just write that F has finite torque dimension over X. So, again, this category is, in general, intermediate between perfect complexes and the category Db of Y, and again, if X is smooth, then this is just the category Db of Y, and if F is smooth, then this is just perfect complexes. Okay, so this different version of category of singularities. So, there's no inclusion from perfect of X over Y over X into Db? Yeah, that's right. And we'll see that there's actually some problem with understanding how they relate to this chest and what's the right version of this guy. Yeah, but still there's this equivalence, and of course, the natural expectation would be that, the natural expectation would be that the analogous results work in this case with perfect complexes. So, we can expect that there are short-exact sequences like this and so on, short-exact, and also that pericyclic homology of these locally-frictionizations should be just the homology of the shift of managing cycles. Just before continuing, we've almost all categories up to maybe the last six categories of the whole type. We have algebra, phonetics, and you can see the perfect object, and you can see the subcategories as perfect object. Well, in one case it's generated by something, in the other case it's conditioned that it's perfect over something, yes. The first, of course, makes it easier because it's easier to produce objects of this kind, but if it's given by some condition, then it's not easy to produce objects and I'll explain what the problem is. So, it's natural to expect that, again, we have the same answer for pericyclic homology as in the smooth case, but yeah, so this is quite natural to expect, but it turns out that this is completely wrong and I'll give a counter-example to this. Okay, so let's consider the following example. Let X be a three-dimensional quadratic column, so just given by a fine column, so just given by equation Z1, Z2 equals Z3, Z4 in the space C4, and let's consider the potential to be the first coordinate. Also, I want to give a counter-example to localization, so I should specify what's the closed subset and let's take the closed subset to be the divisor defined by Z3, so U will be X minus this divisor and then it's not a hard exercise to write down which categories we obtain. First, let's start with coherent factorizations. Then first, let's observe that the pair U, first U is, of course, smooth because we have other similarities, so U is smooth and this log-in model is equivalent just to A2 with coordinate X, Y times A1 minus 0 with function X, Y. This is an identification, just Z1, Z2, Z3, Z4 corresponds to Z1, Z2 over Z3, Z3. For this log-in model, just by non-represent periodicity, we see that the category of matrix factorizations, it doesn't matter because it's smooth, so it's the same as locally free factorizations. It is a superior derivative category of A1 minus 0. So just co-hearing shifts on one-dimensional and now what will be matrix factorizations on X? Well, first let's observe that, again, DW will be just the following. You take a pair of lines, Z1, Z2 equal to 0 and multiply by a fine line and, well, with coordinate T and the potential will be this coordinate and actually by Tom Sebastiani, it's easy to see that the absolute drive category of DW is just 0, but it applies that absolute drive category supported on D is also 0. So therefore we have that the absolute drive category of XW is the same as absolute drive category of UW and both of them are equivalent to this drive category of A1 minus 0, but now if you try to understand what is absolute drive category of locally free factorizations here, then the answer will be quite strange, namely this locally free factorizations on X will be compactly supported complexes to predict complexes on A1 minus 0, so with compact support. Compact means finite just in this case. So you will get complexes where homology are supported in a finite number of points and this is, of course, quite a horrible category which is not generated by one object. It's also a condition just it's kind of supported. It is joined for something and principle can ask you things. It can be bad categories like this. Yeah, but I get bad categories, but the question is how to make them reasonable? No, it's like 4k category of open variety. It's like the rough 4k which are good. Yeah, okay. But still it applies that in particular that this restriction factor is, of course, not a localization. So, right? So the restriction function from X to U is not localization. Yeah, and also it's easy to check that this predictor-sector homology is countable dimensional. Of course, so it's... Define it in the right way. Yeah, but considering this compact support on a finite variety. Yes. It's a super category. It's not more generic object. Yes, yes, yes. In this case, yeah. So if you literally just consider this predictor-sector homology, then dimension will be uncountable. So, yeah. So we get a question. Yeah, but still we have this equivalence, so it means that actually you also... This is counter-example for localization for these relative perfect complexes. Localization fails for both d absolute locally free and relative perfect complexes. Yeah, and so the question arises how to define the right category. Well, maybe, I mean, for some purposes, this category is just... It's the right way. Yeah, they appear, but I mean, there definitely should be some analog of perfect complexes which behaves... I mean, the categories of perfect complexes, they're not behaved like this. They are at least... Because you have this conditional support. Yeah, yeah, yeah. So I'll just formulate the question what is the right notion of perfect matrix vectorizations and also what is the right notion of relatively perfect complexes satisfying... So first, localization. Second, they should satisfy this equality. So that HP for locally free vectorization should correspond to vanishing homology and, again, in the same sense with Riemann-Hilbert correspondence. And also, the psychology of this relatively perfect complexes, this should be just a homology of the extraordinary inverse image of the structure shift on X. Again, this is an object of the constructible draft category. Structure shift. Not the structure shift, constant shift. Yeah, not the dualizing. Yeah, why? Well, for example, if X is a point, then this relative draft category is coherent shift, and this is a dualizing complex on Y. And similarly... So if the morphine is smooth, then this guy is up to shift, is a constant shift here. So this should be the answer for the right version. Well, the right version from the point of view that I explained. Yeah, but at the moment, I don't know the answer, and one more property is that if... Well, say, for matrix particularizations, if the morphism is a potential, is a proper morphism, locally free. Well, this may be considered as a definition in this case, but I don't know how to make it in general. I think it should be just set the perfect DG models over coherent particularizations and vice-versa, just for the usual varieties. Yeah, I'll recall what is set the perfect So by definition, set the perfect DG models over a DG category are just those models, say, right models, which take values in... values being complexes with fine-dimensional homology. So, well, when T is a DG algebra, the DG models with fine-dimensional homology. So for w equal to zero, well, just for usual varieties, this is satisfied, and presumably it should be also the picture for... Yeah, you see, in general, those issues, non-compactness and singularity. Yes, yes. So how one can develop kind of fake picture with conditions of support on some of the pictures, yeah, I suppose. And I don't think it should be people's realizable, because there are naturally, again, like, like, full category of non-compact varieties. Yeah. It has nature of chain classes, and nature of homology, not homology of variety. Oh, yeah. Yeah. And you cannot make... in the categories, not finitely, to be completely generated and technically... But I mean, it's possible that it's just sub-category of... I mean, there's some larger category. No, it's natural, there's no larger category in classes of homology. But I mean, naturally, with the usual definition, by Kenji or...? No, I can't imagine. You still should have objectized some way, finitely, maybe, right? There's nothing new. Nothing new? Yeah, there's nothing new. Oh. I think it's just questions of complex support, essentially, that are in our estimation. Okay, okay, yeah. Well, it's possible. Yeah, no, in general, you should really have to kind of imagine like compact variety and two-close subset and six-supported from one and just joined from another. Yeah, that's kind of a general class of good categories from which you declare HP, whatever it will be. Okay, so you mean that possibly there's no natural category, for example, giving this answer for psychochamology? Oh, I think, no, it's just it's not HP of the category, but you declare a homologer to be or to be followed for this intermediate definition. Oh. Well... Yeah. Yeah. The actual HP will connect to this your finite-dimensional concept. Yeah, I see. Okay, but how however this is the question, at least, what should be the right answer? Well, at least this means that the picture for magical positions is far from the picture. It's far from being similar to the picture of just varieties. And also another related subject which I want to explain is the critical points for maps between single varieties. Yeah, okay. Okay, so again, if x is smooth and if it's smooth and we have a regular function, then there are there are only finite-lemene so the set of constants such that the category of matrix for x and w minus constant is non-zero is finite. This is by Sarge-Lemma just the critical values for a finite set. And actually one can show that if x is singular and again maybe non-reduced, then the set of values such that coherent matrix positions are non-zero is also finite. Yeah, and to show that you just need to take a smooth stratification of x, so i is smooth and then consider critical values of restrictions of w to this strata and the critical values of w and x will be a subset of the union of these critical points. And I'd like to well first we can generalize this to a general smooth base, so this map to a fine line but let's consider a map to something smooth and here again I have some general conjecture well, again we have an analogue of Sarge-Lemma and and some I'd like to propose some general conjecture which should capture all this kind of critical points from f from something singular to something smooth. So if f from x to t is some morphism and suppose t is smooth then by definition point t in the basis of critical value if the relative derived category of let's for simplicity assume that it is just flat. So not just finite torque dimension but just flat. It's just for simplicity it can be generalized by considering dg fibers but in this case we take just the usual fiber over t and this relative derived category is smaller than db of the fiber and one can easily check that if x is smooth then this is just the usual the usual notion of critical points so it agrees and one can again show that there's an analog of third lemma that this the set of critical values inside t is a closed proper subset proper close subset so so this generalization of third lemma to singular varieties and again one can also define the notion of critical points in singular fibers just in the same way and here I'd like to propose some well it's some not very rigid some kind of conjecture that there should be there should exist a natural dg-algebra a over o t well we assume that t is just a fine so this dg-algebra over the algebra functions on t such that derived categories of the fibers of coherent sheets of the fibers are equivalent to perfect complexes of the fibers of this dg-algebra over t where a t is just derived tensor product a of a and structure shift of the point t over t and there should be actually an embedding of db of x into perfect complexes over a which induces the corresponding inclusions yes so if you take the fiber of db of x over t this will be a relative derived category of the fiber over x and and this inclusion should be induced by this fully faithful morphism fully faithful dg-functor similar question was asked to me by Dima Orlov for well for the case of a fine line and for the case of smooth x but of course the natural framework such a general picture and I mean is the point here not the word natural or which one there is exist a natural dg-algebra a over over t there are dg categories that have linear over over t such that such that such that there is there are such equivalences so if you take the fiber of a dg-algebra over t fiber in this sense then it's perfect complex it should be derived categories of coherent shifts on xt so if you take naively just algebra of generator of db of db of x what you will get under restriction to the fiber is the relative derived category but what we want to have is db of the fiber itself and this is not so easy to I don't have actually any general approach to this okay so I think I'll stop for example x subspace c2 you could see this as a complete section first defining equation for this x then you go to c2 to get the discriminant and then whole thing lifts around the discriminant so you have more monodromy than just running around one point yes so that's this where they explain the the differences the difference between this well it's well purposefully yes but well I mean it's just it's just an example it's just an example just an example because you don't take into account that this x is itself in a family yes yeah I see yeah and maybe yeah for this complete intersection I maybe I should mention that for if say suppose that the motion is quasi smooth x is quasi smooth then Gage-Gurion-Daring can define the notion of single support for for D.B. Koch and I mean I don't have much time to define what it is but the intermediate subcategories between perfect complexes and D.B. Koch can be described by means of this single support and this is actually what appears naturally well conjectured of course at the moment for geometrical length yeah and I mean maybe yeah in this pleasant case of quasi smoothness and well it means that it's locally complete intersection it's possible that there's a way to define everything by means of this by specifying a single support yeah yeah but in general it's clear but