 Спасибо большое. Так что, я должен... Я немного извиняюсь. Мой первый разговор сегодня будет немного заборным, потому что я чувствую какую-то responsabilитию, для того, чтобы дать какие-то дефинитии, в связи с делающими категориями. Так что, в первом разговоре, мы видели какие-то интересные геометрии, в связи с делающими категориями, но дайте мне тоже дать brief introduction, объяснить, что делающие категории есть. Ок, так, давайте начнем с категории, а потом эти делающие категории находятся как следы. Вы начнете с категории, скажем, соединения комплексов и объектов. Так что, я думаю, что все знают, что это комплекс. Бунт means, that it has a defining number of non-zero terms. И обычно мы используем геомологический грейдинг, which means that the differential increases the gradient by one. Ок, так, вот эта категория комплексов, она идет с notion of amorphism of complexes. Конечно, amorphism of complexes is the amorphism of underlying graded objects in this abelian category, which commutes with the differential of the complexes. And because of that property, if you have amorphism of complexes from f dot to g dot, say, phi dot, then it induces amorphism between the cohomology objects of these complexes. First of all, for each integer i, one can define the cohomology object of the complex f dot to be the quotient of the kernel of the morphism from f i to f i plus one by the image of the object, of the morphism from f i minus one to f i. Just the usual definition. And then, because amorphism of complexes commutes with the differential, by definition it induces amorphism between cohomology objects of these complexes. So this is denoted by h i of phi. Ок, so for every morphism we have this bunch of morphisms between these cohomologies. And so the crucial definition here is the definition of a quasiisomorphism. So amorphism of complexes phi is called a quasiisomorphism if h i of phi is an isomorphism for every i. So in other words, the condition is that the morphism induces an isomorphism on the cohomology objects. The bounded derived category of an Iberian category A is defined as the localization of the category of complexes with respect to the class of quasiisomorphisms. Which means that we artificially add morphisms to our category which are inverse for every quasiisomorphism. Or saying it informally we just don't distinguish between complexes that are quasiisomorphic. Ок, in fact, you have to do some work to show that this definition makes sense. In fact, to define a reasonable category structure with these properties it is not that simple, I will not discuss this. You can look into some standard textbooks where this process is described in detail. So for instance there is a textbook of Gelfand and Manin dealing with this subject and there are also more modern approaches using for instance differential graded categories and you can look for instance into the paper by Keller where all these are also described. So let me just assume that such operation, such localization operation is well defined on the category of complexes and there is a category in which the objects are the same as the objects of the category of complexes and the morphisms are just so to say formal compositions of usual morphisms and inverse to quasiisomorphisms. Ок, in fact, a bit later I will describe a bit the structure of the derived category and you will see that the definition itself is not very much important. It is much more important to understand how you can work with this category and this I will try to explain in more details. Ок, so this is the definition and let me also mention that by definition there is a natural factor from the category of complexes to this localization category so we can think about this as a factor from the category of complexes to the derived category so every complex can be considered as an object here and also we have a natural from the original from the initial abelian category to the category of complexes given an object we can just consider a complex which has zero terms everywhere except of degree zero and it has just this object as its term so we have this embedding so usually we will consider objects of this initial abelian category also as objects of the derived category by thinking of them as of complexes with only one term in degree zero with only one nonzero term in degree zero Ок so in fact the most important thing is the structure that we have on the derived category and the structure is called triangulated structure so in fact triangulated structure consists of two pieces of data that satisfies some properties so the first thing that should be specified is what is called the shift factor interpolate is also usually called suspension factor so this is just an automorphism from from a category to itself so maybe let me state it as a definition of what a triangulated category is so as a definition triangulated category is an additive category T first of all there should be an automorphism of this category called the shift factor and its iterations are usually denoted by the corresponding integers in the brackets so its kth power is denoted by k and since it is an automorphism you can also take k to be negative so for every integer we have the shift by k so this is the first piece of data and the second piece of data is a class of what is called distinguished triangles which adjust sequences of morphisms that look as follows f1 goes to f2 goes to f3 and the automorphism goes to f1 shifted by one so this shift by one is precisely this shift automorphism which is usually denotation for it is this bracket notation after the object to which it is applied so to specify a class of distinguished triangles means to specify some number of sequences of morphisms and of course they should satisfy a number of axioms list of axioms and I also don't want to list these axioms here you can easily find them in the textbooks or even in Wikipedia or wherever you like let me just mention only some of them so to say maybe to say that the most important is not the right word but those axioms or properties that I used most frequently so the first property or axiom in fact is the following whenever you have a morphism between two objects of your category T it can be extended to a distinguished triangle so maybe every morphism extends a distinguished triangle that means that for any morphism you can find f3 and phi2 and phi3 such that this sequence is a distinguished triangle and the corresponding object f3 that extends this morphism to a distinguished triangle is in fact defined up to a non-canonical isomorphism and it is usually called the cone of the morphism so typically you can write this distinguished triangle in this form so this object called and let me repeat again that the cone of a morphism is defined up to a non-canonical isomorphism which means that any two extensions for any two extensions of this morphism to a distinguished triangle the corresponding objects are isomorphic but there are different choices of such an isomorphism and there is no canonical choice which leads to some of the theory but still the isomorphism class of the object is well defined so so this is the first property which I would like to mention the next property is that whenever you have a distinguished triangle you can construct from it many other distinguished triangles so for instance you can write about its first object and then you can write what you have and then you can add F2 shifted by 1 on the right and the morphism F1 shifted by 1 here and one of the axioms says that this triangle is distinguished if and only if this triangle is distinguished and this second triangle is usually called rotation of the original triangle rotation and so in other words a triangle is distinguished if and only if its rotation is distinguished and in particular this axiom I mean in fact you can using this procedure you can extend your original triangle to an infinite sequence of morphisms like this and so on and it goes both to right and left and such a sequence which you can always construct starting from a distinguished triangle is called a helix associated with this distinguished triangle and it has a property that if you take any three consecutive arrows in this helix this triangle also and one of the main properties is that if you have a helix like that one you can apply to it the function of morphisms from any object of your category or to any object of your category and then you will get a long exact sequence of home spaces so this is also a very important property which is maybe most frequently used that for any object G in T you can just apply the functor from G you will get something like that and so on and this sequence is exact is a long exact sequence and analogously you can apply instead of applying functor home from G you can also apply functor home to G of course this functor is contravariant so you will get a sequence with arrows going in the opposite direction so maybe let me write it like this minus one to G so it is also a longer exact sequence and maybe I will also mention some useful exercises for those who haven't worked with triangulated categories this will be quite useful so first of all assume that you have a distinguished triangle in which the morphism phi3 is 0 then from this that f2 the object f2 in the distinguished triangle is in fact isomorphic to the direct sum of f1 and f3 in such a way that the morphism f1 is the embedding of the direct cement and the morphism f2 is the projection on the other direct cement so this is a useful thing to check and another one is that if the third object of the triangle is 0 then phi1 is an isomorphism so this is if and only if so a morphism in a triangulated category is an isomorphism if and only if its cone is 0 so in some sense is a mixture between the notion of a kernel and of a co-kernel in an abelian category so if you have a morphism in an abelian category such that its kernel and its co-kernel are 0 then the morphism is an isomorphism and this is an analogous property in a triangulated category ok and now let me just say that derived category triangulated structure this is an important theorem so probably due to Verdiere db of a has a triangulated structure such that the shift functor is given by the shift of grading of complexes which means that if you have a complex f dot then if you want to shift it by 1 then its term and degree i should be f i plus 1 which eventually means that you shift the complex to the left so you should keep it in mind and also the differential of the complex changes its sign this is by definition so the shift functor is given by this shift of grading of complexes and with the cone of amorphism defined to be the mapping cone of complexes phi from f dot to g dot being mapping cone which basically means the following so if you take amorphism of complexes g i minus 1 and if you have amorphism of complexes then you have this diagram with commutative squares and so you can consider this diagram as a by complex so there are arrows of two directions and if you imagine that it is not concentrated in two rows but in infinitely many rows just all the other rows being zero then the composition of two horizontal arrows will be zero the composition of two vertical arrows will be zero and the arrows themselves will commute so this is what is called a by complex and whenever you have a by complex you can consider its totalization which just means that you sum up along the diagonals of this complex and use these arrows to put a differential on this totalization this is what is called the cone of amorphism of complexes and so the theorem of Verdiya says that if you define a cone of amorphism in this way and if you define the shift in this way then all this gives you a triangulated structure ok so say a couple of words about how one can think about objects of the derived category so in fact from the definition and this is a kind of naive approach to think about objects as about complexes that have their terms and differential and so on but this is not a very good point of view terms are because you can have two isomorphic objects in the derived category that have completely different terms the term of a complex is not a notion which is invariant under a quasi isomorphism so it is not a good notion but the notion that is good is the notion of the cohomology of a complex so one should think of objects of db of a as cohomology objects are linked to each other and let me explain what does it mean linked together so to explain this let me introduce one more notion which is very useful which is the notion of the canonical filtration so assume that you have a complex fi plus 1 and so on let me define now to other complexes so for each integer k let me define tau less than or equal than k of f dot to be a complex which has the following terms so the term in degree i will be equal to fi if i is less or equal than k minus 1 it will be equal to the kernel of the morphism from fk to fk plus 1 if i equals k and it will be 0 when a is greater or equal than k plus 1 and similarly you can define t greater or equal than k tau greater or equal than k of f to be the complex which has term 0 for i less or equal than k minus 1 so it should have term image from fk to fk plus 1 when a equals k i equals k and fi when i greater or equal than k plus 1 then what it is very easy to see is that for any k there is a morphism from tau less or equal than k of f to f so it is just given by natural embeddings and it is easy to see that everything is compatible with the natural differentials and then there is a morphism from here to tau greater or equal than k plus 1 of f and in fact this gives you a distinguished triangle there is a distinguished triangle of this form for any k also it is very easy to see that if you ok and also a very important property is that if you consider thecachomology of these truncated complexes then thecachomology is the same as thecachomology of your original complex when the index when the degree corresponds to the degree in the notation for this truncation functor and zero otherwise and analogously for the other truncation functor so ok let me check yeah so maybe this is the crucial property that should be satisfied and let us just see whether it holds for this definition or not I mean in fact maybe just to be on the safe side let me write it in a bit different way so let me first do it for this truncation complex and then this quotient of your original complex but what you see here yeah then this is right and here you should take just the quotient of fk by this kernel and this will be precisely the image so I think that this is correct ok sorry ok yeah this canonical truncation distinguished triangle is in some sense the main tool to think about objects of the derived category in some sense you can think about this object as about glued from this part and that part and with this morphism being the gluing data and know that these two objects are more simple than the original object that you started with just because by this property they have smaller number of cahomology shifts and in the end you can somehow to split any object into the bunch of its cahomology objects and that will be linked by morphisms of that type and let me maybe leave it as an exercise to check the following statement so assume that we have an object that has only two cahomologies such that h i of f is 0 or i not equal to k or l and assume that we fix integers k and l k is less than l so maybe let me write it like that so k is less than l and i is not equal all the cahomology except the cahomology in degree k and l are 0 then isomorphism classes of such objects asomorphism class of f is determined first of all hk of f hl of f and amorphism let me call it epsilon which is amorphism from fl of hl of f to hk of f shifted by l minus k plus 1 for instance if l equals k plus 1 if you have cahomology into adjacent degrees then this morphism lives in home from one cahomology to another shifted by two and by the way this is a standard convention to use ex notation for such home spaces so by definition this is just x l minus k plus 1 from one to another so in some sense in the same way as in an abelian category you can construct more complicated objects by considering short exact sequences and so by constructing them from simpler objects by using x1s to link in the same way in a triangulated category or in the derived category you can construct any object from shifts of objects in your original abelian category by using higher x groups and this is the way it is better to think about these objects ok so what goes next now we discussed a bit triangulated categories now let me say a couple of words about triangulated functors so if we have two triangulated categories T1 and T2 then a functor between these categories is called a triangulated functor if I mean basically it should preserve the triangulated structures of our categories which means that it should first compute with the shift functors if it commutes shifts which to be more precise it means that commutativity morphism should be specified but I mean in almost all cases when people discuss triangulated functors this commutativity morphism is left implicit but still you should be careful a bit with it so the first property that the functor should commute with shifts which means that if you shift an object then apply your functor it is the same as you first apply the functor and then shift the object and also the second property that it should take distinguished triangles in the first category to distinguish triangles in the second category keeps triangles distinguished ok so this is the notion of a functor and usually we will at least in my lectures we will discuss derived categories of coherent shifts on some smooth and projective varieties so we will start with some smooth and projective variety over some field and maybe for simplicity it is better to assume that the field is the field of complex numbers but of course in most cases it doesn't matter but let me use this assumption then we consider the category of coherent shifts on x this is a nice abelian category and then we consider its bounded derived category and usually we will discuss its properties and to simplify the notation I will denote it just by d of x I will skip this letter b but I will consider only the bounded category so what kind of functors do we have in geometry between derived categories of coherent shifts let me mention some of the most important functors in fact this is a very good property of the geometry is that it always comes with many functors so what kind of functors do we have Assume that we have a morphism between two algebraic varieties x and y Assume that both are smooth and proper then in this situation one can define the push forward functor between the categories of coherent shifts and the pullback functor and one can extend them to derived categories by considering derived functors so there is derived push forward functor from d of x to d of y left derived pullback functor from d of y to d of x okay and yeah and for instance if you take a coherent shift say f and k of x if you apply derived push forward functor to this object it will no longer be a coherent shift in general and if you compute the cahomology shifts of this object this will be the classical derived push forward functors so in some sense this total derived push forward is a derived category version of this usual derived push forward functors that takes that keeps not only these cahomology shifts but also these linking morphisms between them so to specify this functor is such a specification contains strictly more information than just specifying this sequence of classical derived push forward functors and similarly in the same way for pullback have an object in k of y then if you compute the cahomology of the derived pullback functor then this is l minus e pullback of g so usually for the derived pullback functors people use homological notation and because of that there is this change of sign from cahomological notation to homological notation here ok besides these two functors in the derived world there is one more important functor which is called the twisted pullback functor in fact coming back to this situation a very good property of these two functors is that they form in a joint pair so in fact if you compute homes from the pullback of any object of the derived category g in a so let g be in d of y and f be in d of x then if you compute homes in the derived category of x from the pullback of g to f then this is canonically isomorphic to homes from f to the derived push forward sorry from g to the derived push forward of f so there is this very useful adjunction isomorphism and means that the pullback functor is left joined to the push forward functor the right adjoint functor to the push forward so the next important functor is this twisted pullback functor which goes from d of y to d of x and which is right adjoint to the push forward so in a contrast to the previous situation this functor is not derived functor of some functor between a billion categories it is a nice triangulated functor so this adjunction means that we have a similar isomorphism to this one so let me also write it down so if you compute homes from the push forward of f to g this is the same as homes from f to f upper shriek of g this is this isomorphism which is frequently called the groten-decker junction and in fact in the nice situation if we consider this nice situation when both x and y are smooth and proper then one can even write down an explicit formula for this twisted pullback functor in terms of the usual pullback functor and the canonical class so in fact the formula is that this is the same as derived pullback of g tensored with relative canonical class and with the shift by the difference of the dimensions so this is another important functor okay so next functors that people frequently use are the tensor product and local home functors so let me also mention them so this time you don't need to consider two varieties so just start with one variety and again it is better to assume that it is smooth and then for any two coherent shifts I mean there is this standard tensor product functor on the category of coherent shifts and you can extend it to what is called derived tensor product functor of x to g of x and in the same way tensor product and derived tensor product so this is the notation for functors and also there is another functor so this takes F1, F2 F1 tensor F2 derived tensor product but if you have two objects to coherent shifts you can also consider the shift of local morphisms between F1 and F2 this is also a coherent shift on x so just to distinguish between this vector space of morphisms and this shift of morphisms I will use this calligraphic notation and in the same way if you have two objects of the derived category you can define r-home complex of coherent shifts in d of x and again these two functors are derived by a junction so if you compute homes from F1 derived tensor product F2 to F3 this is the same as home from F1 to r-home from F2 to F3 and again like in this property if you apply this derived tensor product or derived r-home functors to coherent shifts and then compute the cohomology shifts of these objects then you will recover tors of two coherent shifts in the first case and local x shifts in the second case ok so these are and of course these functors satisfy a long list of relations which is very useful to know if you want to work with derived categories of coherent shifts let me mention some of them so relations so first of all there are functoriality relations basically it means that if you have a sequence of two maps F and G then if you compose derived functors derived push forward functors then this is the same as derived push forward functor for the composition of morphisms and the same for the pullback and the same for the twisted pullback so the first are these functoriality isomorphisms second besides these adjunctions that were mentioned before so we have three adjunctions on the blackboard besides them there are also the local versions in fact everywhere we can just replace this global home by appropriate local homes these are called local adjunctions so for instance let me say what will it be in the first case of course you cannot do it just naively you cannot replace this home and this home by local r-home just because this will be shifts on different varieties so the right hand side will live on y but if you apply a push forward then everything will be fine so like if you apply derived push forward to r-home from lf upper star of g to f then this is the same as r-home from g to the push forward of f so there are relations like that and you can do the same for the other two adjunctions that you have on the blackboard then the next list of relations corresponds to the tensor structure to the monoidal structure given by this tensor product this derived tensor product defines a monoidal structure d of x which basically means that in fact it is symmetric monoidal structure which means that tensor product is commutative associative and also the pullback functor is compatible with this monoidal structure is a monoidal functor and on the other hand if you consider the push forward it is not compatible with the monoidal structure it is not true that if you take the tensor product of two objects and then push it forward it will not be equal to the tensor product of the push forwards but instead there is another important relation which is called the projection formula and which says that the push forward not of arbitrary tensor product of objects upstairs but if you take one arbitrary object and if you tensor it with the pullback of another object so if you consider push forward of a tensor product like that then what you will get is the tensor product of the push forward of the first object with the second object so this is also a very important relation and maybe let me finally mention the base change so what's that? Assume that you have amorphism and a fiber product Assume that you have this fiber product diagram this Cartesian diagram so this is vx and this is f sub t then you can always construct a natural morphism between the composition I mean the goal of the base change formula composition of the push forward with the pullback and of the pullback with the push forward and what you can easily do you can easily construct a natural morphism from the composition of the pullback and the push forward to the composition of the push forward and the pullback so there is always a natural morphism like that and so you can ask in which cases this morphism is an isomorphism and in fact you can say very explicitly that it is an isomorphism if the diagram is what is called tor independent and that means basically that if you take any two points on x and t that have the same image in s then if you take the local rings of these points on x and on t and consider the tensor product of the local ring of the corresponding point on s then this tensor product should not have higher tors so tors greater than zero from over o s sub s of o x sub x and o t sub t is zero for any pair of points x and t such that f of x equals s equals v of t so if this condition is satisfied then there is this base change isomorphism and of course this condition is satisfied when one of these two morphisms is flat just because in this case the corresponding ring is flat over the other ring and this tensor product its tors with any other module are zero but there are also other cases when this is correct for instance if both morphisms f and v are closed embeddings and if these sub varieties intersect transversally in s then also this is a tor independent condition t are all smooth varieties and the fiber product has expected dimension then the diagram is tor independent and this is a very useful thing in many cases I see that my short introduction become a bit long so probably I will not be able to do anything besides it so maybe let me introduce one last notion also a very important one and give several more exercises and then we will turn to some interesting stuff tomorrow so one notion is a notion is a more general notion of a functor that is used in most cases so this is what is called Fourier Mochae Functors so I assume that x and y are again smooth projective varieties then I assume that we are given an object some object in the derived category of their product in D of x times y then we can define a functor associated with this object which is called Fourier Mochae associated with k or sometimes one can say with kernel k is a functor which takes the following form so the usual notation is phi sub k from D of x to D of y and the definition is the following you should write the corresponding diagram so we have interactions p and q and then this functor is a composition of the derived pullback then you can tensor it with k in the derived sense and then you push it forward also in the derived sense so you pull back tensor product and push forward then just pullback push forward or tensor product this is like a general combination of these three funters that we have seen before and to finish let me just write some exercises about Fourier Mochae funters which are also quite useful yeah so first of all assume that you have amorphism then we can consider граф so let gamma f be the embedding from x to x times y the graph of f then one can show so this is the exercise is to show that the push forward functor is in fact the Fourier Mochae functor with the kernel being the structure the shift of the graph and similarly if you have if you want to express the pullback functor as a Fourier Mochae functor you can also do it let me just write it in this way derived pullback functor is also a Fourier Mochae functor also if you want to express the tensor product functor then it is also easy to do so let me denote by delta the diagonal embedding of x into its square then if you want to consider the derived tensor product functor with some object e in d of x then it is the same as Fourier Mochae functor with kernel being the push forward of e and also to show that a composition of Fourier Mochae functors is also a Fourier Mochae functor and is that all also adjoins Fourier Mochae functors are also Fourier Mochae so if you take the left or the right adjoint of a Fourier Mochae functor then it is also a Fourier Mochae it is better if you will be able to write the formula, write the kernels in the first case for the composition and in the second case for the adjoint functors ok, let me stop here and I hope next time it will be more interesting