My View of the Current State of the Motivic Homotopy Theory V

My view of the current state of the motivic homotopy theory V. Voevodsky. Work started Jan. 2 2000

0.1 Preface In this text I try to formulate what I believe can and has to be done in the motivic homotopy theory in the nearest few years. I only discuss things which form the foundations of the theory leaving possible applications aside.

0.2 The hierarchy of invariants Invariants of algebraic varieties (as well as of mathematical objects of most other classes) form an hierarchy. Understanding of this hierarchy is crucial for the understanding of the current state and goals of the A1-homotopy theory and I will spend some time on it. There is three levels of invariants in use today - element-level invariants, set-level invariants and category-level invariants. An archetypical example of an element-level invariant is the number of rational points on a variety (= the number of solutions of a system of algebraic equation). Another standard examples are Betti numbers of characteristic numbers. An example of a set-level invariant is the set of rational points of a variety or a cohomology groups. Set-level invariants typically carry more information than element-level invariants. For example Betti numbers are determined by the cohomology groups but not vice versa since cohomology groups may have torsion components. The value of set-level invariants in problems which concern properties of element-level invariants is well recognized. One of the most famous examples of this is the Andre Weil’s suggestion that a number of conjectures about the L-functions (element-level invariants) of varieties over ﬁnite ﬁelds would follow from the existence of a cohomology theory (set-level invariants) for such varieties satisfying some natural conditions. This approach was worked out by A. Grothendieck, P. Deligne and others and is as of today the only way we know to prove all of the Weil Conjectures. Just as element-level invariants do not have to be numbers or collections of numbers but can be elements of some abstract sets the set-level invariants

1 do not have to be sets with structures but can be objects of any category. For example the suspension spectrum of a topological spaces considered as an object of the stable homotopy category is a set-level invariant of this space. I guess that algebraic topology never moved to the next level of invariants partly because this particular set-level invariant turned out to be so unexpectedly rich and complex. An example of a category level invariant is the category of coherent or constructible sheaves over an algebraic variety. Just like set-level invariants usually assign to a variety a set with additional structures a category level invariant usually assigns to a variety a category with an additional structure. A category level invariant is usually more informative than a set-level invariant. For example etale cohomology of a variety can be described as groups of morphisms between appropriately choosen objects in the derived category of constructible sheaves over this variety. Theorems about category-level invariants have corollaries in the form of properties of the associated set-level invariants. For example the properties of the etale cohomology which where necessary for the proof of the Weil Conjectures were themselves deduced from fundamental theorems about the functorial behavior of the derived categories of constructible sheaves. Thus in the case of Weil Conjectures we see how one uses two higher levels of the hierarchy to prove results about the invariants of the lowest one. Another example of the same kind is given by Grothendieck’s proof of the Riemann-Roch theorem. In its original form it is a statement about element- level invariants of vector bundles. Grothendieck first lifted the problem to the set-level introducing the Grothendieck’s group K0. This group is associated to the category of coherent sheaves on a variety which is a category-level invariant and Grothendieck’s proof of the Riemann-Roch theorem uses this in an essential way to reduce the problem from general projective schemes to the projective space. Already a set-level invariant is almost never described just by assigning sets with structure to algebraic varieties. Usually one also assigns maps between these sets to all or some morphisms bewteen varieties. In addition one may have data of other sort such as suspension isomorphisms in the definition of a cohomology theory. Thus speaking of a set-level invariant one should start by specifying the class it belongs to (say it is a contravariant functor or a cohomology theory or a Borel-Moore homology theory) and then specifies the concrete data defining given invariant as a memeber of its class. The same is true for the category-level invariants. To have a useful in-

2 strument it is not enough to specify a category for each variety one also needs additional data such as functoriality with respect to morphisms of varieties, base change transformations etc. Today we have axiomatics only for very basic classes of category-level invariants such as 2-functors or stacks. Already the derived categories of constructible sheaves form a much richer ∗ ! structure with four different types of functors f∗, f , f! and f associated to a morphism and different interdependencies between them for morphisms of specific types. Last year I found what I believe to be a satisfactory axiomatics for category-level invariants of this type which I call 2-theories. Any 2-theory generates four families of set-level invariants analogous to the usual cohomology, homology, cohomology with compact supports and Borel-Moore homology incarnations of a ”cohomology theory” in topology. Axioms of a 2-theory imply that these invariants have have usual properties such as standard co- variant and contravariant functorialities, ”duality” for smooth morphism etc. I see this axiomatics as one of the main techical tools to be used in the next stage of developemnt of the motivic homotopy theory.

0.3 Current state of the motivic homotopy theory The motivic homotopy theory grew on the one hand out of the work of Morel and on the other out of the approach to motivic cohomology which Suslin, Friedlander and I developed in 1992-95. Originally we did not plan to construct a category-level invariant as suggested by Beilinson and Deligne and then derive motivic cohomology as an associated set-level one. Instead we used the ideology coming from topology and the original Grothendieck’s ideas on motives to construct a “universal” set-level invariant. In particular we worked with varieties over fields (except in [?]) instead of general schemes. The first foundations paper on the motivic homotopy theory [] defines the unstable A1-homotopy category over any base scheme. It also contains the proofs of basic functoriality results the most important of which is the glueing theorem [, ]. However in the proof of the Milnor conjecture [] the motivic homotopy theory is used strictly as a machine for dealing with set-level invariants. Thus today we have two lines of development. On the one hand we need to produce detailed proofs of the results used in []. In this paper the motivic homotopy theory is applied only to varieties over field of characteristic zero. It is exactly the context in which one can work in the spirit of [] without

3 going to the category-level invariants. On the other hand I believe that it is necessary to develop the motivic homotopy theory as a theory of category-level invariants. One indication that it is so is that the methods which can be used to prove the fundamental theorems (such as the description of the motivic Steenrod algebra) over fields of charactristic zero fail to generalize to positive characteristics. Another indications is that the only way to prove the motivic analog of the Spanier- Whitehead duality even over fields of characteristic zero which I know uses 2-theories. Finally the 2-theoretic approach may be useful for the future theory of weights in the motivic homotopy. In the next section I will describe the ideal picture of the foundations of the motivic homotopy theory on the next level of development. In the last section I will discuss briefly two possible strategies for proving the conjectures this picture consists of.

0.4 The hypothetical picture The main goal of this stage of development is to clarify the relationships between three types of set-level invariants of Noetherian schemes. These invariants are motivic cohomology, algebraic cobordisms and stable cohomotopy groups. All three of then are associated to a category-level invariant called the motivic stable homotopy 2-theory. The category assigned by this 2-theory to a Noetherian scheme S is called the motivic stable homotopy category over S and denoted below by SH(S). It is a symmetric monoidal category which respect to the smash product. The unit object of this product is called the sphere spectrum and denoted by 1. The category SH(Spec(Z)) has three distinguished objects. The sphere object 1, the Thom spectrum MGL and the Eilenberg-MacLane spectrum HZ. Each of them is deﬁned as by a concrete commutative monoid object in the category of symmetric T -spectra over Spec(Z). They are connected by a sequence of homomorphisms of the form 1 → MGL → HZ. The construction of HZ generalizes and gives a triangulated functor from the derived category of complexes of abelian groups to SH(Spec(Z)) which commutes with direct sums. For any complex of abelian groups C we denote the value of this functor on C by HC . For any Noetherian scheme p : S → Spec(Z) one deﬁnes motivic stable

4 cohomotopy groups, algebraic cobordism and motivic cohomology of as

p,q p,q πs (S)= HomSH(S)(1, Σ 1)

p,q ∗ p,q ∗ MGL (S)= HomSH(S)(p 1, Σ p MGL) p,q ∗ p,q ∗ H (S, Z)= HomSH(S)(p 1, Σ p HZ) ∗ ∗ Below we will denote the objects p MGL and p HC by MGL and HC . The category SH(S) for any S contains a subcategory SHeff (S) generated as a triangulated subcategory closed under direct sums by objects of ! the form p!p (1X ) for smooth schemes p : X → S over S. The suspensions Σ∗,nSHeff (S), n ∈ Z form a filtration of the category SH(S) called the divisibility filtration. The inclusion functors Σ∗,nSHeff (S) → SH(S) have right adjoints whose composition with the inclusion is denoted by d≥n. For any n the cone of the natural transformation d≥n+1 → d≥n is defined up to a canonical isomorphism which gives a functor denoted Sn. For any n the 0,−n rigid eff functor sn = Σ Sn takes values in the subcategory SH (S) of SH (S) which consists of objects orthogonal on the right to Σ∗,1SHeff (S). For any object E of SH(S) the functors d≥n and sn define a Postnikoff tower whose 0,n ∗,n eff stages are Σ sn(E). If E is T-connective i.e. belongs to Σ SH (S) for some n ∈ Z then this tower is bounded on one side and starts with E itself. In this case it provides for any other object F a spectral sequence which p,q+n p,q starts from Hom(F, Σ sn(E)) and “tries to converge” to Hom(F, Σ E).

Conjecture 1 [char] s0(1)= HZ

Consider the canonical morphism 1 → HZ and let H¯Z be its fiber (the desus- pension of its cone). Conjecture ?? is equivalent to the combination of two ∗,1 eff statements. One is that H¯Z belongs to Σ H and another one that HZ is right orthogonal to Σ∗,1SHeff . Let us call the first one the divisibility conjecture and the second one the T-rigidity conjecture. The T-rigidity conjecture (over a given S) asserts that for any smooth scheme X over S one has Hp,q(X, Z)=0 for q < 0. It is a corollary of the Cancellation Theorem which is known when S is a scheme over the spectrum of a field (a proof in the case when resolution of singularities is available can be found in [], the general case using moving lemma for higher Chow groups follows from []). The divisibility part so far is unknown even over a field of characteristic zero.

5 Conjecture 2 [smash] One has an isomorphism

pα,qα HZ ∧ HZ = ∨Σ HAα where Aα are abelian groups.

This is known over a field of characteristic zero and thus over any scheme of characteristic zero. The proof looks as follows. For any S one constructs using finite correspondences (= finite relative cycles [, ]) an additive catgeory of finite correspondences over S and the corresponding abelian category of Nisnevich sheaves with transfers over S. One can define then the unstable A1-homotopy category of sheaves with tranfers DM eff,≤0(S) and the stable homotopy category of sheaves with transfers DM(S). The familiar triangulated category of motives over a field DM eff,−(k) is the full subcategory of connective objects in the s-stable homotopy category of sheaves with transfers. The categories DM(S) and SH(S) are related by a pair of adjoint functors M˜ : SH → DM and K : DM → SH where K is the right adjoint and M˜ is a tensor functor. For any scheme S we have a canonical morphism

[morphism1]HZ → M˜ (1) (1) which is an isomorphism if S is a scheme of characteristic zero or a smooth scheme over Spec(Z). In general this is not clear because it is not clear that the functors M˜ commute with the inverse image functors. Since M˜ is a tensor functor this morphisms deﬁnes for any E in SH(S) a morphism

[morphism2]E ∧ HZ → KM˜ (E) (2)

The four functors formalism for the stable categories SH implies in particular ! ∗ that for a smooth proper morphism p : X → S the objects p!p 1 and p∗p 1 are perfect duals ([, ]) of each other. It is further easy to see that if the morphism (??) is an isomorphism then the morphism (??) is an isomorphism for any E which has a perferct dual. In the case when S is the spectrum of a ﬁeld of characteristic zero the resolution of singularities theorem and the Gysin distinguished squares imply that SH(S) coincides with the smallest triangulated subcategory closed under direct sums which contains objects of ! −1 the form p!p 1 for smooth proper p and T . Thus in this case (??) is an isomorphism for any E.

6 The spectrum HZ is by deﬁnition build out of the suspension spectra of the Eilenberg-MacLane spaces K(Z(n), 2n). Thus it is suﬃcient to prove that ∞ the objects ΣT K(Z(n), 2n)∧HZ are isomorphic to direct sums of suspensions of Eilenberg-MacLane spectra in such a way that the structure morphisms

2,1 ∞ Σ ΣT K(Z(n), 2n) → K(Z(n + 1), 2n + 2) are compatible with the direct sum decompositions. Using isomorphism (??) one reduces this to proving that the objects ˜ ∞ ˜ p,q M(ΣT K(Z(n), 2n)) in DM are direct sums of Tate motives M(S )A where p,q SA are the suspensions of the Moore spectrum corresponding to the abelian group A. This is done in [] and the proof again uses resolution of singularities. It is clear from this outline of the proof of Conjecture ?? that the current approach can not be extended to general S. For example if S is the spectrum of a ﬁeld which is not perfect the catgeory SH(S) is not generated by objects having perfect duals. Conjecture 3 [cobordism] One has isomorphisms

sn(MGL)= HMU2n where MU2n is the abelain group of complex cobordisms of (real) dimenison 2n. ∗ Consider the graded cosimplicial abelian group MU∗ which is the cosimplicial model of the Hopf algebroid of complex cobordisms or equivalently of formal n ∧n group laws where MUm = πm(MU ). For each m the normalization of ∗ the m-th graded part of it is a complex of abelian groups N(MUm). The cohomology groups of this complex are known in topology as the E2-term of the Adams-Novikov spectral sequence. Conjecture 4 One has isomorphisms

2n,0 ∗ sn(1) = Σ HN(MU2n) By the late 80-ies there existed a group of loosely connected conjectures called the motivic conjectures after the Grothendieck’s ”motives”. As far as I understand most of them were originally conjectures about set-level invariants. Conjectures about element-level invariants most notable Beilenson- Lichtenbaum conjectures on L-functions came later. The ﬁrst motivic conjectures were the Grothendieck Standard conjectures which concern properties of the intersection pairing on the groups of algebraic cycles on smooth

7 projective varieties over a field and which can also be formulated in terms of the relations between the Chow groups and Weil cohomology theories. Later came Quillen-Lichtenbaum-Beilinson conjectures about the relations between the algebraic K-theory and the etale cohomology with finite coefficients and Beilinson-Soule vanishing and rigidity conjectures which can also be formulated in terms of the relations bewteen algebraic K-theory and etale cohomology now with rational coefficients. Grothendieck’s definition of motives in the 60-ies was an attempt to construct a set-level invariant for algebraic varieties rich enough to contain information about both groups of algebraic cycles and etale cohomology. The category where this invariant took values was called the category of motives. As far as I understand the first attempt to invent a category-level invariant whose associated set-level invariants would include all the standard cohomology theories known at that time (including the algebraic K-theory) and possibly Grothendieck’s motives as well was made by Deligne and Beilinson and is known as the ”theory of mixed motivic sheaves”. However I would say that it has not yet been systematically studied. To explain what I mean by that I need to say more about category-level invariants in general. In the case of triangulated categories of motives the first question which needs to be addressed is whether or not it is a 2-theory. In fact there are several definitions of triangulated categories of motives which all agree over the spectrum of a perfect field but differ over more general schemes. As far as I can see only one of these vesrions actually forms a 2-theory. In particular the version of [] does not. Once it is established that triangulated categories of motives form a 2-theory the next step will be to address the properties of ∗ ! the corresponding four functors f∗, f , f!, f which do not follow from the axioms of a 2-theory. In particular this concerns the hypothetical property of f ! for regular closed embeddings called the purity conjecture. Of course there are many interesting problems about the triangulated categories of motives which do not directly concern their behavior with respect to the morphisms of schemes but rather their structure over a given scheme. Most importantly we want to know the properties which imply the original motivic conjectures on the set-level. So far we had one big success in dealing with these problems namely the proof of the Milnor Conjecture comparing 2-cotorsion in Milnor’s K-theory and the etale cohomology with Z/2-coefficients. The method employed in this proof leads us directly to the stable A1-homotopy categories which is the next category-level invariant

8 studied by the A1-homotopy theory. The set-level theories associated with the triangulated categories of motives are analogous to the (co-)homology theories in topology which are rep- resentable by generalized Eilenberg-MacLane spectra i.e. to the ordinary cohomology with coefficients in complexes of abelian groups. The stable A1-homotopy categories were invented to accommodate set-level theories on schemes which are not ordinary in this sense. The best known example of such a theory is the algebraic K-theory. One of the reasons such *generalized* theories are interesting is that in many cases one can glue several ordinary theories into a generalized one in such a way that it is easier to compute the value of the resulting compound theory than the value of the theories it was build from. This can happen either because of some formal propereties of the resulting theory or because it happens to admit a particularly convenient geometric description. The former is the case with the Φn theories used in the proof of the Milnor Conjecture. The later is the case with algebraic K- theory and is used for example in the proof of the Bloch-Kato Conjecture for weight 2 by Merkurjev, Suslin and Rost. One of the immediate tasks in the development of the A1-homotopy theory is to work out the details of the proof that the stable A1-homotopy categories form a 2-theory. This is the my main goal for the Fall term of the next academic year (2000). For each scheme S there is a very natural functor going from the stable A1-homotopy category over S to the correspoinding triangulated category of motives. Properties of this functor and its right adjoint over the spectrum of a field of characteristic zero are reasonable well understood. They are described in detail in the forthcoming paper on the cohomological operations in the motivic cohomology which should be out by the end of the spring term of 1999/00. All 2-theories with values in a given 2-category form a 2-category. In particular one can talk about natural transformations of 2-theories. Assuming that triangulated categiers of motives form a 2-theory one may ask whether or not these functors give a natural transformation from the stable homotopy 2-theory to the motivic 2-theory. This is a important open question which is connected with a number of intersting geometric questions. In topology the difference bewteen the derived category of abelian groups which classifies ordinary theories and the stable homotopy category can be understood in terms of ”coefficients”. In fact one can define stable homotopy categories of modules over an appropriate class of ”rings” such that the stable

9 homotopy category itself becomes equivalent to the homotopy category of modules over one ”ring” and the derived category of abelian groups to the homotopy category of modules over another. There is a homomorphism from the first ring to the second which defines the familiar par of adjoint functors bewteen the two categories. The same can be attempted in the A1-homotopy setting. One can define the notion of a ring and there is a clear candidate for the ring modules over which should give the triangulated categories of motives. Over the spectrum of a field of characteristic zero one we know how to show one indeed gets an equivalence between the homotopy category of modules over this ring and the triangulated category of motives. The details of the proof have however yet to be worked out. That the same is true over a general scheme is a conjecture which is very important for our understaning of the general picture of things. The ”ring” related to the triangulated categories of motives as explain above is called the motivic Eilenberg-MacLane spectrum. It defines an object in the stable homotopy category over any scheme which represents the motivic (co-)homology theories. A lot of computations in topology are based on the fcat that the Eilenberg-MacLane spectrum there can be charactersied as the only object which has the same homotopy groups as the sphere spectrum in dimensions ≤ 0 and no homotopy groups in dimensions greater than zero. At the moment it seems likely that an analogopus characterisation of the motivic Eilenberg-MacLane spectrum is possible at least over the spectrum of a fiedl of charactersitc zero. This leads to a number of new ideas concentrated around the idea of the slice filtration and rigid stable homotopy groups which provide much better understaing of the relationship between the motivic and the stable homotopy categories as well as new computations tools. The rigid homotopy groups of many motivic spectra such as the motivic cobordisms spectrum and the sphere spectrum are (modulo the conjectures discussed above) finitely generated abelian groups which do not depend on the base scheme. These groups can be computed using an analog of the Adams spectral sequence. In particular it seems likely that the rigid stable homotopy groups of the sphere spectrum are given by the E2-term of the topological Adams-Novikov spectral sequence and the right homotopy groups of the algebraic cobordism spectrum are the same as the homotopy groups of complex cobordism with the added second grading such that MU2n is in the (2n, n)-component. For general spectra rigid homotopy groups are homotopy invariant sheaves with transfers.

10 There is a spectral sequence which starts from motivic cohomology with coefficients in the rigid homotopy groups and goes to the actual motivic stable homotopy groups. It is called the divisibility spectral sequence. The convergence of this spectral sequence is closely related to the convergence of the motivic Adams spectral sequence. Conjecturally it should converge for finite spectra. The problem is however wide open. A development which may in particular help to prove the convergence of the motivic Adams spectral sequence or of the divisibility spectral sequence is a construction of a geometric model for the stable homotopy category of finite spectra. In topology we know that any element in the stable homotopy groups of spheres can be represented by a manifold with a trivialization of the normal bundle. It should be possible to extend this observation to a construction of a category where morphisms are given by ”correspondences” with trivialized normal bundle of the projection to the first factor whose (unstable) homotopy category will be equivalent to the stable homotopy category of finite complexes. Similarly using complex orientation of the normal bundle instead of a trivialization one should be able to get a model for the stable homotopy category of finite free MU-modules. Some analogs of it should work in the motivic setting. Just as the triangulated categories of motives give (hopefully) geometric models for the homotopy categories of modules over the motivic Eilenberg-MacLane spectrum there should be models for the homotopy catgories of modules over the algebraic cobordisms spectrum and the sphere spectrum. In particular the question about the convergence of the divisibility spectral sequence for algebraic cobordism would benefit greatly from knowing that elements of algebraic cobordism can be represented by some kind of geometric objects. Finally I want to say some words about the unstable A1-homotopy theory. Today it is the least developed part. We know very little. The unstable A1-homotopy categories do not form a 2-theory because the definition of a 2-theory is in a sense a priory stable. The basic functorial properties of these categories are known but there is not good axiomatics. It is unclear whether the unstable A1-homotopy categories constructed in [] are the only possible ones. It may be that there are other unstable theories underlying the same stable one. In particular it is tempting to try to construct an ∗ unstable theory which naturally has functors f! and f and another one ! with f∗ and f which map to the stable theory in a way compatible with the functors f ∗ in the former case and f ! in the later. Even in the case of categories over the spectrum of a field we have very littel understanding

11 about the relationship between the stable and unstable categories. An analog of the Freidental suspension theorem must exist but we have no idea what it is. An analog of the May’s recognition principle for loop spaces remains unknown even hypothetically. The first obstruction here is that it is unclear how to construct a candidate for the motivic En-operades. The situation is somewhat better with the E∞ case because one can use an analog of May’s isometries operad but nothing has been done yet. Eventually I hope to develope a combinatorial model for part of the motivic homotopy theory. This should include a combinatorial category i.e. a category whose objects and morphisms can be explicitly classified together with a functor from this category to schemes (over Spec(Z)). There will be a homotopy theory of objects of this combinatorial category and the corresponding homotiopy categories will map to the motivic homotopy categories. Their image is expected to contain objects such as the Eilenberg-MacLane spectrum, sphere spectrum, cobordisms spectrum and K-theory spectrum and morphisms such as the Steenrod operations. I am toying with a condi- date for this category but it is far from being precise. Another more hypothetical reason why the stable A1-homotopy categories The most important from my point of view open problems in the foundations of this theory Ignoring the historical sequence in which they appeared one can organize them according to our hierarchy as the ones about the element-level, set-level and category-level invariants of algebraic varieties. The triangulated categories of motives provide a framework for the study of algebraic cycles. The groups of morphisms in these categories are closely related to different types of algebraic cycle (co-)homology theories such as the higher Chow groups and Suslin homology. Theories which can be expressed in this way form the class analogous to the class of (co-)homology theories for topological spaces which can be represented by generalized Eilenberg- MacLane spectra. We call them *ordinary* theories. An exmple of an ordinary cohomology theory is the motivic cohomology and in particular Milnor’s K-theory. The higher Chow groups provide an example of an ordinary Borel- Moore homology theory. The Suslin homology form a part of an ordinary homology theory. A basic theory of triangulated categories of motives was developed in http://www.math.uiuc.edu/K-theory/0107/index.html and in http://www.math.uiuc.edu/K-theory/0368/index.html

12 The stable A1-homotopy categories were invented to accommodate (co- )homology theories on schemes which are not ordinary in this sense. The best known example of such a theory is the algebraic K-theory. One of the reasons such *generalized* theories are interesting is that in many cases one can glue several ordinary theories into a generalized one in such a way that it is easier to compute the value of the resulting compound theory than the value of the theories it was build from. As a result generalized theories can be used to obtain information about ordinary ones and ultimately about algebraic cycles. This idea plays major role in the proof of the Milnor conjecture which is essentially a comparison theorem for two ordinary theories. See http://www.math.uiuc.edu/K-theory/0170/index.html or http://www.math.uiuc.edu/K-theory/0316/index.html The triangulated categories of motives and the stable homotopy categories are connected by pairs of adjoint functors. To understand the exact relation bewteen these two categories is one of the major goals of current research. Hypothetically the triangulated category of motives is equivalent to the (homotopy) category of modules over a particular ring spectrum which is by analogy with topology called the Eilenberg-MacLane spectrum. One approach to this problem is based on the use of the analog of the Spanier- Whitehead duality (see below) but it only works in the case when the base is the spectrum of a field and we have resolution of singularities over this field. Another problem is to find an internal characterisation of the Eilenberg- MacLane spectrum. In topology we know that the Eilneberg-MacLane spectra are detrmined by the condition that they are connected and do not have positive homotopy groups. An analogous characterisation hypothetically works in the A1-context but we do not know yet how to prove it. During this year we will concentrate on two topics. One is the extension of the existing theory of triangulated motives from varieties over fields to general schemes. The main remaining problem there can be reformulated in terms of the A1-homotopy theory as the problem of finding a good recognition principle for T-loop spaces. Another one is ordering and reevaluation of many concrete computations obtained for the proof of the Milnor and Bloch-Kato conjectures. Better understanding of both of these topics seems to be necessary for the future construction of the theory of motivic homotopy type. 2. General information. 2. Information on current activities. 3. Plans for the year 2000/01.

13 In the spring term the activities of the A1-homotopy theory program will consist of two weekly lecture series and a weekly seminar. The series by Markus Rost called “Norm varieties” is a continuation of the series with the same title started in the fall. See .... The series by Vladimir Voevodsky called “Motivic Cohomology and Higher Chow Groups” also is a continuation of the “Motivic Cohomology Lectures” started in the fall. The lectures will concentrate on proving the comparison theorem for motivic cohomology introduced in the first part of the series and the higher Chow groups. The proof uses the cohomological theory of homotopy invariant sheaves with transfers in the Nisnevich topology and two other unrelated technical results. The cohomological theory of sheaves with transfers will be reviewed in detail in the first lectures. We will also prove the first of the two technical results - the Suslin’s comparison theorem. The second techical result - the localization theorem for the higher Chow groups, will be used without a proof. The seminar .... The A1-homotopy theory program will continue for the academic year 2000-2001. The formal activities will consist in a lecture series and a seminar. The lectures will be given by Vladimir Voevodsky once or twice a week both in the first and in the second term. A probable topic for the first term is the relationship between the A1- homotopy category and the triangulated category of motives. In particular this includes the theory of cohomological operations in the motivic cohomology. The series will consist of two relatively independent parts. One part starts with the construction of a pair of adjoint functors between the (unstable) A1-homotopy category and an “unstable” version of the triangulated category of motives. Using these functors one proves that over a field of characteristic zero the motives of the objects representing motivic cohomology H2n,n are direct sums of Tate motives. Unfortunately we do not know so far how to extend the methods used in this proof to positive or mixed charactersitc case even if one is willing to assume resolution of singularities. The other part deals with the Steenrod operations in motivic cohomology. One defines Steenrod operations in motivic cohomology over any base and shows that they satisfy the analogs of Cartan formula and Adem relations. Combining the first and the second part one gets in particular a complete description of all bistable cohomological operations in motivic cohomology with Z/l-coefficients over fields of characteristic zero. A probable topic for the second term is the four functors formalism in

14 the stable A1-homotopy theory. The name refers to the two direct image ∗ ! functors f∗, f! and two inverse image functors f , f known in the etale theory and fiber-wise topology. The lectures will start with a a definition of a 2-theory on a category which formalizes standard properties of the four ∗ functors. It will be shown that a *-theory (a theory which only has f and f∗) satisfying certain conditions on the catgeory of schemes can be extended in a unique way to a 2-theory. Applying this construction to the *-theory formed by the stable A1-homotopy categories over Noetherian base schemes one defines the stable A1-homotopy 2-theory. As an application one derives two important corollaries. One is the proper base change theorem which in turn implies the existence of the blow-up long exact sequence for all generalized cohomology theories. The other is a “duality” theorem which implies the Spanier-Whitehead type duality for generalized cohomology theories. It should be noted that two major questions about general functorial properties of the stable A1-homotopy categories remain unavailable to this technique . One is the purity conjecture for regular closed embeddings in the absence of the resolution of singularities. Another one is the finitness conjecture stating that the direct image of the unit with respect to a morphism of finite type is an object of finite type. This year (1999/2000) the homotopy theory seminar was mostly a visitior seminar. Next year we will try to make it function part of the time as a working seminar. It is hard to say exactly what will be happening before the participants are known. One of the possible topics is the computation of the rigid stable homotopy groups in the A1-theory.