the abdus salam united nations educational, scientific and cultural organization international XA0103105 centre international atomic energy agency for theoretical physics *»•**•• AN INTRODUCTION TO PRESHEAVES WITH TRANSFERS AND MOTIVIC COHOMOLOGY Shahram Biglari Available at: http://www.ictp.trieste.it/~pub-off IC/2001/95 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS AN INTRODUCTION TO PRESHEAVES WITH TRANSFERS AND MOTIVIC COHOMOLOGY Shahram Biglari The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. MIRAMARE - TRIESTE August 2001 Introduction The construction of motivic cohomology theories whose existence was predicated through conjec- tures by among others, A. Grothendieck [Kleim], A. Beilinson [Beil], [Bei2] and S. Lichtenbaum [Lichtl],[Licht2], has generated a lot of new research activities in algebraic geometry in the last eight years. The constructions treated in this dissertation have been largely of two types: (1). Motivic cohomology over a field defined as cohomology groups of certain complexes whose terms are given by explicit generators and relations generalizing the Milnor higher K-groups (see [BGSV],[BMS],[Gonch]). This type of construction helped V. Voevodsky in giving a proof for the so-called Milnor conjecture which is a special case of the Bloch-Kato conjecture, see {2.4}. (2). Motivic cohomology of schemes as the cohomology of a complex defined in terms of alge- braic cycles which generalizes the classical definition of Chow groups (see [Bloch], [Bloch2], [BK] and [Grayson]). This last approach, initiated by S. Bloch through his earlier definition of higher Chow groups, has also ramified into several types of motivic theories based on algebraic cycles. The aim of this project is to review some of the central ideas behind these constructions, espe- cially the theory of sheaves with transfers as well as homotopy invariant pretheories discussed in chapter three. We now briefly discuss the contents of this dissertation. Chapter one gives some preliminary ideas of schemes and morphisms, some homological alge- bra in abelian categories and Grothendieck topologies. Chapter two is concerned mainly with K-theory of exact categories, schemes and fields. We indicate briefly the connection between Milnor and Quillen K-theory of fields as well as state the important results of Merkurjev, Suslin, Rost and Voevodsky all of which are special cases of the Bloch-Kato conjecture (see {2.4}). The improved version of relative cycles earlier used in the definition of higher Chow groups by S. Bloch, are defined in chapter three (see {3.1} and {3.2}). This version is due to A. Suslin and V. Voevodsky. We then introduce presheaves with transfers along with their generalization called homotopy invariant pretheories providing examples arising in Etale cohomology and Algebraic K-theory. In the last chapter we define motivic cohomology and Bloch's higher Chow groups as well as indicate some connection between motivic cohomology and higher Chow groups as conjectured by S. Bloch [Bloch]. We also indicate the connection between Quillen K-theory and higher Chow groups which generalizes an earlier theorem of A. Grothendieck. In the process of doing the above, we also mention a link between Suslin's singular homology for schemes and etale cohomology. Chapter 1 Background materials 1.1 Schemes and morphisms Zariski sheaves 1.1.1 Let X be topological space. Set Xxop to be a category whose objects are non-empty open subsets of X with inclusions as morphisms. Call XrOp a Zariski site. Define a (Zariski) sheaf of commutative rings on X to be a contravariant functor (or a presheaf) F: Xxop —> CRings such that the local property holds. That is the following sequence is exact for any covering {Ui}i^\ of an open subset U in X. o _>F (U) -> rii F(Ui)i Uij F(ut n Uj). The exactness of this sequence means that the second arrow is injective and the set of all points at which two pararel arrows take the same value is equal to the image of the second arrow. Similarly a sheaf of "sets with structures" can be denned. Schemes 1.1.2 A locally ringed space consists of a pair (X,A) of a topological space X and a sheaf A of commutativ e rings on (the site) XTOP such that all stalks Ax:=limA{U) xeu are local rings. Let (X, A) be a (locally) ringed space and let /: X —> Y be a continous map into a topological space Y. Then there is a natural way to get a sheaf f*A on Y. This sheaf sends 1 an object V of site YrOp to A(f~ (V)). A morphism between two locally ringed spaces (X,A) and (Y,B) is a continous map f-.X—tY together with a morphism of sheaves 4>:B —> f*A such that the induced map cpx:Bf^ —> Ax is a local homomorphism of rings for any x € X. An affine scheme (Spec(R),R) is a locally ringed space with the undelying topological space as Spec(R): = the set of prime ideals in the commutative ring R equipped with the Zariski topology; a closed set is of the form V{E) :— {p G Spec(A) | p D E} for some subset E in R, and the sheaf R with value Rtf^1} at D(/) := Spec(R) — V(/) for each non-nilpotent element / € R. And finally a scheme is a locally ringed space (X, @x) which is locally affine, that is for each x G X there exists an open neighborhood U of x so that the locally ringed space (U, &x\U) is affine. Let (X, &x) be a scheme. An Ox~module is a sheaf of Z—modules which is campatible with Ox-structure, i.e. for each open subset U of X there is a T(U, &x) -module structure on F(U) so that ifVCU the morphism F(U) -> F(V) is campatible with those structures. The category of all &x~modules will be denoted by Cx — mod. Unramified morphisms 1.1.3. Let S be a scheme. An S—scheme f:X —» 5 is said to be unramified at a ponit x G X if / is locally of finite presentation type at x, the maximal ideal of &x,x is generated by the one of @s,f{x) through the morphism of sheaves 0s -> f*^x, and k(x) is a finite separable extension of k(f{x)). If the base scheme is spectrum of a field k, then an S—scheme X is unramified if and only if X is a disjoint union of finitely many spectra of fields, each of which a separable field extension of k. Etale morphisms 1.1.4 A 5-scheme f:X ->• S is said to be flat at a point x € X just when @x,x is a fl^ ^s./(z)~alget>ra. An etale morphism at a point, is one which is both fiat and unramified at that point. Smooth morphisms 1.1.5 A morphism /: X —> S is called smooth of relative dimension n at a given point x 6 X if there is an open neighborhood U of x in X and a commutative diagram where 3 is etale and p is the canonical projection. A morphism f:X —> 5 is smooth if it is smooth at all points of X. An important property of smooth schemes is that for a smooth morphism f:X -» S if 5 is reduced (respectively normal, respectively regular) then X is also reduced (respectively normal, respectively regular). Proposition 1.1.6 [EGA IV2, 2.4.6] A locally of finite presentation morphism which is flat, is open. Proposition 1.1.7 If X is a smooth scheme over a field k, then the set of closed points x € X so that k(x) is separable over k is dense in X. Proof Let XQ be a point and let U be a neighborhood of x provided by smoothness. Hence the structure morphism factors as U —2-> A£ —-> Spec(k) where g is etale. If a; is in U, then since g is unramified k(x) is separable and finite over k(g(x)), hence the statement follows once we show that g(U) contains a closed point y such that k(y) is a separable extension of k. To prove this, note that by 1.1.6 g(U) is open, being the image of an open set by a flat morphism. Now it is clear that the set of closed points of A£ having separable residue field extension over k is dense; indeed if k is perfect then every closed point of A£ has an algebraic residue field extension over k hence the density of that set becomes clear. If k is not perfect, then it has infinitely many points so the set of k—rational points in A£ is dense. • Proposition 1.1.8 Let X be a locally of finite type scheme over a field. If X is geometrically reduced then the smooth locus of X is open and dense in X. 1.2 Some homological algebra in abelian categories 1.2.1 Abelian Categories An additive category is one for which all Horn sets are Abelian groups, the composition of morphisms is bilinear and finite products and coproducts exist. It is clear by definition that such a category contains a unique zero object. An additive funtor from an additive category to some other additive category is one which induces homomorphisms of abelian groups between Horn sets. An additive functor F: A —>• B is said to be fully faithful if for any objects A and A' in A' the map H.om(A, A') —> H.om(F(A),F(A')) is a bijective homomorphism.