Oriented Bivariant Theories, I

Transcription

1 ESI The Erwin Schrödiner International Boltzmannasse 9 Institute or Mathematical Physics A-1090 Wien, Austria Oriented Bivariant Theories, I Shoji Yokura Vienna, Preprint ESI April 27, 2007 Supported by the Austrian Federal Ministry o Education, Science and Culture Available via

2 ORIENTED BIVARIANT THEORIES, I SHOJI YOKURA ABSTRACT. In 1981 W. Fulton and R. MacPherson introduced the notion o bivariant theory BT, which is a sophisticated uniication o covariant theories and contravariant theories. This is or the study o sinular spaces. In 2001 M. Levine and F. Morel introduced the notion o alebraic cobordism, which is a universal oriented Borel Moore unctor with products OBMF o eometric type, in an attempt to understand better V. Voevodsky s hiher alebraic cobordism. In this paper we introduce a notion o oriented bivariant theory OBT, a special case o which is nothin but the oriented Borel Moore unctor with products. The present paper is a irst one o the series to try to understand Levine Morel s alebraic cobordism rom a bivariant-theoretical viewpoint, and its irst step is to introduce OBT as a uniication o BT and OBMF. 1. INTRODUCTION William Fulton and Robert MacPherson have introduced the notion o bivariant theory as a cateorical ramework or the study o sinular spaces, which is the title o their AMS Memoir book [FM] see also Fulton s book [Fu]. The main objective o [FM] is bivarianttheoretic Riemann Roch s or bivariant analoues o various theorems o Grothendieck Riemann Roch type. Vladimir Voevodsky has introduced alebraic cobordism now called hiher alebraic cobordism, which was used in his proo o Milnor s conjecture [Voe]. Daniel Quillen introduced the notion o complex oriented cohomoloy theory on the cateory o dierential maniolds [Qui] and this notion can be ormally extended to the cateory o smooth schemes in alebraic eometry. Marc Levine and Fabien Morel constructed a universal oriented cohomoloy theory, which they also call alebraic cobordism, and have investiated urthermore see [L1, L2, LM1, LM2, LM3] and also see [Mer] or a condensed review. Recently M. Levine and R. Pandharipande [LP] ave another equivalent construction o the alebraic cobodism via what they call double point deeneration and they ound a nice application o the alebraic cobordism in the Donaldson Thomas theory o 3-olds. In this paper we extend Fulton MacPherson s bivariant theory to what we call an oriented bivariant theory or a eneral cateory, not just or a eometric cateory o, say, complex alebraic varieties, schemes, etc. In most interestin cases bivariant theories such as bivariant homoloy theory, bivariant Chow roup theory, bivariant alebraic K-theory and bivariant topoloical K-theory are already oriented bivariant theories. We show that even in this eneral cateory there exists a universal oriented bivariant theory, whose special case ives rise to a universal oriented Borel Moore unctor with products. Levine Morel s alebraic cobordism requires more eometrical conditions. Indeed, they call alebraic cobordism a universal oriented Borel Moore unctor with products o eometric type [LM3]. In a second paper [Yo1] we will deal with an oriented bivariant theory o eometric type. * Partially supported by Grant-in-Aid or Scientiic Research No and No , the Ministry o Education, Culture, Sports, Science and Technoloy MEXT, and JSPS Core-to-Core Proram 18005, Japan. 1

3 2 SHOJI YOKURA One purpose o this paper is to brin Fulton MacPherson s Bivariant Theory to the attention o people workin on alebraic cobordism and/or subjects related to it. 2. FULTON MACPHERSON S BIVARIANT THEORY We make a quick review o Fulton MacPherson s bivariant theory [FM]. Let V be a cateory which has a inal object pt and on which the iber product or iber square is well-deined. Also we consider a class o maps, called conined maps e.., proper maps, projective maps, in alebraic eometry, which are closed under composition and base chane and contain all the identity maps, and a calss o iber squares, called independent squares or conined squares, e.., Tor-independent in alebraic eometry, a iber square with some extra conditions required on morphisms o the square, which satisy the ollowin: i i the two inside squares in X h X X or Y Y h Y X Y h h X Y Z h are independent, then the outside square is also independent, ii any square o the ollowin orms are independent: Z X id X X X Y where : X Y is any morphism. id X Y idx Y X Y A bivariant theory B on a cateory V with values in the cateory o raded abelian roups is an assinment to each morphism X Y in the cateory V a raded abelian roup in the rest o the paper we inore the radin BX Y which is equipped with the ollowin three basic operations: Product operations: For morphisms : X Y and : Y Z, the product operation id Y is deined. : BX Y BY Z BX Z

4 ORIENTED BIVARIANT THEORIES, I 3 Pushorward operations: For morphisms : X Y and : Y Z with conined, the pushorward operation : BX Z BY Z is deined. Pullback operations: For an independent square the pullback operation X Y X Y, : BX Y BX Y is deined. And these three operations are required to satisy the seven compatibility axioms see [FM, Part I, 2.2] or details: B-1 product is associative, B-2 pushorward is unctorial, B-3 pullback is unctorial, B-4 product and pushorward commute, B-5 product and pullback commute, B-6 pushorward and pullback commute, and B-7 projection ormula. Let B, B be two bivariant theories on a cateory V. Then a Grothendieck transormation rom B to B γ : B B is a collection o homomorphisms BX Y B X Y or a morphism X Y in the cateory V, which preserves the above three basic operations: i γα B β = γα B γβ, ii γ α = γα, and iii γ α = γα. For more details o interestin eometric and/or topoloical examples o bivariant theories e.., bivariant theory o constructible unctions, bivariant homoloy theory, bivariant K-theory, etc., and Grothendieck transormations amon bivariant theories, see [FM]. In this paper we treat with bivariant theories more abstractly rom a eneral viewpoint. A bivariant theory uniies both a covariant theory and a contravariant theory in the ollowin sense: B X := BX pt become a covariant unctor or conined morphisms and B X := BX id X become a contravariant unctor or any morphisms. And a Grothendieck transormation γ : B B induces natural transormations γ : B B and γ : B B. As to the radin, B i X := B i X id X and B j X := B j X id X. In the rest o the paper we assume that our bivariant theories are commutative see [FM, 2.2], i.e., i whenever both

5 4 SHOJI YOKURA W X W Y Y Z X Z are independent squares, then or α BX Z and β BY Z α β = β α. Note: i α β = 1 deαdeβ β α, then it is called skew-commutative. Another assumption in the rest o the paper, which is not in Fulton MacPherson s bivariant theory, is the ollowin additivity: BX Y Z = BX X Z BY Y Z. When we want to emphasize this additivity, we call such a bivariant theory an additive bivariant theory. Deinition 2.1. [FM, Deinition, Part I] Let S be a class o maps in V, which is closed under compositions and containin all identity maps. Suppose that to each : X Y in S there is assined an element θ BX Y satisyin that i θ = θ θ or all : X Y, : Y Z S and ii θid X = 1 X or all X with 1 X B X := BX idx X the unit element. Then θ is called a canonical orientation o. Note that such a canonical orientation makes the covariant unctor B X a contravariant unctor or morphisms in S, and also makes the contravariant unctor B a covariant unctor or morphisms in C S: Indeed, * or a morphism : X Y S and the canonical orientation θ on S the ollowin Gysin homomorphism! : B Y B X deined by! α := θ α is contravariantly unctorial. And ** or a iber square which is an independent square by hypothesis X Y id X idy X Y, where C S, the ollowin Gysin homomorphism! : B X B Y deined by! α := α θ is coavariantly unctorial. The notation should carry the inormation o S and the canonical orientation θ, but it will be usually omitted i it is not necessary to be mentioned. Note that the above conditions i and ii o Deinition 2.1 are certainly necessary or the above Gysin homomorphisms to be unctorial. Deinition 2.2. i Let S be another class o maps in V, called specialized maps e.., smooth maps in alebraic eometry, which is closed under composition and under base chane and containin all identity maps. Let B be a bivariant theory. I S has canonical orientations in B, then we say that S is canonical B-orientable and an element o S is

6 ORIENTED BIVARIANT THEORIES, I 5 called a canonical B-orientable morphism. O course S is also a class o conined maps, but since we consider the above extra condition o B-orientability on S, we ive a dierent name to S. ii Let S be as in i. Let B be a bivariant theory and S be canonical B-orientable. Furthermore, i the orientation θ on S satisies that or a iber square with S the ollowin condition holds X X Y Y θ = θ, which means that the orientation θ preserves the pullback operation, then we call θ a nice canonical orientation and say that S is nice canonical B-orientable and an element o S is called a nice canonical B-orientable morphism. Proposition 2.3. Let B be a bivariant theory and let S be as above. 1 Deine the natural exterior product by : BX pt BY πy pt BX Y pt α β := π Y α β. Then the covariant unctor B or conined morphisms and the contravariant unctor B or morphisms in S are both compatible with the exterior product, i.e., or conined morphisms : X X, : Y Y, α β = α β and or morphisms : X X, : Y Y in S, by! α β =! α! β. 2 Similarly, deine the natural exterior product : BX idx X BY idy Y BX Y idx X X X α β := p 1 α p 2 β where p 1 : X Y X and p 2 : X Y Y be the projections. Then the contravariant unctor B or any morphisms and the covariant unctor B or morphisms in C S are both compatible with the exterior product, i.e., or any morphisms : X X, : Y Y, α β = α β and or morphisms : X X, : Y Y in C S,! α β =! α! β. Proo. The proo is tedious, usin several axioms o the bivariant theory. For the sake o completeness we ive a proo. But, we ive a proo or only 1 and a proo or 2 is let or the reader. For morphisms : X X and : Y Y, consider the ollowin bi commutative diarams

7 6 SHOJI YOKURA X Y Id X Id Y X Y Id X p Id Y pt Y = Y X Y Id Y X Y p Id Y pt Y = Y Id X q q Id X q X = X pt X = X pt p The proo o α β = α β oes as ollows: pt. α β = q α β by deinition = q α β = Id X Id Y q α β = Id X Id Y q α β by B-4 = Id X Id Y q α β by B-6 = Id X q α β by B-6 = q α β by B-7 = α β by deinition Next we show! α β =! α! β. For this, irst we observe that On one hand we have that! := θ θ.! α β = θ θ q α β by deinition On the other hand we have that = θ θ q α β = Id X Id Y q θ p Id Y θ q α β! α! β = θ α θ β by deinition = q θ α θ β = q θ α θ β = Id Y q α q α θ β = Id X Id Y q θ q α θ β = Id X Id Y q θ p Id Y θ q α β The last equality ollows rom the commutativity o the bivariant theory. Thus we et the above equality.

8 ORIENTED BIVARIANT THEORIES, I 7 Proposition 2.4. Let B be a bivaraint theory on a cateory V with a class C o conined morphisms. Let S be a class o nice B-orientable morphisms. Then or any independent square X Y X Y with S and C, the ollowin diaram commutes: B Y B X B Y B X. Here we remark the ollowin act about the covariant and contravariant unctors B and B, which will be needed in later sections. They are what Levine and Morel call Borel Moore unctor with products in [LM3, Mer]. Proposition 2.5. Let the situation be as above. 1-i For a disjoint sum X Y, B X Y = B X B Y, 1-ii or conined morphisms : X Y, the pushorward homomorphisms : B X B Y are covariantly unctorial, 1-iii or morphisms in S, i.e., or nice canonical B-orientable morphisms : X Y, the Gysin pullback homomorphisms are contravariantly unctorial, 1-iv or an independent square! : B Y B X X Y X with C and S, we have the ollowin diaram commutes: B Y! Y B X B Y! B X, 1-v the pushorward homomorphisms : B X B Y or conined morphisms and the Gysin pullback homomorphisms! : B Y B X or morphisms in S are both compatible with the exterior products 2-i or a disjoint sum X Y, : B X B Y B X Y. B X Y = B X B Y,

9 8 SHOJI YOKURA 2-ii or any morphisms : X Y, the pullback homomorphisms : B Y B Y are contravariantly unctorial, 2-iii or conined and specialized morphisms in C S, i.e., or conined and nice canonical B-orientable morphisms : X Y, the Gysin pushorward homomorphisms are covariantly unctorial, 2-iv or an independent square! : B X B Y X Y X with C S, we have the ollowin diaram commutes: B Y Y B X!! B Y! B X, 2-v the pullback homomorphisms : B Y B Y or any morphisms : X Y and the Gysin pushorward homomorphisms! : B X B Y or conined and specialized morphisms in C S are both compatible with the exterior products : B X B Y B X Y. 3. A UNIVERSAL BIVARIANT THEORY The ollowin theorem is about the existence o a universal one o the bivariant theories or a iven cateory V with a class C o conined morphisms, a class o independent squares and a class S o specialized morphisms. The construction is motivated by that o Levine Morel s alebraic cobordism. Theorem 3.1. Let V be a cateory with a class C o conined morphisms, a class o independent squares and a class S o specialized maps. We deine M C SX Y to be the ree abelian roup enerated by the set o isomorphism classes o conined morphisms h : W X such that the composite o h and is a specialized map: h C and h : W Y S. 1 The association M C S is a bivariant theory i the three operations are deined as ollows: Product operations: For morphisms : X Y and : Y Z, the product operation is deined by m V [V hv X] V : M C SX Y M C SY Z M C SX Z W n W [W kw Y ] := V,W m V n W [V h V k W X],

10 ORIENTED BIVARIANT THEORIES, I 9 where we consider the ollowin iber squares V h V X k W k W V h V W k W X Y Z. Pushorward operations: For morphisms : X Y and : Y Z with conined, the pushorward operation : M C SX Z M C SY Z is deined by V n V [V hv X] := V n V [V hv Y ]. Pullback operations: For an independent square X X the pullback operation Y Y, : M C S X Y M C S X Y is deined by V n V [V hv X] := V n V [V h V X ], where we consider the ollowin iber squares: h V V h V V X Y X Y. 2 Let BT be a class o bivariant theories B on the same cateory V with a class C o conined morphisms, a class o independent squares and a class S o specialized maps. Let S be nice canonical B-orientable or any bivariant theory B BT. Then, or each bivariant theory B BT there exists a unique Grothendieck transormation γ B : M C S B such that or a specialized morphism : X Y S the homomorphism γ B : M C S X Y BX Y satisies the normalization condition that γ B [X idx X] = θ B.

11 10 SHOJI YOKURA Proo. For 1, we have to show that the three bivariant operations are well-deined, but we show only the well-deinedness o the bivariant product and the other two are clear. Let [V hv X] M C S X Y and [W kw Y ] M C S Y Z; thus h V : V X is conined and the composite h V : V Y is in S, and also k W : W Y is conined and the composite k W : W Z is in S. By deinition we have [V hv X] [W kw Y ] = [V h V k W X]. We want to show that [V h V k W X] M C S X Z, i.e., h V k W S. From the iber squares iven in Product operations above, we have h V k W = k W h V. h V is in S, because it is the pullback o h V and h V is in S and S is closed under base chane by hypothesis. k W is in S by hypothesis. Thus the composite k W h V is also in S. Thus the bivariant product is well-deined. The rest is to show that these three operations satisy the seven axioms B-1 B-7, which is let or the reader. For 2, irst we show the uniqueness. Suppose that there exists a Grothendieck transormation γ : M C SX Y BX Y such that or any : X Y S the homomorphism γ : M C S X Y BX Y satisies that γ[x idx X] = θ B. Note that or any : X Y S, [X idx X] M C S X Y is a nice canonical orientation, i.e., θ M C S = [X idx X]. Let h V : V X be a conined map such that h V : V Y is in S. We have that [V hv X] = h V [V idv V ], where [V idv V ] M C hv S V Y. Since h V S by hypothesis, it ollows rom the normalization that we et Thus it is uniquely determined. The rest is to show that the assinment γ[v hv X] = γh V [V idv V ] = h V γ[v idv V ] = h V θ B h V. γ : M C SX Y BX Y deined by γ B [V hv X] = h V θ B h V is a Grothendeick transormation, i.e., that it preserves the three bivariant operations: i it preserves the product operation: Lettin the situation be as in i, it suices to show that γ B [V hv X] [W kw Y ] = γ B [V hv X] γ B [W kw Y ].

12 ORIENTED BIVARIANT THEORIES, I 11 Usin the iber squares iven in Product operations, we have γ B [V hv X] [W kw Y ] = γ B [V h V k W X] by the deinition = h V k W θ B h V k W by the deinition = h V k W θ B k W h V = h V k W θb h V θ B k W = h V k W k W θ B h V θ B k W = h V θ B h V k W θ B k W by B-7 projection ormula = h V θ B h V k W θ B k W by B-4 = γ B [V hv X] γ B [W kw Y ]. ii it preserves the pushorward operation: Consider X Y Z and a conined morphsim h V : V X such that the composite h V : V Y is in S. γ B [V hv X] = γ B [V hv Y ] = h V θ B h V = h V θ B h V = γ B [V hv X] iii it preserves the pullback operation: Consider a conined morphsim h V : V X such that the composite h V : V Y is in S and the iber squares iven in Pullback operations above, we have γ B [V hv X] = γ B [V h V X ] = h V θ B h V = h V θ B h V = h V θ B h V by B-5 This completes the proo o the theorem. = γ B [V hv X]. Remark By the deinition o M C S, any class S is nice canonical MC S -orientable. 2 The product operation : M C S X Y M C S Y Z M C S X Z can also be interpreted as ollows. The ree abelian roup MX enerated by the set o isomorphism classes o conined morphisms h V : V X is a commutative rin by the iber product h [V 1 h 1 X] 2 h [V2 X] := [V1 X V 1 Xh 2 2 X]. For a conined morphism : X Y we have the pushorward homomorphism : MX MY and or any morphism : X Y we have the pullback homomorphism : MY MX. Then the product operation is nothin but [V hv X] [W hw Y ] = [V hv X] [W hw Y ]. But in our case we need to chase the morphisms involved, so we just stick to our previous presentation.

13 12 SHOJI YOKURA Let S be a class o specialized morphisms as above and let S be canonical B-orientable or a bivariant theory B. I π X : X pt is in S, in which case we sometimes say, abusin words, that X is specialized, then we have the Gysin homomorphism which, we recall, is deined to be π X! : B pt B X π X! α = θ B π X α. In particular, i we let 1 pt Bpt be the unit, then we have π X! 1 pt = θ B π X 1 pt = θ B π X. This element π X! 1 pt = θ B π X is called the undamental class o X associated to the bivariant theory B c. [LM3, Mer], denoted by [X] B. Corollary 3.3. Let BT be a class o additive bivariant theories B on the same cateory V with a class C o conined morphisms, a class o independent squares and a class S o specialized maps. Let S be nice canonical B-orientable or any bivariant theory B BT. Then, or each bivariant theory B BT, 1 there exists a unique natural transormation Γ B : M C S B such that i π X : X pt is in S the homomorphism Γ B : M C S X B X satisies that Γ B [X idx X] = π! X 1 pt = [X] B, and 2 there exists a unique natural transormation Γ B : M C S B such that or any X the homomorphism Γ B : M C S X B X satisies that Γ B [X idx X] = 1 X B X. Example 3.4. Here we recall some important examples o bivariant theories rom [FM]. 1 Bivariant homoloy theory H: Let V be the cateory o complex varieties embeddable into smooth maniolds, C = Prop be the class o proper morphisms and let S = Lci be the class o local complete intersection abbr., l.c.i. morphisms. A Tor-independent square is an independent square. Then there is a unique Grothendieck transormation γ H : M Prop Lci H such that or : X Y Lci the homomorphism γ H : M Prop Lci X Y HX Y satisies the normalization condition that γ H [X idx X] = U. In particular, we have unique natural transormation: γ H : M Prop Lci H = H BM such that or any l.c.i. variety X, γ H [X idx X] = [X] H BM X and γ H : M Prop Lci H = H such that or any variety X, γ H [X idx X] = 1 X H X. Here H BM X is the Borel Moore homoloy roup and H X is the usual cohomoloy roup. 2 Bivariant Chow roup theory or Operational bivariant Chow roup theory A: Let V be the cateory o schemes, C = Prop be the class o proper morphisms and let S = Lci

14 ORIENTED BIVARIANT THEORIES, I 13 be the class o local complete intersection abbr., l.c.i. morphisms. A Tor-independent square is an independent square. Then there is a unique Grothendieck transormation γ A : M Prop Lci A such that or : X Y Lci, γ A [X idx X] = []. We have unique natural transormations γ A : M Prop Lci A = A or CH such that or any l.c.i. variety X, γ A X] = [X] A X and γ A : M Prop Lci A = A or CH [X idx such that or any variety X, γ A [X idx X] = 1 X A X. Here A = CH is the Chow homoloy roup and A = CH is the Chow cohomoloy roup see [Fu]. 3 Bivariant alebraic K-theory K al : Let the situation be as in 2. Then there is a unique Grothendieck transormation γ Kal : M Prop Lci K al such that or : X Y Lci, γ Kal [X idx X] = O. We have unique natural transormations γ Kal : M Prop Lci K al = K 0 such that or any l.c.i. variety X, γ Kal [X idx X] = [O X ] K 0 X and γ Kal : M Prop Lci K al = K 0 such that or any variety X, γ Kal [X idx X] = 1 X K 0 X. 4 Bivariant topoloical K-theory K top : Let the situation be as in 2 with V bein the cateory o complex varieties. Then there is a unique Grothendieck transormation γ Ktop : M Prop Lci K top such that or : X Y Lci, γ Ktop [X idx X] = Λ. We have unique natural transormation γ Ktop : M Prop Lci K top = K 0 such that or any l.c.i. variety X, γ Ktop [X idx X] = [1 X ] K 0 X and γ Ktop : M Prop Lci Ktop = K 0 such that or any variety X, γ Ktop [X idx X] = 1 X K 0 X. 5 Bivariant theory o constructible unctions F: Let V be the cateory o complex alebraic varieties, C = Prop be the class o proper morphisms and let S = Sm be the class o smooth morphisms not local complete intersection morphisms. A iber square is an independent square. Then there is a unique Grothendieck transormation γ F : M Prop Sm F such that or : X Y Sm, γ F [X idx X] = 1 = 1 X. We have unique natural transormations γ F : M Prop Lci F = F

15 14 SHOJI YOKURA such that or any smooth variety X, γ F [X idx X] = 1 X FX and γ F : M Prop Lci F such that or any variety X, γ F [X idx X] = 1 X F X. Here F X is the abelian roup o locally constant unctions on X. Corollary 3.5. A naïve motivic bivariant characteristic class Let cl : K 0 H R be a multiplicative characteristic class o complex vector bundles with a suitable coeicients R. Then there exists a unique Grothendieck transormation γ cl H : M Prop Lci H R satisyin the normalization condition that or : X Y Lci γh cl idx [X X] = clt U. Corollary 3.6. A naïve motivic characteristic class o sinular varieties Let cl : K 0 H R be a multiplicative characteristic class o complex vector bundles with a suitable coeicients R. Then there exists a unique natural transormation cl : M Prop Lci H R such that or a l.c.i variety V in a smooth variety cl [V idv V ] = clt V [V ]. A urther discussion on motivic bivariant characteristic classes will be done in [Yo2] Remark 3.7. Riemann Roch Theorems We have the ollowin commutative diarams: K al γ Kal M Prop Lci α γktop γ Ktop K top K top ch M Prop Lci td γh i α : K al K top is a Grothendieck transormation such that or a l.c.i morphism : X Y, αo = Λ. ii ch : K top H Q is a Grothendieck transormation such that or an l.c.i morphism : X Y, αλ = tdt U, where T is the virtual relative tanent bundle o the morphism and tdt is the total Todd cohomoloy class o the bundle T. The composite ch α : K al H Q is a bivariant version o Baum Fulton MacPherson s Riemann Roch τ : K 0 H Q see [BFM1, Fu]. Thus one could say that ch α : K al H Q and ch : K top H Q are realizations o a motivic one: γh td : MProp Lci H Q. 4. ORIENTED BIVARIANT THEORIES Here we introduce an orientation to bivariant theories. Deinition 4.1. Let B be a bivariant theory on a cateory. Let L be a class o morphisms in V, called line bundles e.., line bundles in eometry, which are closed under base chane. As in the theory o bundles, or a line bundle L X, we simply denote it by the source object L, unless some conusion is possible. For a line bundle L L, the irst Chern class operator on B associated to the line bundle L, denoted by c 1 L, is an endomorphism which satisies the ollowin properties: c 1 L : BX Y BX Y H Q

16 ORIENTED BIVARIANT THEORIES, I 15 O-1 identity: I two line bundles L X and M X are isomorphic, i.e., there exists an isomorphism L = M over X, then we have c 1 L = c 1 M : BX Y BX Y. O-2 commutativity: Let L X and L X be two line bundles over X, then we have c 1 L c 1 L = c 1 L c 1 L : BX Y BX Y. O-3 compatibility with product: For morphisms : X Y and : Y Z, α BX Y and β BY Z, a line bundle L X and a line bundle M Y c 1 Lα β = c 1 Lα β, c 1 Mα β = α c 1 Mβ O-4 compatibility with pushorward: With bein conined c 1 Mα = c 1 M α. O-5 compatibility with pullback: For an independent square X X Y Y c 1 Lα = c 1 L α. The above irst Chern class operator is called an orientation and a bivariant theory equipped with such a irst Chern class operator is called an oriented bivariant theory, denoted by OB. An oriented Gorthendieck transormation between two oriented bivariant theories is a Grothendieck transormation which preserves or is compatible with the irst Chern class operator. Remark 4.2. In the above deinition, the only requirement on the class L is that morphisms are closed under base chane, and nothin else. The ollowin lemma shows that Levine Morel s oriented Borel Moore unctor with products is a special case o an oriented bivariant theory. Lemma 4.3. Let OB be an oriented bivariant theory. Then the orientation c 1 L on the unctors OB and OB satisies the ollowin properties: 1 Let L X and L X be two line bundles over X, then we have c 1 L c 1 L = c 1 L c 1 L : OB X OB X, c 1 L c 1 L = c 1 L c 1 L : OB X OB X, and i L and L are isomorphic, then we have that c 1 L = c 1 L or both OB and OB. 2 For a line bundle L X and α OB X and β OB Y, we have c 1 Lα β = c 1 p 1 Lα β. Also, or α OB X and β OB Y, we have c 1 Lα β = c 1 p 1 Lα β. Here p 1 : X Y X is the projection. 3 For a conined morphism : X Y or a line bundle L Y, we have c 1 M = c 1 M : OB X OB Y. 4 For a specialized morphism : X Y S here we just require that is canonical OB-orientable and a or line bundle M Y, we have c 1 M! =! c 1 M : OB Y OB X.

17 16 SHOJI YOKURA 5 For a conined and specialized morphism : X Y and a line bundle M Y, we have! c 1 M = c 1 M! : OB X OB Y. 6 For any morphism : X Y and a line bundle M Y, we have c 1 M = c 1 M : OB Y OB X. Proo. 1 ollows rom O-1 and O-2. 2 ollows rom the irst ormula o O-3. 3 ollows rom O-4. 4 ollows rom the second ormula o O-3. 5 ollows rom the irst ormula o O-3 and O-4. 6 ollows rom O-5. Example 4.4. All the examples, except the bivariant constructible unctions F, iven in Example 3.4 are in act oriented bivariant theories. The Chern class operator is by takin the bivariant product with a bivariant element in the bivariant roup o the identity X idx X. And it ollows rom the axioms o the bivariant theory that this operator satisies the properties O-2 O-5. 1 Bivariant homoloy theory H: For a line bundle L X, the irst Chern class operator c 1 L : HX Y HX Y is deined by c 1 Lα := c 1 L α where c 1 L HX idx X = H X is the irst Chern cohomoloy class o the line bundle. 2 Bivariant Chow roup theory or Operational bivariant Chow roup theory A: As in 1 above, or a line bundle L X, the Chern class operator is deined by c 1 L : AX Y AX Y c 1 Lα := c 1 L α where c 1 L AX idx X is the irst Chern cohomoloy class o the line bundle. 3 Bivariant alebraic K-theory K al : For a line bundle L X, the Chern class operator c 1 L : K al X Y K al X Y is deined by c 1 Lα := 1 [L] α where [L] K al X idx X = Kal 0 X the Grothendieck roup o alebriac vector bundles or locally ree sheaves on X. 4 Bivariant topoloical K-theory K top : For a line bundle L X, the Chern class operator is deined by c 1 L : K top X Y K top X Y c 1 Lα := 1 [L] α

18 ORIENTED BIVARIANT THEORIES, I 17 where [L] K top X idx X = KtopX 0 the Grothendieck roup o vector bundles on X. 5 As to the bivariant theory F, as ar as the author knows, there is not available a bivariant class ce F X = FX idx X associated to a vector bundle E over X, which is just the multiplication by locally constant unctions. Even i such a Chern class is available, we do not know what ce would be. So, it is not clear whether there is a reasonable Chern class operator in the case o the bivariant theory F. In act, ollowin Levine Morel s construction [LM3], we show the existence o a universal one o such oriented bivariant theories or any cateory. Let us consider a morphism h V : V X equipped with initely many line bundles over the source variety V o the morphism h V : V hv X; L 1, L 2,, L r with L i bein a line bundle over V. This amily is called a cobordism cycle over X [LM3, Mer]. Then V hv X; L 1, L 2,, L r is isomorphic to W hw X; M 1, M 2,, M r i and only i h V and h W are isomorphic, i.e., there is an isomorphism : V = W over X, there is a bijection σ : {1, 2,, r} = {1, 2,, r } so that r = r and there are isomorphisms L i = M σi or every i. Theorem 4.5. A universal oriented bivariant theory Let V be a cateory with a class C o conined morphisms, a class o independent squares, a class S o specialized morphisms and a class L o line bundles. We deine OM C SX Y to be the ree abelian roup enerated by the set o isomorphism classes o cobordism cycles over X such that the composite o h and [V h X; L 1, L 2,, L r ] h : W Y S. 1 The association OM C S becomes an oriented bivariant theory i the our operations are deined as ollows: Orientation: For a morphism : X Y and a line bundle L X, the irst Chern class operator is deined by c 1 L : OM C S X Y OM C S X Y c 1 L[V hv X; L 1, L 2,, L r ] := [V hv X; L 1, L 2,, L r, h V L]. Product operations: For morphisms : X Y and : Y Z, the product operation : OM C S X Y OM C S Y Z OM C S X Z is deined as ollows: The product on enerators is deined by [V hv X; L 1,, L r ] [W kw Y ; M 1,, M s ] := [V h V k W X; k W L1,, k W Lr, h V M 1,, h V M s ],

19 18 SHOJI YOKURA and it extends bilinearly. Here we consider the ollowin iber squares V h V X k W k W V h V W k W X Y Z. Pushorward operations: For morphisms : X Y and : Y Z with conined, the pushorward operation is deined by V : OM C SX Z OM C SY Z n V [V hv X; L 1,, L r ] := V Pullback operations: For an independent square the pullback operation is deined by V X Y X Y, : OM C S X Y OM C S X Y n V [V hv X; L 1,, L r ] := V n V [V hv Y ; L 1,, L r ]. n V [V h V X ; L1,, Lr ], where we consider the ollowin iber squares: h V V V h V X Y X Y. 2 Let OBT be a class o oriented bivariant theories OB on the same cateory V with a class C o conined morphisms, a class o independent squares, a class S o specialized morphisms and a class L o line bundles. Let S be nice canonical OB-orientable or any oriented bivariant theory OB OBT. Then, or each oriented bivariant theory OB OBT there exists a unique oriented Grothendieck transormation γ OB : OM C S OB such that or any : X Y S the homomorphism γ OB : OM C SX Y OBX Y satisies the normalization condition that γ OB [X idx X; L 1,, L r ] = c 1 L 1 c 1 L r θ OB.

20 ORIENTED BIVARIANT THEORIES, I 19 Proo. 1: It is easy to see that the above our operations are well-deined. Here we note that the above Chern class operator is nothin but the bivariant product with the motivic irst Chern class o L, [X idx X; L] OM C SX idx X, i.e., we have c 1 L[V hv X; L 1, L 2,, L r ] := [X idx X; L] [V hv X; L 1, L 2,, L r ]. 2: Suppose that there is an oriented Grothendieck transormation γ : OM C S OB satisyin that or any : X Y S the homomorphism γ : OM C SX Y OBX Y satisies that γ[x idx X] = θ OB. It suices to show that the value o any enerator [V hv X; L 1,, L r ] OM C SX Y is uniquely determined. For that, irst we notice the ollowin: [V hv X; L 1,, L r ] = h V [V idv V ; L 1,, L r ] = h V c 1 L 1 c 1 L r [V idv V ]. Since [V idv V ] OM C SV hv Y and h V S by hypothesis, we have γ[v hv X; L 1,, L r ] = γ h V c 1 L 1 c 1 L r [V idv V ] = h V c 1 L 1 c 1 L r γ[v idv V ] = h V c 1 L 1 c 1 L r θ OB h V Thus the uniqueness ollows. Next, we show the existence o such an oriented Grothendieck transormation satisyin the above normalization condition. We deine the association deined by γ OB : OM C SX Y OBX Y γ OB [V hv X; L 1,, L r ] := h V c 1 L 1 c 1 L r θ OB h V. This certainly satisies the normalization condition. The rest is to show that it is an oriented Grothendieck transormation. i it preserves the product operation:it suices to show that γ OB [V hv X; L 1,, L r ] [W kw Y ; M 1,, M s ] = γ OB [V hv X; L 1,, L r ] γ OB [W kw Y ; M 1,, M s ] Usin some parts o the proo o Theorem 3.1, we have γ OB [V hv X; L 1,, L r ] [W kw Y ; M 1,, M s ] = γ OB [V h V k W X; k W L1,, k W Lr, h V M 1,, h V M s ] = h V k W c1 k W L1 c 1 k W Lr c 1 h V M 1 c 1 h V M s θ OB h V θ OB k W. Here we use the property O-4 compatibility with pushorward and O-3 compatibility with product [ c 1 Mα β = α c 1 Mβ], the above equality continues as ollows:

21 20 SHOJI YOKURA = h V c1 L 1 c 1 L r k W c1 h V M 1 c 1 h V M s θ OB h V θ OB k W. = h V c 1 L 1 c 1 L r k W θob h V c 1 M 1 c 1 M s θ OB k W. = h V c1 L 1 c 1 L r k W = h V c 1 L 1 c 1 L r k W θ OB h V c 1 M 1 c 1 M s θ OB k W. θ OB h V k W c 1 M 1 c 1 M s θ OB k W by B-7 projection ormula. Furthermore, usin O-3 compatibility with product [ c 1 Lα β = c 1 Lα β ] and by B-4, it continues as ollows: = h V c 1 L 1 c 1 L r θ OB h V k W c 1 M 1 c 1 M s θ OB k W = γ OB [V hv X; L 1,, L r ] γ OB [W kw Y ; M 1,, M s ] ii it preserves the pushorward operation:consider X Y Z and a conined morphsim h V : V X such that the composite h V : V Y is in S. For a enerator we have [V hv X; L 1,, L r ] OM C S X Z, γ OB [V hv X; L 1,, L r ] = γ OB [V hv Y ; L 1,, L r ] = h V c 1 L 1 c 1 L r θ OB h V = h V c 1 L 1 c 1 L r θ OB h V = γ OB [V hv X; L 1,, L r ]. iii it preserves the pullback operation: Consider a conined morphsim h V : V X such that the composite h V : V Y is in S and the iber squares iven in Pullback operations above, we have γ OB [V hv X; L 1,, L r ] = γ OB [V h V X ; L1,, Lr ] = h V c1 L1 c 1 Lr θ OB h V = h V c1 L1 c 1 Lr θ OB h V = h V c 1 L 1 c 1 L r θ OB h V = h V c 1 L 1 c 1 L r θ OB h V by B-5 = γ OB [V hv X; L 1,, L r ].

22 ORIENTED BIVARIANT THEORIES, I 21 Corollary 4.6. The abelian roup OM C S X := OMC S X pt is the ree abelian roup enerated by the set o isomorphism classes o cobordism cycles [V hv X; L 1,, L r ] such that h V : V X C and V pt is a specialized map in S and L i is a line bundle over V. The abelian roup OM C S X := OM C S X idx X is the ree abelian roup enerated by the set o isomorphism classes o cobordism cycles [V hv X; L 1,, L r ] such that h V : V X C S and L i is a line bundle over V. Both unctor OM C S and OM C S are oriented Borel Moore unctors with products in the sense o Levine Morel. Corollary 4.7. A universal oriented Borel Moore unctor with products Let BT be a class o oriented additive bivariant theories B on the same cateory V with a class C o conined morphisms, a class o independent squares, a class S o specialized maps and a class o line bundles. Let S be nice canonical OB-orientable or any oriented bivariant theory OB OBT. Then, or each oriented bivariant theory OB OBT, 1 there exists a unique natural transormation o oriented Borel Moore unctors with products Γ OB : OM C S OB such that i π X : X pt is in S idx Γ OB [X X; L 1,, L r ] = c 1 L c 1 L r π! X 1 pt, and 2 there exists a unique natural transormation o oriented Borel Moore unctors with products such that or any object X Γ OB : OM C S OB Γ OB [X idx X; L 1,, L r ] = c 1 L c 1 L r 1 X. Remark Let k be an arbitrary ield. In the case when V k is the admissible subcateory o the cateory o separated schemes o inite type over the ield k, C = Proj is the class o projective morphisms, S = Sm is the class o smooth equi-dimensional morphisms and L is the class o line bundles, then OM Proj Sm X = OMProj Sm X pt is nothin but the oriented Borel Moore unctor with products Z X iven in [LM3]. In this sense, our associated contravariant one OM Proj Sm X = OM Proj idx Sm X X is a cohomoloical counterpart o Levine Morel s homoloical one Z X. Note that this cohomoloical one OM Proj Sm X or any scheme X is the ree abelian roup enerated by [V hv X; L 1,, L r ] such that h V : V X is a projective and smooth morphism. 2 One can see that in 1 Proj can be replaced by Prop. And urthermore one can consider OM Proj Lci and OM Prop Lci, which will be treated in [Yo1] An oriented bivariant theory can be deined or any kind o cateory as lon as we can speciy classes o conined morphisms, specialized morphisms toether with nice canonical orientations, line bundles, and independent squares. The above oriented bivariant theory is the very basis o other oriented bivariant theories o more eometric natures. In [Yo1] we will deal with a more eometrical oriented bivariant theory, i.e., what could be called bivariant alebraic cobordism or alebraic bivariant cobordism, which is a bivariant version o Levine Morel s alebraic cobordism.

23 22 SHOJI YOKURA Acknowledements. The author would like to thank the sta o the Erwin Schrödiner International Institute o Mathematical Physics or their hospitality durin his stay in Auust, 2006, durin which the present research was initiated. The author ave a talk on a very earlier version o this paper at the Kinosaki Alebraic Geometry Symposium Oct. 24 Oct.27, 2006: The author would like to thank the oranizers H. Kaji, F. Kato, S.-I. Kimura o the symposium or invitin him to ive a talk, and also thank M. Hanamura, S.-I. Kimura, Y. Shimizu, T. Terasoma or their interests on this subject. REFERENCES [BFM1] P. Baum, W. Fulton and R. MacPherson, Riemann Roch or sinular varieties, Publ. Math. I.H.E.S , [BFM2] P. Baum, W. Fulton and R. MacPherson, Riemann Roch and topoloical K-theory or sinular varieties, Acta Math , [Fu] W. Fulton, Intersection Theory, Spriner Verla, [FM] W. Fulton and R. MacPherson, Cateorical rameworks or the study o sinular spaces, Memoirs o Amer. Math. Soc. 243, [L1] M.Levine, Alebraic cobordism, Proceedins o the ICM Beijin 2002, Vol.II, 57 66, [L2] M.Levine, A survey o alebraic cobordism, Proceedins o the International Conerence on Alebra, Alebra Colloquium , 79 90, [LM1] M. Levine and F. Morel, Cobordisme alébrique I, C. R. Acad. Sci. Paris, 332 Séri I 2001, , [LM2] M. Levine and F. Morel, Cobordisme alébrique II, C. R. Acad. Sci. Paris, 332 Séri I 2001, , [LM3] M. Levine and F. Morel, Alebraic Cobordism, Spriner Monoraphsin Mathematics, Spriner-Verla 2007, [LP] M. Levine and R. Pandharipande, Alebraic cobordism revisited, math.ag/ , [Mer] A. Merkurjev, Alebraic cobordism theory Notes o mini-course in Lens, June 2004, math.ucla.edu/ merkurev [Qui] D. Quillen, Elementary proos o some results o cobordism theory usin Steenrod operations, Advances in Math , 29 56, [Voe] V. Voevodsky, The Milnor Conjecture, preprint Dec. 1996, /index.html [Yo1] S. Yokura, Oriented bivariant theory, II, in preparation. [Yo2] S. Yokura, On a naïve motivic bivariant characteristic class, in preparation. DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, FACULTY OF SCIENCE, UNIVERSITY OF KAGOSHIMA, KORIMOTO 1-CHOME, KAGOSHIMA , JAPAN address: yokura@sci.kaoshima-u.ac.jp