Schubert calculus for algebraic cobordism

SCHUBER T CALCULUS FOR ALGEBRA IC COBORDISM JENS HORNBOSTEL AND V ALENTINA KIRITCHENKO Abstract. W e establish a Sc hu b ert calculus for Bott-Samelson re solutions in the a lgebraic cob ordism ring of a complete ﬂag v ariety G/B extending the results of Bressler –Evens [4] to the algebro -geometric setting . 1. Intr oduction W e ﬁx a base ﬁeld k of c haracteristic 0. Algebraic cob ordism Ω ∗ ( − ) has b een in v en ted some y ears ago b y Levine and Morel [14] as the univers al orien ted algebraic cohomolog y theory on smo oth v arieties o ver k . In particular, its co eﬃcien t ring Ω ∗ ( k ) is isomorphic to the Lazard ring L (in tro duced in [12 ]). In a recen t article [15 ], L evine and Pandharipande sho w that algebraic cob ordism Ω n ( X ) allo ws a presen tation with generators being pro je ctiv e morphisms Y → X of relativ e co dimension n (:= dim( X ) − dim( Y )) b et wee n smo o th v arieties and relations giv en b y a reﬁnemen t o f the naiv e a lgebraic cob ordism relation (in v olving double p oint relations). A recen t result o f L evine [13] which relies on unpublished work of Hopkins and Morel asserts an isomorphism Ω n ( − ) ∼ = M GL 2 n,n ( − ) b etw een Levine-Morel and V o ev o dsky algebraic cob ordism for smo oth quasipro jectiv e v a rieties. In particular, a lgebraic cob ordism is represen table in the motivic stable homoto p y category . In short, algebraic cob o r dism is to algebraic v arieties what complex cob ordism M U ∗ ( − ) is to top olog ical manifo lds. The ab o ve fundamen tal results b eing established, it is high time for computations, whic h ha v e b een carried out only in a v ery small n um b er of cases (see e.g. [22] and [23]). The presen t article fo cuses on cellular v arieties X , for whic h the additiv e structure of Ω ∗ ( X ) is easy to describe: it is the free L -mo dule generated by the cells (see the next section fo r mo r e precise deﬁnitions, statemen ts, pro of s and references). So additiv ely , a lgebraic cob ordism for cellular v arieties b eha ve s exactly a s Chow groups do. O f course, algebraic K-theory also b eha v es in a similar wa y , but we will restrict our comparisons here and b elow to Cho w groups. There is a ring homomorphism Ω ∗ ( X ) → M U 2 ∗ ( X ( C ) an ) whic h fo r cellular v arieties is a n isomorphism, see Section 2.2 and the app endix. Ho w ev er, computatio ns in Ω ∗ ( X ) b ecome mor e transparent and suitable for algebro-geometric applications if they a r e done b y algebro-geometric metho ds rather than by a translation of the already existing results f or M U 2 ∗ ( X ( C ) an ) (e. g. those of Bressler and Ev ens, see [4] and b elow), esp ecially if the latter w ere obtained b y top ological metho ds whic h do not hav e coun terparts in algebraic geometry . Let us concen trate on complete ﬂag v arieties X = G/B where B is a Borel subgroup o f a connected split reductiv e g roup G ov er k . In the case where G = GL n ( k ), the cob o rdism ring The second autho r would like to thank Jaco bs Universit y Br emen, the Hausdorﬀ Center for Mathematics and the Ma x Pla nck Institute for Mathematics in Bonn for hospita lit y and supp or t. The second author was also partially supp or ted by the Dynast y F o undation fellowship and RFBR gr ant 10-0 1-005 40-a. 1 2 JENS HOR NBOSTEL AND V ALENTINA KIRITCHENKO Ω ∗ ( X ) ma y b e describ ed as the quotien t of a free p olynomial ring ov er L with generators x i b eing the ﬁrst Chern classes of certain line bundles on X and explicit relations. More precise ly , w e show (see Theorem 2 .6 ): Theorem 1.1. The c ob or dism ring Ω ∗ ( X ) is isomo rp hic to the gr a de d ring L [ x 1 , . . . , x n ] of p olynomials with c o eﬃcients in the L azar d ring L and deg x i = 1 , quotient by the ide al S gener ate d by the homo gene ous symmetric p olynom ials of strictly p ositive de gr e e: Ω ∗ ( X ) ≃ L [ x 1 , . . . , x n ] /S. This g eneralizes a theorem o f Borel [2] o n the Cho w ring (or equiv a len tly the singular coho- mology ring) of a ﬂag v ariety to its a lgebraic cob ordism ring. The Cho w ring of the ﬂag v ariety has a natural basis giv en b y the Schub ert c ycles . The cen tral problem in Sc h ub ert calculus was to ﬁnd p olynomials (later called Sc h ub ert p olyno- mials) represe n ting the Sc h ub ert cycles in the Borel presen ta tion. This problem w as solv ed indep enden tly b y Bernstein–Gelfand–Gelfand [1] and Demazure [10] using divide d diﬀer enc e op er ators on the Chow ring (most of the ingredien ts w ere already con tained in a manus cript of Chev alley [8], whic h for many y ears remained unpublished). Explicit formulas for Sch ub ert p olynomials giv e an a lgorithm for decomp osing the pro duct of any tw o Sc h ub ert cycles in to a linear combination o f other Sc h ub ert cycles with integer co eﬃcien ts. The complex (as w ell as the algebraic) cob o rdism ring of the ﬂa g v ariet y also has a natur a l generating set give n b y the Bott-Samelson r e s olutions of the Sc h ub ert cycles (note that the latter are not alwa ys smo oth and so, in g eneral, do not deﬁne any cob ordism classes). F or the complex cob ordism r ing , Bressler and Ev ens describ ed the cob or dism classes of Bott - Samelson resolutions in t he Borel presen tation using gener a l i z e d div i d e d diﬀer enc e op er a tors o n the cob or- dism ring [3, 4] (w e thank Burt T otaro from whom w e ﬁrst learned ab out this reference). Their form ulas for these op erators are not a lg ebraic and in volv e a passage to the classifying sp ace of a compact torus in G and homoto p y theoretic considerations (see [3, Corollary-Deﬁnition 1.9, Remark 1.1 1 ] and [4, Prop o sition 3]). One of the goals of the presen t pap er is to prov e an algebraic f o rm ula for the generalized divided diﬀerence operato rs (see Deﬁnition 2.2 a nd Corollary 2 .3). This formula in turn implies explicit purely algebraic formulas for the p olyno- mials (now with co eﬃcien ts in the Lazard r ing L ) represen ting the classes of Bott- Sa melson resolutions. Note that each suc h p olynomial con tains the resp ectiv e Sc h ub ert p olynomial as the low est degree term (but in most cases also has non- trivial higher order terms). W e also giv e an a lgorithm for decomp o sing the pro duct of tw o Bott-Sa melson resolutions in to a linear com binatio n of o ther Bo t t - Samelson resolutions with co eﬃcien ts in L . W e no w form ulate our main theorem (compare Theorem 3.2), whic h can b e view ed as an algebro-geometric analo gue of the results of Bressler-Ev ens [4, Corollary 1, Prop osition 3]. Let I = ( α 1 , . . . , α l ) b e an l -tuple of simple ro ots of G , and R I the corresp onding Bott-Sa melson resolution of the Sc hubert cycle X I (see Section 3 for the precise deﬁnitions). Recall that t here is an isomorphism betw een the Picard group of the ﬂag v ariety and the w eight lattice of G suc h that v ery ample line bundles map to strictly dominant w eigh ts (se e, for instance, [5, 1.4.3]). W e den ote b y L ( λ ) the line bundle on X corresp onding to a w eigh t λ , and b y c 1 ( L ( λ )) its ﬁrst Chern class in algebraic cob ordism. F or eac h α i , w e deﬁne the op erator A i on Ω ∗ ( X ) in a purely alg ebraic w a y (see Section 3.2 for the rigorous deﬁnition for arbitrary reductiv e groups). SCHUBER T CALCULUS FOR A LGEBRAIC COBOR DISM 3 Informally , the o p erator A i can b e deﬁned in the case G = GL n b y the fo r m ula A i = ( 1 + σ α i ) 1 c 1 ( L ( α i )) , where σ α i acts on the v aria bles ( x 1 , . . . , x n ) b y the transp o sition corresp o nding to α i . Here w e use that the W eyl gro up of GL n can b e identiﬁe d with the symmetric gr oup S n so that the simple reﬂections s α i corresp ond to elemen tary transp ositions (see Section 2 f or more details). Note that t he c 1 ( L ( α i )) can b e written explicitly as p olynomials in x 1 ,. . . , x n using the formal group la w (see Section 2). Theorem 1.2. F o r any c omplete ﬂag variety X = G/B and a n y tuple I = ( α 1 , . . . , α l ) of simple r o ots of G , the class of the Bott-Same ls on r esolution R I in the al g e br aic c ob or dism ring Ω ∗ ( X ) is e qual to A l . . . A 1 R e , wher e R e is the class of a p oint. This theorem reduces the computation o f the pro ducts of t he geometric Bot t-Samelson classes to the pro ducts in the p olynomial ring giv en by the previous theorem. Note that in the co- homology case analogously deﬁned op erators A i coincide with t he divid e d diﬀer en c es op er ators deﬁned in [1, 10 ], so our theorem g eneralizes [1, Theorem 4.1] and [10, Theorem 4.1] f o r Sc h u- b ert cycles in cohomology and Chow ring, r esp ective ly , to Bott- Samelson classes in a lgebraic cob ordism. Note that in the case o f Cho w ring, the theorem analogous to Theorem 1.2 has t wo diﬀeren t pro ofs. A more algebraic pro of using the Chev alley-Pieri form ula was giv en b y Bernstein– Gelfand–Gelfand ([1, Th eorem 4.1], see also Section 4 f or a short ov erview). Demazure ga v e a more geometric pro of b y iden tifying the divided diﬀerence op erator s with the push-forward morphism for ce rtain Cho w rings ([10, Theorem 4.1], see also Section 3). At ﬁrst glance, it seems that the former pro of is easier to extend to t he algebraic cob o rdism. Indeed, we w ere able to extend the main ingredien t of t his pro of, namely , the alg ebraic Chev a lley–Pieri fo rm ula (see Prop osition 4.3). How ev er, the rest of the Bernstein–Gelfand–Gelfand argumen t fails for cob ordism (see Section 4 for more details) wh ile the mor e geometric arg umen t of D emazure can b e exten ded to cob ordism with some extra work. F or the complex cobo rdism ring this was done b y Bressler and Ev ens [3, 4]. T o describ e the push-forw ard morphism they used r esults from ho motop y theory , whic h are not ( yet) applicable to algebraic cob ordism. In our article, w e also follo w Demazure’s approac h. A k ey ingredien t for extending this approac h to algebraic cob ordism is a form ula fo r the push-forward in algebraic cob o r dism for pro jectiv e line ﬁbrations due to Vishik, see Prop osition 2.1. W e provide a new pro of of this form ula using the double p oint r elation in cob ordism intro duced b y Levine a nd P andharipande [15]. In general, push-forwards (sometimes also called “transfers” or “Gysin homomorphisms”) for algebraic cob ordism are considerably more in t r icate than the o nes f or Chow groups. Consequen tly , their computation, whic h applies to any orien table coho molo gy theory , is more complicated. Using the ring isomorphism Ω ∗ ( X ) ≃ M U 2 ∗ ( X ( C ) an ) for cellular v arieties, it seems p ossible to deduce o ur Theorem 1.2 from the results of Bressler–Ev ens [3, 4] on complex cob ordism ( t he main task w ould b e to compare our a lg ebraically deﬁned op erators A i with t heirs). W e will not exploit this approach. Instead, all our pro ofs are purely algebraic or algebro-geometric. Con vers ely , w e note that all our pro ofs concerning algebraic cob ordism ring of the ﬂag v ariet y 4 JENS HOR NBOSTEL A ND V ALENTINA KIRITCHENKO (suc h a s the pro of of Propo sition 4.3) ma y b e easily translated to pro ofs for the analogue statemen ts concerning the complex cob ordism ring. The a rticle [4] do es no t con tain any computations. It w o uld b e in teresting to do some com- putation using their algorit hm and then compare them with our approa ch, whic h we consider to b e the easier one due to our explicit formula fo r the pro duct o f a Bott- Sa melson class with the ﬁrst Chern class (see formula 5.1) based on our algebraic Chev alley-Pieri formula. (Note also that the notatio ns o f [4] a re essen tially consisten t with [1], but not a lwa ys with [16]. W e rather stic k to the former than to the latter.) This pap er is organized as follo ws. In the next section, w e giv e some further bac kground on algebraic cob ordism, in particular, t he f orm ula for the push-forward mentione d ab o v e. In the case of the ﬂa g v ariety for GL n , w e describ e the mu ltiplicativ e structure of its algebraic cob ordism ring. In the third section, we recall the deﬁnition of Bott-Samelson resolutions and then express t he classe s of Bott-Samelson resolutions as p olynomials with coeﬃcien t s in the Lazard ring. Section 4 con ta ins an algebraic Chev a lley-Pieri formula and a short discussion of why the pro of of [1] for singular cohomology do es not carr y o v er to algebraic cobordism. The ﬁnal section contains a n algorithm for computing the pro ducts of Bott - Samelson classes in terms o f other Bott-Sa melson classes as well as some examples and explicit computations. Our main r esults are v alid for the ﬂag v ariety of an arbitrary reductiv e group G , but can b e made more explicit in t he case G = GL n using Borel presen tation g iven by Theorem 2.6. So w e will use t he ﬂag v ariet y for GL n as the main illustrating example whenev er p ossible. One might conjecture tha t the a lgebraic cob ordism rings of ﬂag v arieties with resp ect to o t her reductiv e groups G a lso allow a Borel presen ta t ion as p olynomial rings o ve r L in certain ﬁrst Chern classes mo dulo t he p olynomials ﬁxed b y the appropriate W eyl g r o ups (at least when passing to rationa l co eﬃcien ts), b ecause the corresp onding statemen t is v alid for singular cohomology resp. Cho w groups (compare [2] resp. [9]). After most of o ur preprin t was ﬁnishe d, w e learned that Calm` es, P etro v and Zainoulline are also working on Sch ub ert calculus f o r algebraic cob ordism. It will b e interesting to compare their results and pro ofs to ours (their preprint is no w av ailable, see [7]). W e are grateful to P aul Bressler and Nicolas P errin for useful discussions and to Miche l Brion and t he referee for v aluable commen ts on earlier v ersions of this ar ticle. 2. Algebraic cobordism groups, push- f or w ards and ce llular v arieties W e brieﬂy recall the geometric deﬁnition of algebraic cob or dism [15 ] and some of its basic prop erties as established in [1 4]. F o r more details se e [14, 15]. Recall that (up to sign) any elemen t in the algebraic cob ordism group Ω n ( X ) for a sc heme X (separated, of ﬁnite type o v er k ) ma y be represen ted b y a pro jectiv e morphism Y → X with Y smo oth and n = dim( X ) − dim( Y ), the relations b eing the “double p oin t relations”, whic h w e explain further b elo w. In particular, Ω ∗ ( X ) only liv es in degrees ≤ dim X , which w e will use sev eral times throughout the pap er. Similar to the Cho w ring C H ∗ , algebraic cob ordism Ω ∗ is a functor on the category of smo oth v arieties ov er k , cov ariant f o r pro jectiv e and con tr av aria nt for smo oth and mor e generally lci morphisms, whic h allo ws a theory of Chern classes. Ho w eve r, the map from the Picard group of a smoo th v ariety X to Ω 1 ( X ) given by the ﬁrst Chern class is neither a bijection nor a homomorphism an ymore (unlike the corresponding map in the Chow ring SCHUBER T CALCULUS FOR A LGEBRAIC COBOR DISM 5 case). Its failure o f b eing a group homomorphism is enco ded in a formal gr oup l a w tha t can b e constructed f r om Ω ∗ . More precisely , a n y algebraic orientable coho mo lo gy theory allows b y deﬁnition a calculus of Chern classes, and conse quen tly the construction of a formal group law . A formal group la w is a formal p o w er series F ( x, y ) in t w o v ariables suc h tha t f or an y t w o line bundles L 1 and L 2 w e hav e the fo llo wing iden tit y relating their ﬁrst Chern classes: c 1 ( L 1 ⊗ L 2 ) = F ( c 1 ( L 1 ) , c 1 ( L 2 )) . E. g. t he for mal g roup la w for C H ∗ is additiv e, that is, F ( x + y ) = x + y . Algebraic cob ordism is the univ ersal o ne among the algebraic orien table cohomology theories. In what follo ws, F ( x, y ) will alw a ys denote the univers al formal group la w corresp onding to alg ebraic cob ordism unless stated otherwise. In this and in man y other w a ys - as the computations below will illustrate - algebraic cob ordism is a reﬁnem en t of Ch o w ring, a nd one has a na t ural isomorphism of functors Ω ∗ ( − ) ⊗ L Z ∼ = C H ∗ ( − ) (see [14] where a ll these results are pro v ed). Here and in the se- quel, L denotes the Lazard ring , whic h classiﬁes one-dimensional comm utativ e formal group la ws and is isomorphic t o the graded p olynomial ring Z [ a 1 , a 2 , . . . ] in coun tably many v ariables [12], where w e put a i in degree − i . When considering po lynomials p ( x 1 , ....x n ) o v er L with deg( x i ) = 1, w e will distinguish the (total) de gr e e and the p ol yno m ial de gr e e o f p ( x 1 , ..., x n ). Note that the La zard ring is isomorphic to the algebraic (as w ell as complex) cob ordism r ing of a p oin t. In particular, its elemen ts can b e represen ted by the cob ordism classes of smo oth v arieties. In what fo llows, w e use this geometric in terpretation. W e are also going t o use a g eometric in terpretation of the formal group law , namely , the double p oint r elation . This is an equalit y for elemen ts in the a lg ebraic cob o rdism ring established in [15]. W e recall the deﬁnition for the reader’s conv enience. Double p oin t relation: Assume that w e hav e three smo ot h h yp ersurfaces A , B a nd C on a smo oth v ariet y Z suc h that the fo llo wing conditions hold (1) C is linearly equiv a len t to A + B (2) A , B and C ha v e tra nsv erse pair wise interse ctions (3) C do es not in tersect A ∩ B Then w e hav e the follo wing double p oint r elation . Denote by D the inters ection A ∩ B . W e ha v e [ C → Z ] = [ A → Z ] + [ B → Z ] − [ P D → Z ] in Ω ∗ ( X ), whe re P D = P ( O D ⊕ N A/D ) = P D ( N B /D ⊕ O D ) and t he map P D → Z is the comp osition of the natural pro jection P D → D with the embedding D ⊂ Z . Here N A/D and N B /D are the normal bundles to D in A and B , respectiv ely . The second conditio n ensures that P ( O D ⊕ N A/D ) = P D ( N B /D ⊕ O D ) (since L ( C ) | D = ( L ( A ) ⊗ L ( B )) | D = N A/D ⊗ N B /D is trivial). This formulation is a sp ecial case of the extended double p o in t relation in [15, Lemma 5.2]. The double p oin t relatio n allows to expre ss geometrically the discrepancy b etw een the additive formal gro up law and the univers al one. Namely , since C = F ( A, B ) by the ﬁrst conditio n, we get A + B − F ( A, B ) = [ P D → Z ] . 6 JENS HOR NBOSTEL A ND V ALENTINA KIRITCHENKO W e will use this equation when proving Prop o sition 2.1. W e will a lso use rep eatedly the pro jectiv e bundle form ula, whic h w e recall below for the reader’s con venie nce. F or more details see [14, Section 1.1] and [16 , 3.5.2]. Pro ject iv e bun dle form ula : Let E → X b e a v ector bundle of rank r ov er X . Denote b y Y = P ( E ∗ ) the v ariety of h yp erplanes of E , and b y π the natural pro jection π : Y → X . The v ariet y P ( E ∗ ) is a ﬁbratio n o ver X with ﬁb ers isomorphic to P r − 1 . Note that equiv alen tly P ( E ∗ ) can b e deﬁned as the v a riet y of one-dimensional quotien ts of E since there is a canonical isomorphism b et we en the v ariety o f h yp erplanes and the v ariety o f quotien ts b y h yp erplanes in a v ector space. This is how P ( E ∗ ) is deﬁned in [14, Sec tion 1.1] (where it is denoted b y P ( E )). Let A ∗ ( − ) b e an y orien ted cohomology theory . Denote b y ξ the ﬁrst Chern class of the tauto logical quotien t line bundle O E (1) on Y whose restriction o n eac h ﬁb er of Y ov er X coincides with O P r − 1 (1). The ﬁrst Chern can b e deﬁned a s ξ = s ∗ s ∗ (1 Y ) where s : Y → O E (1) is the zero se ction and 1 Y ∈ A 0 ( Y ) is the m ultiplicative unit ele men t. Then there is a ring isomorphism: A ∗ ( Y ) = A ∗ ( X )[ ξ ] / ( r X j =0 ( − 1) j c j ( π ∗ E ) ξ r − j ) . The isomorphism iden t iﬁes a p olynomial b 0 + b 1 ξ + . . . + b n − 1 ξ n − 1 in A ∗ ( X )[ ξ ] with the elemen t π ∗ b 0 + ( π ∗ b 1 ) ξ + . . . + ( π ∗ b n − 1 ) ξ n − 1 in A ∗ ( Y ). In particular, A ∗ ( Y ) splits into the direct sum π ∗ A ∗ ( X ) ⊕ ξ π ∗ A ∗ ( X ) ⊕ . . . ⊕ ξ n − 1 π ∗ A ∗ ( X ). Note tha t the relation r X j =0 ( − 1) j c j ( π ∗ E ) ξ r − j = 0 admits t he f o llo wing alternative description. Consider a short exact sequence o f v ector bundles on Y : 0 → τ E → π ∗ E → O E (1) → 0 , where τ E is the tautolo gical hyperplane bundle o n Y . By the Whitney sum for mula w e hav e that the t o tal Chern class c ( π ∗ E ) is equal to the pro duct c ( τ E ) c ( O E (1)). Since c ( O E (1)) = 1 + ξ w e ha v e c ( π ∗ E ) = c ( τ E )(1 + ξ ) . W e now divide t his identit y by (1 + ξ ) (that is, multiply by P r +dim X − 1 j =0 ( − 1) j ξ j ) and get that c ( τ E ) = c ( π ∗ E )( P r +dim X − 1 j =0 ( − 1) j ξ j ). In particular, c r ( τ E ) = ( − 1) r r X j =0 ( − 1) j c j ( π ∗ E ) ξ r − j , so w e can inte rpret the relation ab ov e as the v anishing of the r - t h Chern class of the bundle τ E (whic h ha s rank r − 1). 2.1. A formu la for the push-forw ard. Let X b e a smo o t h algebraic v ariet y , and E → X a v ector bundle of rank tw o on X . Consider the pro jectiv e line ﬁbra t ion Y = P ( E ) deﬁned as the v ariet y of all lines in E . W e ha v e a natural pro jection π : Y → X whic h is pro jectiv e and hence induces a push-forwar d (or tr ansfer , sometimes also called Gysin map ) π ∗ : Ω ∗ ( Y ) → Ω ∗ ( X ). W e now state a form ula f o r this push-forw ard. Note that this formula is true not only for algebraic cob ordism but fo r an y orien table cohomology theory , as the pro of s remain true in this more general case. SCHUBER T CALCULUS FOR A LGEBRAIC COBOR DISM 7 Consider the r ing of formal p ow er series in t w o v ariables y 1 and y 2 with co eﬃcien ts in Ω ∗ ( X ). Deﬁne the op era t o r A on this ring by the form ula A ( f ) = (1 + σ ) f F ( y 1 , χ ( y 2 )) , where [ σ ( f )]( y 1 , y 2 ) := f ( y 2 , y 1 ). Here F is the univ ersal f o rmal group la w (or more g enerally , the one of the o rien ta ble cohomology theory one considers) and χ is the in v erse for the formal group law F , that is, χ is uniquely determined b y the equation F ( x, χ ( x )) = 0 (we use notatio n from [1 4, 2.5]) . The op erator A is an analog of the d i v i de d d i ﬀ er enc e op er ator in tro duced in [1, 10]. In the case of Cho w rings, o ur deﬁnition coincides with t he classical divided diﬀerence op erator, since the fo rmal gr oup la w for Cho w ring is additiv e, that is, F ( x, y ) = x + y and χ ( x ) = − x . Though A ( f ) is deﬁne d as a fraction, it is easy to write it a s a formal p ow er series as w ell (see Section 5). Suc h a pow er series is unique since F ( y 1 , χ ( y 2 )) = y 1 − y 2 + . . . is clearly not a zero divisor. E.g. w e ha v e A (1) = x + χ ( x ) xχ ( x ) = q ( x, χ ( x )) = − a 11 − a 12 ( x + χ ( x )) + . . . , where x = F ( y 1 , χ ( y 2 )), and q ( x, y ) is the p o w er series uniquely determined by the equation F ( x, y ) = x + y − xy q ( x, y ). In particular, since F ( x, χ ( x )) = 0 by deﬁnition of the p o w er series χ ( x ), w e hav e x + χ ( x ) − xχ ( x ) q ( x, χ ( x )) = 0 whic h justiﬁes the second equality . F or the last equalit y , w e used computation of the ﬁrst few terms of F ( x, y ) and χ ( x ) fro m [14, 2.5 ]. Here a 11 , a 12 etc. denote the co eﬃcien ts o f the univ ersal formal gr o up law, that is, F ( x, y ) = x + y + a 11 xy + a 12 xy 2 + . . . . The co eﬃcien ts a ij are the elemen ts of the Lazard ring L ∗ , e.g. a 11 = − [ P 1 ], a 12 = a 21 = [ P 1 ] 2 − P 2 (see [1 4, 2.5]). W e a lso hav e A ( y 1 ) = y 2 A (1) + F ( x, y 2 ) − y 2 x = y 2 q ( x, χ ( x )) − y 2 q ( x, y 2 ) + 1 = 1 + a 12 y 1 y 2 + . . . . The pull-back π ∗ : Ω ∗ ( X ) → Ω ∗ ( Y ) giv es Ω ∗ ( Y ) the structure of an Ω ∗ ( X )-mo dule. Recall that b y the pro j ectiv e bundle for mula w e hav e an isomorphism of Ω ∗ ( X )-mo dules Ω ∗ ( Y ) ∼ = π ∗ Ω ∗ ( X ) ⊕ ξ π ∗ Ω ∗ ( X ) , where ξ = c 1 ( O E (1)). Since the push-fo rw ar d is a homomorphism of Ω ∗ ( X )-mo dules, it is enough to determine the action of π ∗ on 1 Y and o n ξ . The follo wing result is a sp ecial case of [21, Theorem 5.30], which giv es an explicit fo rm ula f o r the push-forw ard π ∗ for ve ctor bundles of arbitra ry rank. Prop osition 2.1. [21, Theorem 5.30] L et ξ 1 and ξ 2 b e the Chern r o ots of E , that is, formal variables satisfying the c onditions ξ 1 + ξ 2 = c 1 ( E ) and ξ 1 ξ 2 = c 2 ( E ) . Then the push-forwar d acts on 1 Y and ξ a s fol lows: π ∗ (1 Y ) = [ A (1)]( ξ 1 , ξ 2 ) , π ∗ ( ξ ) = [ A ( y 1 )]( ξ 1 , ξ 2 ) , wher e A (1) and A ( y 1 ) ar e the formal p ower series in two v a riables deﬁ n e d ab ove. Sinc e A (1) an d A ( y 1 ) ar e s ymm etric in y 1 and y 2 , they c an b e written as p ow er series in y 1 + y 2 and y 1 y 2 . Henc e, the right hand sides a r e p ower series in c 1 ( E ) and c 2 ( E ) and e ven 8 JENS HOR NBOSTEL A ND V ALENTINA KIRITCHENKO p olynomials (as al l terms of de gr e e gr e ater than dim X wil l vanish by [15] ). So the right h and sides inde e d deﬁne elements in Ω ∗ ( X ) . F or the Chow ring and K 0 , analo g ous statemen ts w ere prov ed in [10, Propositions 2 .3 ,2.6] for certain morphisms Y → X . Note that for b o t h of these theories, the fo rm ula for π ∗ ( ξ ) reduces to π ∗ ( ξ ) = 1 since the corresp onding formal group laws do not contain terms of degree greater than t w o. As Vishik sho w ed (see [21, Theorem 5.35]), his form ula is eq uiv alen t to Quillens form ula [18] for complex cob ordism, as also pro ve d b y Shinder in the algebraic setting [19]. W e giv e a new geometric pro of of Prop osition 2.1 (that is, o f Vishik’s formula for r a nk tw o bundles) based on the double p oint relation in a lg ebraic cob ordism. Pr o of. First, note t hat replacing E with E M = E ⊗ M for an arbitra r y line bundle M on X do es not c hange the v ariet y Y a nd the map π . Ho w ev er, this do es c hange the tautolo gical quotien t line c 1 ( O E (1)). More precisely , we hav e the following isomorphism of line bundles on Y (compare e.g. [15, Pro of of Lemma 7.1]): π ∗ M ⊗ O E (1) = O E M (1) . Let us denote b y ξ M the ﬁrst Chern class of the tautological quotien t line bundle O E M (1). The iden tity ab ov e implies that ξ M = F ( ξ , π ∗ c 1 ( M )) or equiv alen t ly ξ = F ( ξ M , π ∗ c 1 ( M ∗ )). Hence, to compute π ∗ ξ it is enough to compute π ∗ 1 Y and π ∗ ξ M for some M . It is conv enien t to c ho ose M = L ∗ 1 so that E M has a trivial summand, and hence one of the Chern ro ots of E M is zero. Th us we can a ssume t ha t E = O X ⊕ L . In this case, the second formula of Prop osition 2.1 reduces to π ∗ ξ = 1, whic h is easy to sho w b y similar metho ds as for the Cho w ring. Namely , consider the natural em b edding i : X = P ( O X ) → Y = P ( E ). Then ξ = i ∗ 1 X b y [1 4, Lemma 5.1.11]. Hence, π ∗ ( ξ ) = π ∗ i ∗ 1 X = 1 X since π ◦ i = id X . It is more diﬃcult to compute π ∗ 1 Y , whic h is the cob ordism class of [ π : Y → X ]. F or the Cho w ring, it is zero b y degree reasons, but for cob ordisms it is not . E.g. ev en f or a trivial bundle E we ha v e [ π : Y → X ] = [ π : X × P 1 → X ] = − a 11 1 X . W e compute [ π : Y → X ] b y represen ting it as one of the terms in a suitable double p oint relation. Namely , consider the v ariet y Z = Y × X Y and deﬁne three smo o th hypersurfaces A , B and C on Z as follows: A = { y × i ( π ( y )) : y ∈ Y ) } , B = { i ′ ( π ( y )) × y : y ∈ Y } and C = { y × y : y ∈ Y } . Here i : X → Y and i ′ : X → Y are the em b eddings P ( O X ) → P ( E ) and P ( L ) → P ( E ), resp ectiv ely . Then it is easy t o chec k (using aga in [14, Lemma 5.1.11 ]) tha t A = c 1 ( p ∗ 1 O E (1)), B = c 1 ( p ∗ 2 O E ⊗ L ∗ (1)) and C = c 1 ( O E (1) ⊠ O E ⊗ L ∗ (1)), where p 1 , p 2 are the pro jections o f Z on t o the ﬁrst and the second factor, resp ectiv ely . Hence, w e hav e the FGL iden tity C = A + B − AB q ( A, B ) on Z , from whic h w e can easily get the double p oin t relation w e need. Namely , apply π ∗ p 1 ∗ to b o t h sides and get that [ π : Y → X ] = π ∗ p 1 ∗ AB q ( A, B ) , b ecause the other t wo terms cancel out. The righ t hand side can b e computed b y the pro jection form ula using that AB = σ ∗ 1 X , where σ : X → Z sends x to i ( x ) × i ′ ( x ).  If w e iden tify Ω ∗ ( Y ) with the p olynomial ring Ω ∗ ( X )[ ξ ] / ( ξ 2 − c 1 ( E ) ξ + c 2 ( E )) by the pro jectiv e bundle form ula, we can refor m ulat e Prop osition 2 .1 as follows : π ∗ ( f ( ξ )) = [ A ( f ( y 1 ))]( ξ 1 , ξ 2 ) for a n y p olynomial f with co eﬃcien t s in Ω ∗ ( X ) (where f ( y 1 ) in the righ t hand side is regarded as an elemen t in Ω ∗ ( X )[[ y 1 , y 2 ]]). In this form, Prop osition 2 .1 is consisten t with the classical SCHUBER T CALCULUS FOR A LGEBRAIC COBOR DISM 9 form ula f o r the push-forward in the case o f Cho w ring (cf. [16, Remark 3.5 .4]). Indeed, since the f ormal group law for Chow ring is additiv e we hav e A ( 1 ) = 1 y 1 − y 2 + 1 y 2 − y 1 = 0 and A ( y 1 ) = y 1 y 1 − y 2 + y 2 y 2 − y 1 = 1 . Deﬁnition 2.2. We deﬁne an Ω ∗ ( X ) -line ar op er ator A π on Ω ∗ ( Y ) as fol lows. We have an isomorphism Ω ∗ ( X )[[ y 1 , y 2 ]] / ( y 1 + y 2 − c 1 ( E ) , y 1 y 2 − c 2 ( E )) ∼ = Ω ∗ ( Y ) given by f ( y 1 , y 2 ) 7→ f ( ξ , c 1 ( E ) − ξ ) ) . Then the op er ator A on Ω ∗ ( X )[[ y 1 , y 2 ]] desc ends to a n op er ator A π on Ω ∗ ( Y ) , which c an b e d escrib e d using the ab ove is omorphism as fol low s A π : f ( ξ , c 1 ( E ) − ξ ) → [ A ( f ( y 1 , y 2 ))]( ξ , c 1 ( E ) − ξ ) . We als o deﬁne a Ω ∗ ( X ) -line ar en d omorphism σ π of Ω ∗ ( Y ) by the formula: σ π : f ( ξ , c 1 ( E ) − ξ ) = f ( c 1 ( E ) − ξ , ξ ) . The op erato r A π is w ell-deﬁned since A preserv es the ideal ( y 1 + y 2 − c 1 ( E ) , y 1 y 2 − c 2 ( E )). Indeed, for an y p ow er series f ( y 1 , y 2 ) symmetric in y 1 and y 2 (in particular, for y 1 + y 2 − c 1 ( E ) and y 1 y 2 − c 2 ( E )) and an y p ow er series g ( y 1 , y 2 ) we ha v e A ( f g ) = f A ( g ). The op erator A π decreases degrees b y one, and its image is contained in π ∗ Ω ∗ ( X ) ⊂ Ω ∗ ( Y ), whic h c an b e iden tiﬁed using the ab ov e isomorphism f o r Ω ∗ ( X ) with the subring of symmetric p olynomials in y 1 and y 2 . Prop osition 2.1 tells us that the push-for ward π ∗ : Ω ∗ ( Y ) → Ω ∗ ( X ) is the comp osition of A π with the isomorphism π ∗ Ω ∗ ( X ) ∼ = Ω ∗ ( X ), whic h sends (under the a b o v e iden tiﬁcations) a symme tric p olynomial f ( y 1 , y 2 ) into the polynomial g ( c 1 ( E ) , c 2 ( E )) suc h that g ( y 1 + y 2 , y 1 y 2 ) = f ( y 1 , y 2 ). Hence, w e get the follo wing corollary , whic h w e will use in the sequel. Corollary 2.3. The c o mp osition π ∗ π ∗ : Ω ∗ ( Y ) → Ω ∗ ( Y ) is e qual to the op er ator A π : π ∗ π ∗ = A π . In the sp ecial case Y = G/B and X = G / P i (and t his is the main application w e hav e, see Section 3.2), the top ological analogue of this fo rm ula app eared in [3, Corolla ry-Deﬁnition 1.9] for a diﬀerent deﬁnition of A π . 2.2. Algebraic cob or dism groups of cellular v arieties. W e start with the deﬁnition of a cellu lar v ariet y . The follo wing deﬁnition is tak en from [11, Example 1.9.1], other authors sometimes consider sligh t v ariat io ns. Deﬁnition 2.4. We say that a smo o th variety X over k is “c el lular” or “admits a c e l lular de c omp osition ” if X has a ﬁltr ation ∅ = X − 1 ⊂ X 0 ⊂ X 1 ⊂ ... ⊂ X n = X by close d subvarieties such that the X i − X i − 1 ar e isomorphic to a disjoint union of aﬃne s p ac es A d i for al l i = 0 , ..., n , which ar e c al l e d the “c el ls” o f X . Examples of cellular v a rieties include pro jectiv e spaces a nd more general Grassmannians, and complete ﬂag v arieties G/ B where G is a split reductiv e g r oup and B is a Borel subgroup. The f o llo wing theorem is a corollary of [22, Corolla r y 2.9]. W e thank Sasc ha Vishik for ex- plaining to us how it can b e deduced using the pro jectiv e bundle form ula. The main p oin t is that for d = dim X and i an arbitrary in t eger, one has for A = Ω that Ω i ( X ) = : Ω d − i ( X ) is isomor- phic to H om ( A ( d − i )[2( d − i )] , M ( X )) using that in the notation of lo c. cit. H om ( A ( d − i )[2 d − 10 JENS HOR NBOSTEL A ND V ALENTINA KIRITCHENKO 2 i ] , M ( X )) is a direct summand in H om ( M ( P d − i ) , M ( X )) = A d − i ( P d − i × X ) = ⊕ d − i j =0 A d − i − j ( X ), and it is not diﬃcult to see that it corresp onds to the summand with j = 0. Theorem 2.5. L et X b e a variety with a c el l ular de c om p osition as in the deﬁnition ab o v e . Then we hav e an isomorphi s m of gr ade d ab elian gr oups (and eve n of L -mo dules) Ω ∗ ( X ) ∼ = ⊕ i L [ d i ] wher e the sum is take n ove r the c el ls of X . Ther e is a b asis in Ω ∗ ( X ) giv en b y r esolutions of c el l closur es (cho ose one r esolution for e ach c el l). The second statemen t of this theorem follows from the ﬁrst one if w e sho w that the cob o r - dism classes o f resolutions o f the cell closures generate Ω ∗ ( X ). This can b e deduced from the analogous statemen t fo r the Cho w ring using [1 4, Theorem 1.2.19, Remark 4.5.6]. F or complex cob ordism of top ological complex cellular spaces , the corresp onding theorem simply follows from an it era t ed use o f the long exact lo calization sequence whic h alwa ys splits as ev erything in sight has M U ∗ -groups concen trated in ev en degrees only . No t e also that in the t op ological case, the Atiy ah-Hirzebruc h sp ectral sequence degenerates for these spaces , whic h allo ws to transp ort information from singular cohomology to complex cob ordism. As Morel p oin ts out, the analo gous motivic sp ectral sequence inv en t ed by Hopkins-Morel (unpublished) conv erging to algebraic cob ordism do es not in g eneral degenerate eve n fo r the p oint S pec ( k ), b ecause the one conv erging to algebraic K -theory do es not. W e no w turn to the ring structure. First, w e not e that if k = C , then there is a map of g raded rings and ev en of L -algebras Ω ∗ ( X ) → M U 2 ∗ ( X ( C ) an ) by univ ersality of algebraic cob ordism [14, Example 1.2.10]. Using the geometric description of push-fo rw ar ds b ot h for Ω ∗ and M U ∗ and the fact that the ab ov e morphism resp ects push-forw a rds [14] as w ell as [15], we ma y describe this map explicitly b y mapping an elemen t [ Y → X ] of Ω ∗ ( X ) to [ Y ( C ) an → X ( C ) an ] in M U 2 ∗ ( X ( C ) an ). As b oth pro duct structures a re deﬁned b y taking cartesian pro ducts of the geometric represen tativ es and pulling it back along the diagonal o f X r esp. X ( C ) an , w e see that this map do es indeed prese rv e the graded L -algebra structure. Also, for any em b edding k → C w e obtain a r ing homomorphism from algebraic cob o rdism ov er k to alg ebraic cob ordism ov er C . F or the ﬂag v ariety of GL n , this is an isomorphism by Theorem 2.6 b elow whic h is also v alid for M U ∗ , as b oth base c hange fro m k to C and complex to p ological realization resp ect pro ducts and ﬁrst Chern classes. F or general cellular v arieties, it is still an isomorphism. This is probably kno wn t o the exp erts, w e provide a pro o f in the app endix. F or some v arieties X , the ring structure of Ω ∗ ( X ) can b e completely determined using the pro jectiv e bundle form ula [14, Section 1.1]. This is the case for the v ariety of complete ﬂags for G = GL n (see Theorem 2.6 b elow) and a lso for Bott- Samelson resolutions of Sc hubert cycles in a complete ﬂag v ariet y fo r any reductiv e g roup G (see Section 3). 2.3. Borel presen tation for the ﬂag v ariet y of GL n . W e no w turn to the case of the complete ﬂag v ariety X for G = GL n ( k ). The p oints of X are identiﬁed with c om p lete ﬂags in k n . A c omplete ﬂag is a strictly increasing sequence of subspaces F = {{ 0 } = F 0 ⊂ F 1 ⊂ F 2 ⊂ . . . ⊂ F n = k n } SCHUBER T CALCULUS FOR A LGEBRAIC COBOR DISM 11 with dim ( F k ) = k . The group G a cts transitiv ely on the set o f all ﬂags, and the stabilizer of a p oin t is isomorphic to a Bo rel subgroup B ⊂ G , whic h mak es X = G/B into a homogeneous space under G . By this deﬁnition, X has structure of a n algebraic v ariet y . Note that o v er C , one may equiv a len tly deﬁne the ﬂag v ariety X t o be the homogeneous space K /T under the maximal compact subgroup K ⊂ G , where T is a maximal compact tor us in K (t ha t is, the pro duct of sev eral copies of S 1 ) [2]. E. g., for G = GL n ( C ) (resp. S L n ( C )), the maximal compact subgroup is U ( n ) (resp. S U ( n )). This is the language in which man y of the deﬁnitions and results in [1], [2] and [4] are stated. W e sometimes allow ourselv es to use those deﬁnitions a nd results whic h do carry o ver to the “algebraic” case (reductiv e gro ups ov er k ) without men t io ning explicitly the o b vious changes that hav e to b e carried out. There are n natura l line bundles L 1 ,. . . , L n on X , namely , the ﬁb er of L i at the p oint F is equal to F i /F i − 1 . Put x i = c 1 ( L i ), where the ﬁrst Chern class c 1 with resp ect to algebraic cob ordism is deﬁned in [14]. Note that o ur deﬁnition of x i diﬀers by sign from the one in [16]. The following result on the algebraic cob o r dism ring is an analog of the Borel presen ta tion for the singular cohomology ring of a ﬂag v ariet y . In fact, it holds for any orien t a ble cohomology theory since its pro of only uses the pro jectiv e bundle formula. Theorem 2.6. L et A ∗ ( − ) b e any o ri e n table c oh omolo gy the ory (e.g. C H ∗ ( − ) or Ω ∗ ( − ) ). Then the ring A ∗ ( X ) is isomorph ic as a gr ade d ring to the ring of p olynom i als in x 1 ,. . . , x n with c o eﬃcients in the c o eﬃcient ring A ∗ ( pt ) and deg ( x i ) = 1 , quotient by the ide al S gen er ate d b y the symm e tric p ol ynom ials of strictly p osi tive p ol yno m ial de gr e e: A ∗ ( X ) ≃ A ∗ ( pt )[ x 1 , . . . , x n ] /S. Mor e gener al ly, let E b e a ve ctor b und le of r ank n over a smo oth va riety Y and F ( E ) b e the ﬂag va riety r elative to this bund le. The n we h a ve an isomorp hism of gr ade d rin gs A ∗ ( F ( E )) ≃ A ∗ ( pt )[ x 1 , ..., x n ] /I wher e I is the ide al gener ate d by the r elations e k ( x 1 , .., x n ) = c k ( E ) for 1 ≤ k ≤ n w i th e k denoting the k -th eleme n tary symmetric p olynomial. Pr o of. The pro of of [16, Theorem 3.6.15] for the Cho w ring case can b e sligh tly mo diﬁed so that it b ecomes a pplicable to any other o r ientable theory A ∗ . Namely , f or an arbitrary oriented cohomology theory A ∗ , it is more conv enien t to dualize the geometric argumen t in [16 , Theorem 3.6.15] b ecause we can no longer use that c i ( E ) = ( − 1) i c i ( E ∗ ) for a ve ctor bundle E (whic h is used implicitly sev eral times in t he pro of of [16, Theorem 3.6.15]) . That is, w e start with the v ariet y of par tial ﬂag s P i = { F n − i ⊂ F n − i +1 ⊂ . . . ⊂ F n = k n } (e. g. P 1 is t he v ariet y of h yp erplanes in k n and P n − 1 = X ). The rest of the a rgumen t is completely analogo us to the pro of of [16 , Theorem 3.6.15]. W e giv e the details b elow for the r eader’s con v enience. Denote b y W j the corresp onding tauto lo gical v ector bundle of rank j ov er P i , where j ≥ n − i (that is, the ﬁb er of W j o v er a p o int { F n − i ⊂ F n − i +1 ⊂ . . . ⊂ F n } is equal to F j ). In particular, L i deﬁned ab o ve is equal to W i / W i − 1 . Put x i = c 1 ( L i ). As P i = P (( W n − i +1 ) ∗ ) is the pro jectiv e bundle o ve r P i − 1 and the line bundle O W n − i +1 (1) is isomorphic to L n − i +1 , the pro jectiv e bundle form ula f o r orientable cohomology theories [14, Section 1 .1] yields A ∗ ( P i ) ∼ = A ∗ ( P i − 1 )[ x n − i +1 ] / ( n − i +1 X j =0 ( − 1) j c j ( W n − i +1 ) x n − i +1 − j n − i +1 ) . 12 JENS HOR NBOSTEL A ND V ALENTINA KIRITCHENKO Or, using the a b o v e interpretation of the relat io n in the pro jectiv e bundle form ula and t he short exact sequence of v ector bundles on P i 0 → W n − i → W n − i +1 → L n − i +1 → 0 w e g et A ∗ ( P i ) ∼ = A ∗ ( P i − 1 )[ x n − i +1 ] / ( c n − i +1 ( W n − i )) . It remains to compute c n − i +1 ( W n − i ). This can b e done by induction on i starting from i = 0 (in whic h case W n is a trivial line bundle) a nd applying the Whitney sum form ula to the short exact sequenc e of v ector bundles ab ov e. W e get c ( W n − i ) = Q n j = n − i +1 c ( L j ) − 1 = Q n j = n − i +1 (1 + x j ) − 1 = P k ≥ 0 ( − 1) k h k ( x n − i +1 , . . . , x n ), where h k ( x n − i +1 , . . . , x n ) denotes the sum of a ll monomi- als of degree k in x n − i +1 ,. . . , x n . In par t icular, c n − i +1 ( W n − i ) = ( − 1) n − i +1 h n − i +1 ( x n − i +1 , . . . , x n ). F rom this w e deduce that A ∗ ( P i ) ∼ = A ∗ ( P i − 1 )[ x n − i +1 ] / ( h n − i +1 ( x n − i +1 , . . . , x n )) , and hence A ∗ ( P n ) ∼ = A ∗ ( pt )[ x 1 , . . . , x n ] / ( h n ( x n ) , h n − 1 ( x n − 1 , x n ) , . . . , h 1 ( x 1 , . . . , x n )) . The ideal generated by the relations h n − i +1 ( x n − i +1 , . . . , x n ) is exactly S , whic h is easy to c hec k starting with the recurrence relation h i ( x 1 , . . . , x n ) = h i ( x i , . . . , x n ) + X j j ( x i − x j ) mo d S (whic h is easy to sho w b y induction on n using tha t ( x n − x n − 1 ) · · · ( x n − x 1 ) = nx n − 1 n mo d S ) w e also hav e that the class of a p oin t can b e represen ted b y the p olynomial ∆ n = 1 n ! Q i>j ( x i − x j ). Note that the pro of also give s an exp licit form ula for the classes o f one- dimensional Schub ert cycles X 1 = X s γ 1 , . . . , X n − 1 = X s γ n − 1 in X corresp onding to the simple ro o t s γ 1 ,. . . , γ n − 1 of GL n (see the b eginning of Section 3 for t he deﬁnition of the Sc h ub ert cycles X w for w in the W eyl group of G ). The cycle X k consists of ﬂag s F = {{ 0 } = F 0 ⊂ F 1 ⊂ F 2 ⊂ . . . ⊂ F n = k n } suc h that all F i except f or F k are ﬁxed. Then the class of X k is equal to the class of a p oint divided b y x k +1 . Indeed, to get the class of X k ⊂ X we should take the p oint in P n − k − 1 corresp onding to the ﬁxed partia l ﬂag { F k +1 ⊂ F k +2 ⊂ . . . ⊂ F n = k n } and then take a line in a ﬁb er of the pro jectiv e bundle P n − k → P n − k − 1 o v er this p oint. Namely , the line will consist of all h yp erplanes in F k +1 that contain the ﬁxed co dimension tw o subspace F k − 1 . Again it is easy to show by induction on n that the p olynomial x n − 1 n x n − 2 n − 1 · · · x 2 /x k is equal to 2∆ n / ( x k +1 − x k ) mo dulo the ideal S . Note that the Borel presen tation for singular cohomolog y implies, in pa rticular, that Picard group o f the ﬂag v ariet y is generated (as an a b elian group) by the ﬁrst Chern classes o f the line bundles L 1 ,. . . , L n the only nontrivial relation b eing P c 1 ( L i ) = 0. In what f ollo ws, w e SCHUBER T CALCULUS FOR A LGEBRAIC COBOR DISM 13 will also use the follow ing a lternativ e description of the Picard g r o up of X . Recall that eac h strictly dominant w eight λ of G deﬁnes an irreducible represen ta t io n π λ : G → GL ( V λ ) and an em b edding G/B → P ( V λ ). Henc e, to each strictly dominan t weigh t λ of G w e can a ssign a ve ry ample line bundle L ( λ ) on X b y taking the pull-back of the line bundle O P ( V λ ) (1) on P ( V λ ). The map λ 7→ L ( λ ) extended to non-do minan t we igh ts by linearity giv es an isomorphism b et w een the Picard group of X and the w eight lattice of G [5 , 1.4.3]. In part icular, for the line bundles ab ov e we ha ve L i = L ( − e i ) where e i is the weigh t of GL n giv en b y the i -th en try of the diagonal torus in GL n . W e now compute c 1 ( L ( α i )) as a p olynomial in x 1 ,. . . , x n . Let γ 1 ,. . . , γ n − 1 b e the simple ro ots o f G (that is, γ i = e i − e i +1 ). W e can express the line bundles L ( γ i ) in terms of the line bundles L 1 ,. . . , L n . Since L i = L ( − e i ) and γ i = e i − e i +1 , w e hav e that the line bundle L ( γ i ) is isomorphic to L − 1 i ⊗ L i +1 . In pa rticular, w e can compute c 1 ( L ( γ i )) = c 1 ( L − 1 i ⊗ L i +1 ) = F ( χ ( x i ) , x i +1 ) . E.g. b y the formulas for F ( x, y ) and χ ( x ) from [14, 2.5] the ﬁrst few terms of c 1 ( L ( γ i )) lo ok as follo ws c 1 ( L ( γ i )) = − x i + x i +1 + a 11 x 2 i − a 11 x i x i +1 + . . . , where a 11 = − [ P 1 ]. In what follows, w e will use the isomorphism W ∼ = S n . The simple reﬂec tion s α for any ro ot α = e i − e j acts on the w eigh t lattice (spanned by the w eights e 1 ,. . . , e n , whic h form an orthonormal ba sis) by the reﬂection in the pla ne p erp endicular to e i − e j and hence p erm utes the we igh ts e 1 ,. . . , e n b y the transp osition ( i j ). 3. Schuber t calculus f or algebraic cobordism of flag v arieties In this section, w e assume that G is an arbitrary connected split reductiv e gro up unless w e explicitly mention that G = GL n ( k ), and X = G/ B is the complete ﬂag v ariet y for G . W e now in ves tigate the ring structure of Ω ∗ ( X ) in more geometric terms. 3.1. Sc h ub ert cycles and Bot t-Samelson r esolutions. Recall that the ﬂag v ariet y X is cellular with t he follo wing cellular decomp o sition in to Bruhat c el ls . Let us ﬁx a Borel subgroup B . F or eac h elemen t w ∈ W of the W eyl g r o up of G , deﬁne the Bruhat (or Sc hub ert) c el l C w as the B –orbit of the the p oint w B ∈ G/B = X (w e iden tify the W eyl group with N ( T ) /T for a maximal torus T of G inside B ). The Schub ert cycle X w is deﬁned as the closure of C w in X . The dimension of X w is equal to the length of w [1]. Recall that the length of an elemen t w ∈ W is deﬁned as the minimal num b er of factors in a decomp osition o f w into the pro duct of simple reﬂections. Recall also that for each l - tuple I = ( α 1 , . . . , α l ) of simple ro o t s of G , one can deﬁne the Bott-Samel s on r eso lution R I (whic h has dimension l ) to g ether with the map r I : R I → X . Bot t -Samelson resolutions a re smo oth. Consequ en tly , for any I the map r I : R I → X represen ts a n elemen t in Ω ∗ ( X ) whic h w e denote by Z I . Denote b y s α ∈ W the r eﬂection corresponding to a r o ot α , and by s I the pro duct s α 1 · · · s α l . If the decomp osition s I = s α 1 · · · s α l deﬁned by I is reduced (that is, s I can not b e written a s a pro duct o f less than l simple reﬂections, or equiv alently , the length of s I is equal to l ), then the image r I ( R I ) coincides with the Sc h ub ert cycle X s I (whic h we will also den ote b y X I ). The dimension of X I in this case is also equal t o l and the map r I : R I → X I is a birational 14 JENS HOR NBOSTEL A ND V ALENTINA KIRITCHENKO isomorphism. In this case, the v ariety R I is a resolution of singularities for the Sch ub ert cycle X I . Bott-Samelson resolutions w ere in tro duced b y Bott and Samelson in the case of compact Lie groups, and b y Demazure in the case of a lgebraic semisimple groups [10]. There are sev eral equiv alent deﬁnitions, see e. g. [6, 10, 16]. W e will use the deﬁnition b elow ( which follow s easily fro m [6, 2.2]), since it is most suited to our needs. Namely , R I is deﬁned b y the follow ing inductiv e pro cedure starting from R ∅ = pt = S pec ( k ) (in what follow s w e will rather denote R ∅ b y R e ). F or eac h j -tuple J = ( α 1 , . . . , α j ) with j < l , denote by J ∪ { j + 1 } the ( j + 1)- tuple ( α 1 , . . . , α j , α j +1 ). D eﬁne R J ∪{ j +1 } as the ﬁber pro duct R J × G/P j +1 G/B , where P j +1 is t he minimal para b olic subgroup corresp onding to the ro ot α j +1 . Then the map r J ∪{ j +1 } : R J ∪{ j +1 } → X is deﬁned as the pro jection to the second factor. In what follows , we will use that R J can b e em b edded into R J ∪{ j +1 } b y sending x ∈ R J to ( x, r J ( x )) ∈ R J × G/P j +1 G/B . In particular, o ne-dimensional Bott- Samelson resolutions are isomorphic to the corresp onding Sc hubert cycles. It is easy to show that an y tw o-dimensional Bott-Sa melson r esolution R I for a reduced I is also isomorphic to the corresp onding Sc h ub ert cycle. More generally , R I is isomorphic to X I if and only if all simple ro ots in I ar e pair wise distinct (in particular, the length of I should not exceed t he rank of G ). The simplest example where R I and X I are not isomorphic for a reduced I is G = GL 3 and I = ( γ 1 , γ 2 , γ 1 ) (where γ 1 , γ 2 are tw o simple ro ots of GL 3 ). It is easy to sho w that R J ∪{ j +1 } is the pro jectivization of the bundle r ∗ J π ∗ j +1 E , where E is the rank t wo v ector bundle on G/P j +1 deﬁned in the next subsection and π j +1 : G/B → G/P j +1 is the natural pro jection. This is the deﬁnition used in [4]. In the to p ological setting, the v ector bundle r ∗ J π ∗ j +1 E splits in to the sum of t w o line bundles [4] but in in the algebro- geometric setting this is no long er true (though r ∗ J π ∗ j +1 E still con tains a line subbundle as f ollo ws fro m the pro of of Lemma 3.4). This deﬁnition o f R I allo ws to describ e easily (b y rep eat ed use of the pro jectiv e bundle form ula) the ring structure of the cob ordism ring Ω ∗ ( R I ). It a lso implies that R I is cellular with 2 l cells lab eled by all subindices J ⊂ I . The cobo r dism classes Z I of Bo t t-Samelson resolutions generate Ω ∗ ( X ) but do not form a basis. The following prop osition is an immediate corolla r y of The orem 2.5. An analogo us statemen t fo r complex cob ordism is prov ed in [4, Prop o sition 1] by using the Atiy ah-Hirzebruc h sp ectral sequence (as men tioned in Section 2). Prop osition 3.1. As an L -mo d ule, the algebr aic c ob or dism ring Ω ∗ ( X ) of the ﬂag v a riety is fr e ely gen e r ate d by the B ott-Samelson classes Z I ( w ) wher e w ∈ W and I ( w ) deﬁnes a r e duc e d de c omp osition for w (we cho os e exactly one I ( w ) for e ach w ). There is no canonical c hoice for a decomp osition I ( w ) of a give n elemen t w in the W eyl group. F rom the geometric viewp oint it is more na t ural t o consider all Bott - Samelson classes at o nce (including those for non-reduced I ) ev en though they are not linearly indep enden t o v er L . So throughout the rest of the pap er w e will no t put an y restrictions on the mu ltiindex I . 3.2. Sc h ub ert calculus. W e will now describ e the cob ordism classes Z I as p olynomials in the ﬁrst Chern classes of line bundles on X . This allows us to compute pro ducts of Bott-Samelson resolutions and hence achiev es the goal o f a Sc hubert calculus for algebraic cob ordism. SCHUBER T CALCULUS FOR A LGEBRAIC COBOR DISM 15 W e ﬁrst deﬁne op erators A i on Ω ∗ ( X ) following the approac h of t he previous section (see Deﬁnition 2.2). These op erato rs generalize t he divide d diﬀer enc e o p er ators on the Cho w ring C H ∗ ( X ) deﬁned in [1, 10, 8 ] to algebraic cob ordism. W e ﬁrst deﬁne op erators A i for GL n since in this case the Borel presen tation allo ws to ma ke them more explicit. W e start with the subgroup B of upp er triangular matr ices and the diagonal torus, whic h yields a n isomorphism W ∼ = S n . Under this isomorphism, the reﬂection s α with resp ect to a ro ot α = e i − e j go es t o the transp o sition ( i j ) (see the end of Section 2). F or eac h p o sitiv e ro ot α of G , w e deﬁne the o p erators σ α and ˆ A α on the ring of formal p o w er series L [[ x 1 , . . . , x n ]] as follow s: ( σ α f )( x 1 , . . . , x n ) = f ( x s α (1) , . . . , x s α ( n ) ) , ˆ A α = ( 1 + σ α ) 1 F ( x i +1 , χ ( x i )) . It is easy to che c k that ˆ A α is w ell-deﬁned o n the whole ring L [[ x 1 , . . . , x n ]] (see Section 5). Note also that under t he homomorphism L [[ x 1 , . . . , x n ]] → L [ x 1 , . . . , x n ] /S ∼ = Ω ∗ ( X ) the pow er series F ( x i +1 , χ ( x i )) maps to c 1 ( L ( γ i )) (see the end of Section 2) , so our deﬁnition for additiv e formal group law reduces to the deﬁnition of divided diﬀerence o p erator on the p olynomial ring Z [ x 1 , . . . , x n ] (see [16, 2.3.1 ]). Finally , w e deﬁne the op erato r A α : Ω ∗ ( X ) → Ω ∗ ( X ) using the Borel presen tation by the formu la A α ( f ( x 1 , . . . , x n )) = ˆ A α ( f )( x 1 , . . . , x n ) for each p olynomial f ∈ L [ x 1 , . . . , x n ]. Again, by degree reasons the rig h t hand side is a p oly- nomial. The op erator A α is we ll deﬁned (that is, do es not dep end on a ch oice of a p olynomial f represen ting a giv en class in L [ x 1 , . . . , x n ] /S ) since for an y p olynomial h and an y symmetric p olynomial g we ha v e ˆ A α ( g h ) = g ˆ A α ( h ). W e now deﬁne A i = A α i for an a r bitrary reductiv e group G a nd a simple ro ot α i . Denote b y P i ⊂ G the minimal parabo lic subgroup corresponding to the ro ot α i . Then X = G/B is a pro jectiv e line ﬁbration o v er G/P i . Indeed, consider the pro jection π i : G/B → G/P i . T ake the line bundle L ( ρ ) on G/B corresp onding to the w eigh t ρ , where ρ is the half - sum of all p ositiv e ro ots or equiv alen tly the sum of all fundamen tal w eights of G (the w eight ρ is uniquely ch aracterized b y the prop erty that ( ρ, α ) = 1 f or all simple ro ots α ) . Then it is easy to c hec k that the v ector bundle E := π i ∗ L ( ρ ) on G/P i has rank tw o and G/B = P ( E ). Note that tensoring E with any line bundle L on G/P i do es not c hange P ( E ) = P ( E ⊗ L ) so t he prop erty P ( E ) = X do es not uniquely deﬁne the bundle E . How ev er, the c hoice E = π i ∗ L ( ρ ) (suggested to us b y Miche l Brion) is the only uniform c hoice for all i , since L ( ρ ) is t he only line bundle o n X with the prop erty P ( π i ∗ L ( ρ )) = X f o r all i . W e no w use D eﬁnition 2 .2 to deﬁne an Ω ∗ ( G/P i )-linear op erator A i := A π i on Ω ∗ ( X ). F or G = GL n , this deﬁnition coincides with the one g iven ab ov e. This is easy to show using that G/P i for α i = γ i is the partial ﬂag v ariety whose p oin t s are ﬂags F = {{ 0 } = F 0 ⊂ . . . ⊂ F i − 1 ⊂ F i +1 ⊂ . . . ⊂ F n = k n } . Let I = ( α 1 , . . . , α l ) b e a n l -tuple of simple ro ots of G . D eﬁne t he elemen t R I in Ω ∗ ( X ) by the for m ula R I := A l . . . A 1 Z e . 16 JENS HOR NBOSTEL A ND V ALENTINA KIRITCHENKO In the case G = GL n , w e can also regar d R I as a p olynomial in L [ x 1 , . . . , x n ] /S . Similar to [1, Theorem 3 .15] or [4, pag e 80 7 ], one ma y describ e Z e for general G using the formula Z e = R e := 1 | W | Y α ∈ R + c 1 ( L ( α )) , where R + denotes the set of p o sitiv e ro ots of G (recall t ha t | R + | = dim X =: d ). As in the Cho w r ing case, there is also the formula Z e = 1 d ! L ( ρ ) d . Both formulas immediately follow from the analogous fo r m ulas for the Cho w ring [1, Theorem 3.15, Corollary 3.16] since Ω d ( X ) ≃ C H d ( X ) (as follows fro m [14, Theorem 1.2.19 , Remark 4.5.6]). Note that f o r GL n the form ula for R e reduces to R e = ∆ n since c 1 ( L ( e i − e j )) = x j − x i +higher order terms, and hence the equalit y Z e = R e follo ws f r o m Remark 2.7. In pa r t icular, by the same remark R e mo dulo S has a denominator- free expression x n − 1 n x n − 2 n − 1 · · · x 2 . W e now pro v e an algebro-geometric v ersion of [4, Corollary 1, Prop o sition 3 ] using our algebraic op erators A i . Theorem 3.2. The c ob or dism class Z I = [ r I : R I → X ] of the Bott-Samels o n r esolution R I is e qual to R I . Pr o of. The essen tial part of the pro of is the form ula for the push-forward as stated in Corollary 2.3. Once this f orm ula is established it is not har d to sho w that A i Z I = Z I ∪{ i } for a ll I by exactly the same metho ds as in the Cho w ring case [16] and in the complex cob ordism case [4]. Namely , we hav e the f o llo wing cartesian square G/B × G/P i G/B p 2 − − − → G/B p 1   y   y π i G/B π i − − − → G/P i . . E.g., if G = GL n w e get exactly the diagram of [1 6, pro of of Lemma 3.6.20]. Using this comm utative diag ram a nd the deﬁnition of Bott-Samelson resolutions it is easy to show that π i ∗ π i ∗ Z I = Z I ∪{ i } [4, pro of o f Prop o sition 2.1]. W e no w apply Corollary 2.3 and get that A i = π i ∗ π i ∗ . It follows b y induction on the length of I that Z I = A l . . . A 1 Z e .  Remark 3.3. Note that if w e apply the base c hange form ula [14, Deﬁnition 1.1 .2 (A2)] to the cartesian diagram f rom the pro of of Theorem 3.2 , w e get p 1 ∗ p ∗ 2 = π ∗ i π i ∗ , where the righ t hand side is precisely t he deﬁnition of the “geometric” o p erator denoted A i in [4], while the left hand side is the op erato r denoted δ i in [16, pro o f of Theorem 3.6.18]. Hence Maniv el and Bressler–Ev ens consider the same op erators. W e now compute the action of the op erator A i on p olynomials in the ﬁrst Chern class es (this computation will b e used in Sections 4 and 5). C onsider the op erator σ i := σ π i again deﬁned as in Deﬁnition 2.2. Note that σ i corresp onds to the simple reﬂection s i := s α i in the follo wing sense. SCHUBER T CALCULUS FOR A LGEBRAIC COBOR DISM 17 Lemma 3.4. F or any line bund le L ( λ ) on X , we ha ve σ i ( c 1 ( L ( λ ))) = c 1 ( L ( s i λ )) . Pr o of. Since X = P ( E ) (recall that E = π i ∗ L ( ρ )), the bundle π ∗ i E on X admits the usual short exact sequence 0 → τ E → π ∗ i E → O E (1) → 0 , where τ E is the tautolog ical line bundle on X (that is, the ﬁb er of τ E at the p oint x ∈ X = P ( E ) is the line in E represen ted b y x ). Note that in o ur case P ( E ) = P ( E dual ) since E is of rank t w o (th us h yp erplanes in E are the same as lines in E ). It is easy to sho w that there is an isomorphism of line bundles τ − 1 E ⊗ O E (1) = L ( α i ) . (Moreo v er, one can sho w that τ E = L ( ρ − α i ) and O E (1) = L ( ρ ).) Indeed, τ − 1 E ⊗ O E (1) ≃ H om ( τ E , O E (1)) can b e though t of as the bundle o f tangen ts along the ﬁb ers of π i . The latter is the line bundle asso ciated with t he B –mo dule p i / b , whic h has w eigh t − α i (see [5, Remark 1.4.2] for an alternativ e deﬁnition of the line bundles L ( λ ) in terms of the one-dimensional B –mo dules). Here p i and b denote t he Lie algebras of P i and B , resp ectiv ely . By deﬁnition, σ i switc hes c 1 ( τ E ) and c 1 ( O E (1)). Hence, σ i ( c 1 ( L ( α i ))) = c 1 ( L ( − α i )). Since the Picard group of G/P i can b e iden tiﬁed with with the sublattice { λ | ( λ, α i ) = 0 } of the w eight lattice of G (this follo ws f rom [5, remark af ter Prop osition 1.3.6] combine d with [5, Propo sition 1.4.3]) we also hav e σ i ( c 1 ( L ( λ ))) = c 1 ( L ( λ )) for all λ p erp endicular to α i . These t w o iden tities imply the statemen t of the lemma.  This lemma allo ws us to describe explicitly the a ctio n of σ i and hence of A i on an y p olynomial in the ﬁrst Chern classes. Indeed, since for an y weigh t λ w e hav e s i λ = λ + k α i for some in teger k , w e can compute c 1 ( L ( σ i λ )) = c 1 ( L ( λ ) ⊗ L ( α i ) k ) as a p o wer series in c 1 ( L ( λ )) and c 1 ( L ( α i )) using the formal group law. This will be us ed in the pro of of Prop osition 4.3 b elo w and in Subsection 5.1. 4. Chev alley-Pie ri formulas A key ingredien t for the classical Sc h ub ert calculus is the Che v alley-Pieri form ula for the pro duct of the Sc h ub ert cycle with the ﬁrst Chern class of the line bundle o n X , see e. g. [1, Prop osition 4 .1] and [10, Prop osition 4.2]. W e no w establish analogous fo r mulas for the pro ducts of Z I and R I with c 1 ( L ( λ )) (without using that Z I = R I ). A t the end of this section, w e explain wh y in t he case of algebraic cob ordism this alone is not enough to show tha t Z I = R I , hence justifying our diﬀeren t approach of the previous tw o sections. By L ( D ) denote the line bundle corresponding to the divisor D . F or eac h l - t uple I as abov e, denote b y I j the ( l − 1)– tuple ( α 1 , . . . , ˆ α j , . . . , α l ). F or each ro ot α , deﬁne the linear f unction ( · , α ) (that is, the c or o ot ) on the weigh t lattice of G b y the pro p ert y s α λ = λ − ( λ, α ) α fo r a ll w eights λ . (The pairing ( a, b ) is o f ten denoted b y h a, b ∨ i or by h a, b i .) Note that by deﬁnition ( λ, α ) = ( w λ, w α ) for all elemen ts w of the W eyl gr oup. Prop osition 4.1. Ge ometric Cheval ley-Pieri f ormula (1) (for Bott-Sa melson resolutions) In the Pic ar d gr oup of R I we have r ∗ I L ( λ ) = ⊗ l j =1 L ( R I j ) ( λ,β j ) 18 JENS HOR NBOSTEL A ND V ALENTINA KIRITCHENKO wher e β j = s l · · · s j +1 α j . (2) (for Sc h ub ert cycles)[1, Prop osition 4.1] , [10, Prop osition 4.4] , [8] In the Chow ring of X we have c 1 ( L ( λ )) X I = X j ( λ, β j ) X I j wher e the sum is taken ove r j ∈ { 1 , . . . , l } for which the de c omp osition deﬁne d by I j is r e duc e d. The ﬁrst part of this prop osition w as prov ed in [4, Prop osition 4] in the top olo g ical setting (for ﬂag v arieties o f compact Lie groups). It is not hard to c hec k that the pro of carries ov er to algebro-geometric setting. W e instead prov ide a shorter pro of along the same lines. Our pro of is based on the f ollo wing lemma. Lemma 4.2. [10, Prop osition 2.1] L et p : R I → R I l b e the natur al p r o j e ction (c oming fr om the fact that we deﬁne d R I as a pr oje ctive bund le ove r R I l ). Then we h ave an isomo rphism r ∗ I L ( λ ) ∼ = p ∗ r ∗ I l L ( s l λ ) ⊗ L ( R I l ) ( λ,α l ) of l i n e bund les on R I . Prop osition 4.1(1) now follow s f r om Lemma 4.2 by induction on l . The base l = 1, that is r ∗ 1 L ( λ ) = O P 1 (1) ( λ,α 1 ) , follows from the fact that r 1 : R 1 → X maps R 1 isomorphically to P 1 /B ∼ = P 1 , whic h can b e regarded as the ﬂag v ariet y for S L 2 . Then the w eigh t λ restricted to S L 2 is equal t o ( λ, α 1 ) times the highest w eigh t of the tautolo gical represen tation of S L 2 , whic h corresp onds to the line bundle O P 1 (1) on P 1 . T o prov e the induction step plug in the induction h yp o thesis for r ∗ I l L ( s l λ ) = ⊗ l − 1 j =1 L ( R I j,l ) ( s l λ,s l − 1 ··· s j +1 α j ) in to the lemma and use that ( s l λ, s l − 1 · · · s j +1 α j ) = ( λ, β j ) (since s 2 l = e ) and p ∗ R I j,l = R I j . Prop osition 4.1(1) w as used in [4] to establish an algorithm for computing c 1 ( L ( λ )) Z I in Ω ∗ ( X ) [4]. W e no w brieﬂy recall this algorithm. By the pro jection form ula w e hav e c 1 ( L ( λ )) Z I = ( r I ) ∗ ( c 1 ( r ∗ I L ( λ ))) . Note that the usual pro jection formula with resp ect to smo oth pro jectiv e morphisms f : X → Y holds fo r algebraic cob ordism as we ll. This follo ws from the deﬁnition of pro ducts via pull- bac ks along the diagonal and the base ch ange axiom ( A 2) of [14] applied to the cartesian square obtained f r om Y diag → Y × Y p × id ← X × Y . One can now use Prop o sition 4.1(1) and the f o rmal group la w to compute c 1 ( r ∗ I L ( λ )) in terms of the Bott-Samelson classes in Ω ∗ ( R I ) by an iterativ e pro cedure (since the mu ltiplica- tiv e structure of Ω ∗ ( R I ) can b e determined b y the pro jectiv e bundle for mula and t he Chern classes arising this w a y again ha v e form c 1 ( L ( λ )) f o r some λ ). After c 1 ( r ∗ I L ( λ )) is written as P J ⊂ I a J [ R J ] f o r some a J ∈ L it is easy to ﬁnd ( r I ) ∗ ( c 1 ( r ∗ I L ( λ ))) since ( r I ) ∗ [ R J ] = Z J . Ho w ev er, this pro cedure is rather lengthy , and w e will not use it . Instead, we will prov e a more explicit fo r mula for c 1 ( L ( λ )) Z I (see form ula 5.1 b elow) using our algebraic Chev alley-Pieri form ula to gether with Theorem 3.2. Prop osition 4.3. A l gebr aic Cheval ley-Pieri for mula: SCHUBER T CALCULUS FOR A LGEBRAIC COBOR DISM 19 (1) (cob ordism v ersion) L et A 1 = A α 1 ,. . . , A l = A α l b e the op er ators on Ω ∗ ( X ) c orr esp onding to α 1 ,. . . , α l . The n we h ave c 1 ( L ( λ )) A 1 . . . A l R e = l X j =1 A 1 . . . A j − 1 c 1 ( L ( λ j )) − c 1 ( L ( s j λ j )) c 1 ( L ( α j )) A j +1 . . . A l R e in Ω ∗ ( X ) , wher e λ j = s j − 1 · · · s 1 λ and s j = s α j is the r eﬂe ction c orr esp onding to the r o ot α j . (2) (Cho w ring ve rsion) [1, Corollary 3.7] L e t A 1 = A α 1 ,. . . , A l = A α l b e the op er ators on C H ∗ ( X ) c o rr esp onding to α 1 , . . . , α l . Then c 1 ( L ( λ )) A 1 . . . A l R e = l X j =1 ( λ, s 1 · · · s j − 1 α j ) A 1 . . . ˆ A j . . . A l R e in C H ∗ ( X ) . Pr o of. First, note that c 1 ( L ( λ j )) − c 1 ( L ( s j λ j )) c 1 ( L ( α j )) is a w ell-deﬁned elemen t in Ω ∗ ( X ) b ecause s j λ = λ − ( λ, α j ) α j (and hence L ( λ ) = L ( s j λ ) ⊗ L ( α j ) ( λ,α j ) ) a nd the formal group law expansion for c 1 ( L 1 ⊗ L k 2 ) − c 1 ( L 1 ) is divisible b y c 1 ( L 2 ) for an y in teger k [14, (2.5 .1)]. Next w e sho w t hat c 1 ( L ( λ )) A 1 − A 1 c 1 ( L ( s 1 λ )) = c 1 ( L ( λ )) − c 1 ( L ( s 1 λ )) c 1 ( L ( α 1 )) , where b oth sides are rega rded as o p erators on Ω ∗ ( X ). Indeed, by deﬁnition A 1 = (1 + σ 1 ) 1 c 1 ( L ( α 1 )) and c 1 ( L ( λ )) σ 1 = σ 1 c 1 ( L ( s 1 λ )) by Lemma 3.4. Hence, we can write c 1 ( L ( λ )) A 1 . . . A l R e = c 1 ( L ( λ )) − c 1 ( L ( s 1 λ )) c 1 ( L ( α 1 )) A 2 . . . A l R e + A 1 c 1 ( L ( s 1 λ )) A 2 . . . A l R e , and then con tin ue moving c 1 ( L ( s 1 λ )) to the r ig h t until w e ar e left with with the t erm A 1 . . . A l c 1 ( L ( s l . . . s 1 λ )) R e . This term is equal to ze ro since c 1 ( L ( s l . . . s 1 λ )) R e is the pro d- uct of mo r e than dim X ﬁrst Chern classes, and hence its degree is greater than dim X . The Cho w r ing case follows immediately from the cob or dism case since c 1 ( L ( λ )) − c 1 ( L ( s j λ )) c 1 ( L ( α j )) = ( λ, α j ) in the Cho w r ing . The last iden tity holds b ecause the fo r ma l gro up la w for the Chow ring is additiv e, and hence c 1 ( L ( λ )) − c 1 ( L ( s j λ )) = ( λ, α j ) c 1 ( L ( α j )).  The sec ond part of this propo sition w as pro ved in [1] b y more in v olv ed calculations. A calculation similar to ours w as used in [17] to deduce a combinatorial Chev alley-Pieri form ula for K -theory . It would b e intere sting to ﬁnd an analogous com binatorial interpretation of our Chev alley-Pieri form ula in t he cob o rdism case. Note tha t in the case o f Cho w g roups, the algebraic Chev alley-Pieri formula fo r A l . . . A 1 R e is exactly the same as the geometric one for the Sch ub ert cycle X I . T ogether with the Bo rel presen tation this easily implies that the p olynomial A l . . . A 1 R e represen ts the Sc hubert cycle X I whenev er I deﬁnes a reduced decomp osition [1]. Inde ed, w e can pro ceed b y the induction on l . Algebraic and geometric Chev alley-Pieri formulas allow to compute the in tersection indices of A l . . . A 1 R e and of X I , resp ective ly , with the pro duct of k ﬁrst Chern classes, a nd the result is 20 JENS HOR NBOSTEL A ND V ALENTINA KIRITCHENKO the same in b oth cases by the induction h yp o thesis (for all k > 0). By the Borel presen tation w e kno w that the pro ducts of ﬁrst Chern classes span ⊕ d i =1 C H i ( X ). Hence, b y the non-degeneracy of the in tersection form on C H ∗ ( X ) (t ha t is, by Poincar ´ e dualit y) w e ha ve t ha t A l . . . A 1 R e − X I m ust lie in C H d ( X ) = Z [ pt ] (that is, in the o r thogonal complemen t to ⊕ d i =1 C H i ( X )). This is only p ossible if A l . . . A 1 R e − X I = 0 (unless l = 0, whic h is the induction base). Note that the only geometric input in this pro o f is the g eometric Chev alley-Pieri formula. In the cob ordism case, it is no t immediately clear why geometric and algebraic Chev alley- Pieri formulas a r e the same (t ho ugh, of course, it follows fro m Theorem 3.2). But ev en without using that R I = Z I it migh t b e p ossible to sho w that b oth formulas ha v e the same structure co eﬃcien ts, t ha t is, if c 1 ( L ( λ )) Z I = P J ⊂ I a J Z J then necessarily c 1 ( L ( λ )) R I = P J ⊂ I a J R J with the same co eﬃcien ts a J ∈ L . Ho w ev er, this do es no t lead to the pro of of R I = Z I as in the case of the Cho w ring. The reason is that ev en though there is an analog of P oincar ´ e dualit y fo r the cob ordism r ing s of cellular v arieties, this only yields an equality R I = Z I up to a m ultiple of [ pt ] (as in Lemma 4.4 b elow in the case of GL n /B ), and this is not enough to carry out the desire d induction argumen t. F or the Cho w ring, P oincar ´ e dualit y also yields only an equality up to t he class of a p oin t, but unless I = ∅ , the diﬀerence R I − Z I (where now Z I means t he Sc hubert cycle and no t the Bott- Samelson class) can not b e a non-zero multiple of [ pt ] b ecause the co eﬃcien t ring C H ∗ ([ pt ]) = C H ∗ ( k ) ∼ = Z is concen trated in degree zero, hence has no nonzero elemen ts in the corresp onding degree l − d . Ho wev er, for algebraic cob ordism the co eﬃcien t ring Ω ∗ ( k ) ∼ = L do es contain plen ty o f elemen ts of negativ e degree, so one can not deduce R I = Z I . Lemma 4.4. L et X = GL n /B and let c ∈ Ω ∗ ( X ) ∼ = L [ x 1 , . . . , x n ] /S b e a homo g e ne ous element of de gr e e l such that the pr o duct of c with any non-c onstant monomial in x 1 ,. . . , x n is zer o. Then c b elong s to L l − d [ pt ] , wh er e d = n ( n − 1) / 2 . Pr o of. W e will use that the ideal S con ta ins all homogeneous p olynomials of degree greater than d with integer co eﬃcien ts [16, Corollary 2.5.6]. Let ˆ c b e a homogenous elemen t in L [ x 1 , ..., x n ] that represen ts c . Recall that L is isomorphic to the gra ded p olynomial ring Z [ a 1 , a 2 , . . . ] in coun ta bly man y v ariables, where a i has degree − i . Therefore ˆ c has a unique decomp osition a s a sum of in tegral p olynomials with co eﬃcien t s b eing monomials in the a i , tha t is ˆ c = c 0 + a 1 c 1 + a 2 c 2 + a 2 1 c 1 , 1 + a 3 c 3 + a 1 a 2 c 1 , 2 + a 3 1 c 1 , 1 , 1 + . . . where c i 1 ,...,i s is a p olynomial of degree l − P deg( a i j ) with in teger co eﬃcien ts. Note that we migh t c ho ose a ˆ c such that t he sum is ﬁnite since c i 1 ,...,i s v anishes mo dulo S if l − P deg( a i j ) > d . No w w e m ultiply ˆ c with an arbitrary monomial m d − l in the x i of degree d − l . Since m d − l c = 0 it follows that m d − l c 0 is zero mo dulo S . By algebraic Poincar ´ e dua lity [16, Prop osition 2.5.7] it follows that c 0 = 0 mo dulo S . Next, w e multiply with monomials of degree d − l − 1 to deduce that c 1 equals zero mo dulo S , and then deduce inductiv ely that all the c i 1 ,...,i s of degree strictly les s tha n d are zero. It remains to not e that each c i 1 ,...,i s of degree d is equal to an in teger m ultiple of [ pt ] (since all homo g eneous p olynomials of degree d with integer co eﬃcien ts are equal to a multiple o f R e mo dulo the ideal S [16, 2.5.2]) Hence ˆ c = a [ pt ] for some a ∈ L , whic h must hav e degree l − d b y homog eneity of ˆ c .  SCHUBER T CALCULUS FOR A LGEBRAIC COBOR DISM 21 5. Comput a tions and examples Un til no w, w e used the formal gro up law of algebraic cob ordism (i.e., the univ ersal one) as little as po ssible in order to mak e our presen tation simpler. In this section, w e make the results of the previous section more explicit using this for mal group la w. In particular, we giv e an explicit formula for the pro ducts of a Bott-Samelson resolution with the ﬁrst Chern class of a line bundle in terms of other Bott-Samelson resolutions (see formula 5.1 b elow ). Using this form ula, we giv e a n alg orithm for computing the pro duct of tw o Bott- Samelson resolutions. First, w e sho w that the op erator A from Section 2 a nd the op erator ˆ A α from Section 3 are w ell-deﬁned. W e use notatio n of Subsection 2.1, so F ( u , v ) is the univ ersal formal group la w and χ ( u ) is t he inv erse for the univ ersal formal group la w deﬁned by the iden tit y F ( u , χ ( u )) = 0. T o sho w that the op erator A = (1 + σ ) 1 F ( y 1 ,χ ( y 2 )) is w ell deﬁned on Ω ∗ ( X )[[ y 1 , y 2 ]] it is enough to sho w that A ( m ) is a formal p o w er series for any monomial m = y k 1 1 y k 2 2 . W e compute A ( y k 1 1 y k 2 2 ) using that y 1 = F ( x, y 2 ) = y 2 + χ ( x ) p ( x, y 2 ) and y 2 = F ( χ ( x ) , y 1 ) = y 1 + χ ( x ) p ( χ ( x ) , y 1 ) where x = F ( y 1 , χ ( y 2 )) and p ( u, v ) = F ( u,v ) − u v is a we ll-deﬁned pow er series (since F ( u, v ) − u con tains only terms u i v j for j ≥ 1 ). W e get A ( y k 1 1 y k 2 2 ) = (1+ σ ) y k 1 1 y k 2 2 x = y k 1 1 y k 2 2 x + y k 2 1 y k 1 2 χ ( x ) = ( y 2 + χ ( x ) p ( x, y 2 )) k 1 ( y 1 + χ ( x ) p ( χ ( x ) , y 1 )) k 2 x + y k 1 2 y k 2 1 χ ( x ) = = y k 1 2 y k 2 1 q ( x, χ ( x )) + terms divisible by x or by χ ( x ) x . The second term in the last expression is a p ow er series since the formal group law expansion for χ ( x ) is divisible b y x [14, (2.5.1 )]. A similar argumen t shows t ha t the op erato r ˆ A α from Section 3 is indeed w ell-deﬁned on the whole ring L [[ x 1 , . . . , x n ]] for an y ro ot α . Indeed, by relab eling x 1 ,. . . , x n w e can assume that α = e 1 − e 2 . Then for any monomial m = x k 1 1 x k 2 2 . . . x k n n w e hav e ˆ A α ( m ) = x k 3 3 . . . x k n n ˆ A α ( x k 1 1 x k 2 2 ) . Then exactly the same arg umen t as the one a b o v e for A shows that ˆ A α ( x k 1 1 x k 2 2 ) is a p ow er series in x 1 and x 2 . 5.1. Algorithm for computing the products of Bott -Samelson resolutions. W e now pro duce an explicit algorithm for computing the pro duct of the Bott-Samelson classes Z I in terms of other Bo tt-Samelson classes, where I = ( α 1 , ..., α l ). The k ey ing r edient is our algebraic Chev alley-Pieri form ula (Prop osition 4.3) whic h can b e reform ulated as follow s c 1 ( L ( λ )) A 1 . . . A l Z e = l X j =1 A 1 . . . A j − 1 A ∗ j ( c 1 ( L ( λ j ))) A j +1 . . . A l Z e , where λ j = s j − 1 · · · s 1 λ (in other w ords, c 1 ( L ( λ j )) = [ σ j − 1 . . . σ 1 ]( c 1 ( L ( λ )))) and the op erato r A ∗ j is deﬁned as follows A ∗ j = A ∗ α j = 1 c 1 ( L ( α j )) (1 − σ α j ) . 22 JENS HOR NBOSTEL A ND V ALENTINA KIRITCHENKO W e can compute A ∗ j on a n y p olynomial in the ﬁrst Chern classes b y the same metho ds as A j (see the end of Section 3). Note that for the Cho w ring A j = A ∗ j (this follo ws fro m Lemma 3.4 and the fact that s j α j = − α j and c 1 ( L ( α j )) = − c 1 ( L ( − α j )) for the additiv e forma l group la w), but for the a lgebraic cob ordism ring this is no longer true. More generally , for a n y p olynomial f = f ( c 1 ( L ( µ 1 )) , . . . , c 1 ( L ( µ k ))) in the ﬁrst Chern classes of some line bundles on X , w e can compute its pro duct with A 1 . . . A l Z e b y exactly the same argumen t as in the pro of of Prop osition 4.3: f · A 1 . . . A l Z e = l X j =1 A 1 . . . A j − 1 [ A ∗ j σ j − 1 . . . σ 1 ]( f ) A j +1 . . . A l Z e + A 1 . . . A l [ σ l . . . σ 1 ]( f ) Z e (5 . 0) Note that the last term on the right hand side is equal to the constan t term of the p o lynomial [ σ l . . . σ 1 ]( f ) (whic h is of course the same as the constan t term of f ) times A 1 . . . A l Z e . In particular, f o r f = c 1 ( L ( λ )) this t erm v anishes mo dulo S . Here and b elo w, by the “constant term” of a p olynomial in L [ x 1 , . . . , x n ] w e mean the t erm of p olynomial degree zero (the tot a l degree of suc h a constant term might b e negativ e since the L azard ring L contains elemen ts of negativ e degree). Note that a ll elemen ts of L ⊂ L [ x 1 , . . . , x n ] are in v arian t under the op erators σ i , and hence comm ute with the op erators A i . F or a n arbitrary reductiv e g roup, the constant term of a n elemen t f ∈ Ω ∗ ( X ) is deﬁned as the pro duct of f with the class of a p oint. It is now easy to sho w b y induction on l that f A 1 . . . A l Z e = X J ⊂ I a J ( f )[ Y i ∈ I \ J A i ] Z e , where a J ( f ) for the k -subtuple J = ( α j 1 , . . . , α j k ) of I is the constan t term in the expansion for [ σ l . . . σ j k +1 A ∗ j k σ j k − 1 . . . σ j 1 +1 A ∗ j 1 σ j 1 − 1 . . . σ 1 ] f , whic h is in v arian t under σ i (for all i ) and hence equal to [ A ∗ j k σ j k − 1 . . . σ j 1 +1 A ∗ j 1 σ j 1 − 1 . . . σ 1 ] f . Indeed, w e ﬁrst use form ula (5.0) ab ov e and then apply the induction h yp othesis to all t erms in the right hand side except for the last t erm, whic h alr eady has fo rm a J ( f )[ Q i ∈ I \ J A i ] Z e for J = ∅ . W e g et A 1 . . . A j − 1 [ A ∗ j σ j − 1 . . . σ 1 ]( f ) A j +1 . . . A l Z e = = A 1 . . . A j − 1 X J ⊂ I \{ 1 ,...,j } a J ([ A ∗ j σ j − 1 . . . σ 1 ]( f ))[ Y i ∈ I \ ( J ∪{ 1 ,...,j } ) A i ] Z e = = X J ′ ⊂ I a J ′ ( f )[ Y i ∈ I \ J ′ A i ] Z e , where the last summation go es ov er a ll subsets J ′ of I that do con tain j but do not con tain 1,. . . , j − 1. Plugging this back into form ula (5.0) w e get the desired fo rm ula . Com bining this with Theorem 3.2, w e get the follo wing formula in Ω ∗ ( X ) for the pro duct of the Bott-Sa melson class Z I with the ﬁrst Chern class c 1 ( L ( λ )) in terms of other Bott-Samelson classes c 1 ( L ( λ )) Z I = X J ⊂ I b J ( λ ) Z I \ J , (5 . 1) where b J ( λ ) is the constan t term in the expansion for [ A ∗ j 1 σ j 1 +1 . . . σ j k − 1 A ∗ j k σ j k +1 . . . σ l ]( c 1 ( L ( λ ))) . SCHUBER T CALCULUS FOR A LGEBRAIC COBOR DISM 23 W e c hanged the order of the σ i when passing f rom a J to b J since Z I = A l . . . A 1 Z e . Note that for J = ∅ we hav e b J = 0, and f o r J = ( α j ) we hav e b J = ( λ, β j ) since t he constant term in A ∗ j ( c 1 ( L ( s j +1 . . . s l λ ))) = c 1 ( L ( s j +1 ...s l λ )) − c 1 ( L ( s j s j +1 ...s l λ )) c 1 ( L ( α j )) is equal to ( s j +1 . . . s l λ, α j ) (see the pro of of Prop osition 4.3, and Prop osition 4.1 fo r the deﬁnition of β i ), whic h is equal to ( λ, β j ). So the lo w est order terms (with resp ect to the p olynomial grading) of this formula giv e an a nalogous form ula f o r the Cho w ring as exp ected. W e now ha v e assem bled all necessary to ols for actually performing the desired Sc h ub ert calculus. Namely , to compute the pro duct Z I Z J w e apply the fo llo wing pro cedure (which is formally similar to the one for the Chow ring ) . W e replace Z J with the resp ectiv e p olynomial R J in the ﬁrst Chern classes (using Theorem 3.2 together with the form ula for Z e ) and then compute the pro duct of Z I with eac h monomial in R J using r ep eatedly f orm ula (5.1) . Note that form ula (5 .1) a llows us to mak e this a lgorithm more explicit than the one giv en in [4] (see an example b elo w). The naiv e approac h to represen t b o th Z I and Z J as fractions of p o lynomials in ﬁrst Chern classes and then computing their pro duct is less useful. In particular translating the pro duct of the f ractions bac k in to a linear com bination of Bott-Samelson classes will b e v ery hard, if p ossible at all. 5.2. Examples. W e no w compute the Bott-Sa melson classes Z I in terms o f the Chern classes x i for the example X = S L 3 /B where B is the subgroup of upp er- t riangular matrices. W e then compute certain pro ducts of Bott- Sa melson classes in t w o w ays, b y hand and then using the algorithm ab o ve together with formula (5.1 ) . Note that only the second approa c h generalizes to higher dimensions. In S L 3 , there are tw o simple ro ot s γ 1 and γ 2 . In X , there are six Sch ub ert cycles X e = pt , X 1 , X 2 , X 12 , X 21 and X 121 = X (here 12 is a short ha nd notation for ( γ 1 , γ 2 ), etc.). E ac h X I except for X 121 coincides with its Bott-Samelson resolution R I . Note that in general R I and X I do not coincide eve n when X I is smo ot h. (By t he w a y , for G = GL n the ﬁrst non-smo o t h Sc hubert cycles show up for n = 4.) Computing Z I as a p olynomial in the ﬁrst Chern classes. W e w ant to express Z I as a p olynomial in x 1 , x 2 , x 3 using the formulas Z s i 1 ...s i l = A i l . . . A i 1 R e ; R e = 1 6 c 1 ( L ( γ 1 )) c 1 ( L ( γ 2 )) c 1 ( L ( γ 1 + γ 2 )) . Note that in computations in v olving the op erators A α it is more conv enien t not to replace c 1 ( L ( α )) with its expression in terms of x i un til the v ery end. Let us f o r instance compute R 1 as a p olynomial in x 1 , x 2 , x 3 mo dulo the ideal S g enerated b y the symmetric p olynomials of p ositiv e degree: R 1 = A 1 R e = 1 6 (1 + s 1 ) c 1 ( L ( γ 2 )) c 1 ( L ( γ 1 + γ 2 )) = 1 3 c 1 ( L ( γ 2 )) c 1 ( L ( γ 1 + γ 2 )) = 1 3 F ( χ ( x 2 ) , x 3 ) F ( χ ( x 1 ) , x 3 ) . 24 JENS HOR NBOSTEL A ND V ALENTINA KIRITCHENKO W e ha v e χ ( u ) = − u + a 11 u 2 − a 2 11 u 3 and F ( u, v ) = u + v + a 11 uv + a 12 u 2 v + a 21 uv 2 , wh ere a 11 = − [ P 1 ] a nd a 12 = a 21 = [ P 1 ] 2 − [ P 2 ] [1 4, 2.5]. Th us 1 3 F ( χ ( x 2 ) , x 3 ) F ( χ ( x 1 ) , x 3 ) = 1 3 F ( − x 2 + a 11 x 2 2 − a 11 x 3 2 , x 3 ) F ( − x 1 + a 11 x 2 1 − a 11 x 3 1 , x 3 ) = = 1 3 ( − x 2 + x 3 + a 11 x 2 2 − a 11 x 2 x 3 )( − x 1 + x 3 + a 11 x 2 1 − a 11 x 1 x 3 ) = x 2 3 , since ( x 3 − x 2 )( x 3 − x 1 ) = 3 x 2 3 mo d S , and ( x 2 + x 1 )( x 2 − x 3 )( x 1 − x 3 ) = − 3 x 3 3 = 0 mo d S . So t he answ er agrees with the one w e got in Remark 2.7. Here are the p olynomials for the ot her Bott- Samelson resolutions: R 212 = 1 + a 12 x 2 1 ; R 121 = 1 + a 12 x 1 x 2 R 12 = − x 1 − [ P 1 ] x 2 1 R 21 = x 3 = − x 1 − x 2 R 1 = x 2 3 = x 1 x 2 R 2 = x 2 1 R e = − x 2 1 x 2 . Note tha t t he Bott- Samelson resolutions R I in this list coincide with the Sc h ub ert cycles they resolv e if I has length ≤ 2. The corresp onding p olynomials R I are the classical Sc h ub ert p olynomials (see e.g. [16] and k eep in mind that his x i is equal to our − x i ) ex cept for the p olynomial R 12 . In general, p olynomials R I can b e computed b y induction on the length of I . E.g. to compute R 212 w e can use that R 212 = A 2 R 21 and R 21 = x 3 . Hence, R 212 = A 2 ( x 3 ) = 1 + a 12 x 2 x 3 = 1 + a 12 x 2 1 The middle equation is obtained using the form ula A ( y 1 ) = 1 + a 12 y 1 y 2 + . . . from Section 2.1 and the observ at io n that all symmetric p olynomials in x 2 and x 3 of degree greater than 2 v anish mo dulo S . Computing pro ducts of the B ott-Samelson resolutions. Let us for instance compute Z 12 Z 21 . First, w e do it b y hand. Denote by ω 1 , ω 2 the fundamen tal w eigh ts of S L 3 . Apply ing Prop osition 4.1(2) to X 121 = X w e g et L ( λ ) = L ( X 21 ) ( λ,γ 2 ) ⊗ L ( X 12 ) ( λ,γ 1 ) . Hence, c 1 ( L ( ω 1 )) = X 12 and c 1 ( L ( ω 2 )) = X 21 . (Note that if w e instead applied Prop osition 4.1(1) t o R 121 , w e w o uld obta in the more complicated expression r ∗ 121 L ( λ ) = L ( R 21 ) ( λ,γ 2 ) ⊗ L ( R 11 ) ( λ,γ 1 + γ 2 ) ⊗ L ( R 12 ) ( λ,γ 1 ) , whic h do es not a llo w us to express X 12 = R 12 as the Chern class of the line bundle L ( λ ) on R 121 .) Hence, Z 12 Z 21 = c 1 ( L ( ω 1 )) Z 21 = r 21 ∗ c 1 ( r ∗ 21 L ( ω 1 )) b y the pro jection fo rm ula : c 1 ( L ( λ )) · Z I = r I ∗ c 1 ( r I ∗ L ( λ )) . W e now apply Prop o sition 4.1(1) to R 21 and L ( ω 1 ) and get r ∗ 21 L ( ω 1 ) = L ( R 1 ) ⊗ L ( R 2 ) . Using the formal gr oup law we compute c 1 ( L ( R 1 ) ⊗ L ( R 2 )) = R 1 + R 2 − [ P 1 ] R e . F inally , we use that r J ∗ [ R J ] = Z J and get that Z 12 Z 21 = Z 1 + Z 2 − [ P 1 ] Z e . SCHUBER T CALCULUS FOR A LGEBRAIC COBOR DISM 25 Similarly , w e can easily compute t he following pro ducts: Z 12 Z 12 = Z 2 ; Z 21 Z 21 = Z 1 Z 12 Z 1 = Z 21 Z 2 = Z e , Z 12 Z 2 = Z 21 Z 1 = 0 , whic h in particular gives us a nother wa y to compute p olynomials R I . So the only pro duct that diﬀers from the analogous pro duct in the Chow ring case is the pro duct Z 12 Z 21 . W e no w compute the pro duct Z 12 Z 21 using f orm ula (5.1) . W e ha ve Z 12 = c 1 ( L ( ω 1 )) by Prop osition 4.1(1). Hence, according to f o rm ula ( 5 .1) Z 12 Z 21 = c 1 ( L ( ω 1 )) Z 21 = b 1 ( ω 1 ) Z 2 + b 2 ( ω 1 ) Z 1 + b 21 ( ω 1 ) Z e , where b 1 , b 2 and b 21 are the constant t erms in A ∗ 1 ( c 1 ( L ( ω 1 ))), [ A ∗ 2 s 1 ]( c 1 ( L ( ω 1 ))) and [ A ∗ 2 A ∗ 1 ]( c 1 ( L ( ω 1 ))), resp ectiv ely . W e already kno w that b 1 ( λ ) = ( λ, γ 1 ) and b 2 ( λ ) = ( λ, s 1 γ 2 ). It remains to compute b 21 ( λ ). First, by using t ha t L ( λ ) = L ( s 1 λ ) ⊗ L ( γ 1 ) ( λ,γ 1 ) and the formal group la w we write A ∗ 1 ( c 1 ( L ( λ ))) = c 1 ( L ( λ )) − c 1 ( L ( s 1 λ )) c 1 ( L ( γ 1 )) = = ( λ, γ 1 ) + a 11 ( λ, γ 1 )[ c 1 ( L ( s 1 λ )) + ( λ, γ 1 ) − 1 2 c 1 ( L ( γ 1 ))] + terms of deg ≥ 2 Hence, [ A ∗ 2 A ∗ 1 ]( c 1 ( L ( λ ))) = a 11 ( λ, γ 1 ) A ∗ 2 [ c 1 ( L ( s 1 λ )) + ( λ, γ 1 ) − 1 2 c 1 ( L ( γ 1 ))] + terms of deg ≥ 1 = = a 11 ( λ, γ 1 )[( λ, s 1 γ 2 ) − ( λ, γ 1 ) − 1 2 ] + terms o f deg ≥ 1 , and b 21 = a 11 ( λ, γ 1 )[( λ, s 1 γ 2 ) − ( λ,γ 1 ) − 1 2 ]. W e get c 1 ( L ( λ )) Z 21 = ( λ, γ 1 ) Z 2 + ( λ, s 1 γ 2 ) Z 1 + a 11 ( λ, γ 1 )[( λ, s 1 γ 2 ) − ( λ, γ 1 ) − 1 2 ] Z e . In particular, c 1 ( L ( ω 1 )) Z 21 = Z 2 + Z 1 + a 11 Z e (whic h coincides with the answ er we ha v e fo und ab ov e b y hand). Finally , note that it tak es more w ork to compute c 1 ( L ( λ )) Z 21 using t he algorithm in [4] b ecause a pa rt fro m certain formal gro up law calculations (whic h are more in v olved than the calculations we used to ﬁnd b 21 ) one has also to compute the pro ducts R 2 1 and R 2 2 in C H ∗ ( R 21 ). 6. Appendix: Complex realiza tion for c e llular v arietie s W e will no w prov e the follow ing result stated in Section 2: Theorem 6.1. F or any smo oth c e l lular v ariety X over k and any emb e dding k → C , the c om- plex g e ometric r e aliz a tion functor of L -al g ebr as r : Ω ∗ ( X ) → M U ∗ ( X ( C ) an ) is an isomorphism. Pr o of. Recall (see ab o ve ) that the geometric realizatio n functor coincides with the map giv en b y the univ ersal prop erty of Ω ∗ , and that b oth sides are freely generated b y (resolutions of the closures of ) the cells. Thus it suﬃces to sho w that it is an isomorphism if we pass to the induce d morphism after taking ⊗ L Z o n b oth sides, whic h w e denote by r ′ . Now b y a theorem of T o taro [20, Theorem 3.1] (compare also [14, Remark 1 .2.21]), for cellular v arieties 26 JENS HOR NBOSTEL A ND V ALENTINA KIRITCHENKO the classical cycle class map c : C H ∗ ( X ) → H ∗ ( X ( C ) an ) (whic h is an isomorphism fo r cellular v arieties X ) deﬁne d using fundamental classe s and P oincar ´ e duality (see e. g. [16 , sec tion A.3]) factors as C H ∗ ( X ) → M U ∗ ( X ( C ) an ) ⊗ L Z ∼ = H ∗ ( X ( C ) an ), and the left arro w in this factorization is giv en b y ﬁrst taking an y resolution of singularities of the algebraic cycle a nd then applying ( C ) an . W e also hav e a morphism q : Ω ∗ ( X ) → C H ∗ ( X ) whic h induces a n isomorphism q ′ : Ω ∗ ( X ) ⊗ L Z → C H ∗ ( X ) by Levine-Morel [14, Theorem 1.2.19] and corresp onds to r esolution of singularities [14, Section 4.5.1]. Putting ev erything t ogether, w e obtain a commutativ e square Ω ∗ ( X ) ⊗ L Z ∼ = q ′   r ′ / / M U ∗ ( X ( C ) an ) ⊗ L Z f ′ ∼ =   C H ∗ ( X ) c ∼ = / / H ∗ ( X ( C ) an ) with the v ertical maps a nd c b eing isomorphisms, whic h ﬁnishes the pro of.  Reference s [1] I. N . Bernstein, I. M. G elf and, S. I. Gelf and , S chub ert c el ls, and the c ohomolo gy of the s p ac es G/P , Russian Math. Surveys 28 (197 3 ), no . 3 , 1–26. [2] A . Borel , Sur la c ohomolo gie des esp ac es ﬁbr´ es princip aux et des esp ac es homo g` enes de gr oup es de Lie c omp acts , Ann. of Math., 57 (19 53), 115–2 07 [3] P . Bressl er, S. E vens , The S chub ert c alculus, br aid r elations, and gener alize d c ohomolo gy, T rans . of the Amer. Math. So c., 317 , no. 2 (19 90), 7 99–8 11 [4] P . Bressler, S. Evens , Schub ert c alculus in c omplex c ob or dism , T rans. Amer. Ma th. So c. 3 31 (1992 ), no. 2, 799 – 813. [5] M. 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