Диссертация (1136188), страница 24

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A more algebraic proof using the Chevalley–Pieri formula was givenby Bernstein–Gelfand–Gelfand ([1], Theorem 4.1, see also Section 4 for a short overview).Demazure gave a more geometric proof by identifying the divided di¤erence operators withthe push-forward morphism for certain Chow rings ([10], Theorem 4.1, see also Section 3).At first glance, it seems that the former proof is easier to extend to the algebraic cobordism. Indeed, we were able to extend the main ingredient of this proof, namely, the algebraic Chevalley–Pieri formula (see Proposition 4.3). However, the rest of the Bernstein–Gelfand–Gelfand argument fails for cobordism (see Section 4 for more details) while themore geometric argument of Demazure can be extended to cobordism with some extrawork.

For the complex cobordism ring, this was done by Bressler and Evens [3], [4]. Todescribe the push-forward morphism, they used results from homotopy theory, which areBereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:4662Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordismnot (yet) applicable to algebraic cobordism. In our article, we also follow Demazure’sapproach. A key ingredient for extending this approach to algebraic cobordism is a formula for the push-forward in algebraic cobordism for projective line fibrations due toVishik (see Proposition 2.1).

In general, push-forwards (sometimes also called ‘‘transfers’’or ‘‘Gysin homomorphisms’’) for algebraic cobordism are considerably more intricate thanthe ones for Chow groups. Consequently, their computation, which applies to anyorientable cohomology theory, is more complicated.Using the ring isomorphism W ðX Þ F MU 2 X ðCÞ an for cellular varieties, it seemspossible to deduce our Theorem 1.2 from the results of Bressler–Evens [3], [4] on complexcobordism (the main task would be to compare our algebraically defined operators Ai withtheirs). We will not exploit this approach. Instead, all our proofs are purely algebraic oralgebro-geometric. Conversely, we note that all our proofs concerning the algebraic cobordism ring of the flag variety (such as the proof of Proposition 4.3) may be easily translated to proofs for the analogue statements concerning the complex cobordism ring.The article [4] does not contain any computations.

It would be interesting to dosome computation using their algorithm and then compare them with our approach,which we consider to be the easier one due to our explicit formula for the product of aBott–Samelson class with the first Chern class (see formula 5.1) based on our algebraicChevalley–Pieri formula.

(Note also that the notations of [4] are essentially consistentwith [1], but not always with [16]. We rather stick to the former than to the latter.)This paper is organized as follows. In the next section, we give some further background on algebraic cobordism, in particular the formula for the push-forward mentionedabove.

In the case of the flag variety for GLn , we describe the multiplicative structure of itsalgebraic cobordism ring. In the third section, we recall the definition of Bott–Samelsonresolutions and then express the classes of Bott–Samelson resolutions as polynomials withcoe‰cients in the Lazard ring. Section 4 contains an algebraic Chevalley–Pieri formula anda short discussion of why the proof of [1] for singular cohomology does not carry over toalgebraic cobordism. The final section contains an algorithm for computing the products ofBott–Samelson classes in terms of other Bott–Samelson classes as well as some examplesand explicit computations.Our main results are valid for the flag variety of an arbitrary reductive group G, butit can be made more explicit in the case G ¼ GLn using the Borel presentation given byTheorem 2.6.

So we will use the flag variety for GLn as the main illustrating example whenever possible. One might conjecture that the algebraic cobordism rings of flag varieties withrespect to other reductive groups G also allow a Borel presentation as polynomial ringsover L in certain first Chern classes modulo the polynomials fixed by the appropriateWeyl groups (at least when passing to rational coe‰cients), because the correspondingstatement is valid for singular cohomology resp.

Chow groups (compare [2] resp. [9]).After most of our preprint was finished, we learned that Calmès, Petrov and Zainoulline are also working on Schubert calculus for algebraic cobordism. It will be interesting tocompare their results and proofs to ours (their preprint is now available, see [7]).We are grateful to Paul Bressler and Nicolas Perrin for useful discussions and toMichel Brion and the referee for valuable comments on earlier versions of this article.Bereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:46Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordism632.

Algebraic cobordism groups, push-forwards and cellular varietiesWe briefly recall the geometric definition of algebraic cobordism [15] and some of itsbasic properties as established in [14]. For more details see [14], [15]. Recall that (up tosign) any element in the algebraic cobordism group W n ðX Þ for a scheme X (separated, offinite type over k) may be represented by a projective morphism Y ! X with Y smoothand n ¼ dimðX Þ dimðY Þ, the relations being the ‘‘double point relations’’ established in[15]. In particular, W ðX Þ only lives in degrees e dim X , which we will use several timesthroughout the paper.

Similar to the Chow ring CH , algebraic cobordism W is a functoron the category of smooth varieties over k, covariant for projective and contravariant forsmooth and more generally lci morphisms, which allows a theory of Chern classes. However, the map from the Picard group of a smooth variety X to W 1 ðX Þ given by the firstChern class is neither a bijection nor a homomorphism any more (unlike the correspondingmap in the Chow ring case).

Its failure of being a group homomorphism is encoded in aformal group law that can be constructed from W . More precisely, any algebraic orientablecohomology theory allows by definition a calculus of Chern classes, and consequently theconstruction of a formal group law. A formal group law is a formal power series F ðx; yÞ intwo variables such that for any two line bundles L1 and L2 we have the following identityrelating their first Chern classes:c1 ðL1 n L2 Þ ¼ F c1 ðL1 Þ; c1 ðL2 Þ :E.g. the formal group law for CH is additive, that is, F ðx þ yÞ ¼ x þ y. Algebraic cobordism is the universal one among the algebraic orientable cohomology theories. In what follows, F ðx; yÞ will always denote the universal formal group law corresponding to algebraiccobordism unless stated otherwise.In this and in many other ways—as the computations below will illustrate—algebraiccobordism is a refinement of Chow ring, and one has a natural isomorphism of functorsW ðÞ nL Z G CH ðÞ (see [14] where all these results are proved).

Here and in the sequel,L denotes the Lazard ring, which classifies one-dimensional commutative formal grouplaws and is isomorphic to the graded polynomial ring Z½a1 ; a2 ; . . . in countably manyvariables [12], where we put ai in degree i. When considering polynomials pðx1 ; . . . ; xn Þover L with degðxi Þ ¼ 1, we will distinguish the (total) degree and the polynomial degree ofpðx1 ; . . . ; xn Þ.Note that the Lazard ring is isomorphic to the algebraic (as well as complex) cobordism ring of a point. In particular, its elements can be represented by the cobordism classesof smooth varieties. In what follows, we use this geometric interpretation.We will also use repeatedly the projective bundle formula, which can be found e.g. in[14], Section 1.1, and [16], 3.5.2.2.1.

A formula for the push-forward. Let X be a smooth algebraic variety, andE ! X a vector bundle of rank two on X . Consider the projective line fibration Y ¼ PðEÞdefined as the variety of all lines in E. We have a natural projection p : Y ! X which is projective and hence induces a push-forward (or transfer, sometimes also called Gysin map)p : W ðY Þ ! W ðX Þ. We now state a formula for this push-forward. Note that this formulaBereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:4664Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordismis true not only for algebraic cobordism but for any orientable cohomology theory, as theproofs remain true in this more general case.Consider the ring of formal power series in two variables y1 and y2 with coe‰cientsin W ðX Þ.

Define the operator A on this ring by the formulaf;Að f Þ ¼ ð1 þ sÞ F y1 ; wðy2 Þwhere ½sð f Þðy1 ; y2 Þ :¼ f ðy2 ; y1 Þ. Here F is the universal formal group law (or more generally, the one of the orientable cohomology theory one considers) and w is the inverse forthe formal group law F , that is, w is uniquely determined by the equation F x; wðxÞ ¼ 0(we use the notation from [14], 2.5). The operator A is an analog of the divided di¤erenceoperator introduced in [1], [10].

In the case of Chow rings, our definition coincides with theclassical divided di¤erence operator, since the formal group law for Chow rings is additive,that is, F ðx; yÞ ¼ x þ y and wðxÞ ¼ x. Though Að f Þ is defined as a fraction, it is easy towrite it as aformal power series as well (see Section 5). Such a power series is unique sinceF y1 ; wðy2 Þ ¼ y1 y2 þ is clearly not a zero divisor. E.g.

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