Диссертация (1136188), страница 25
Текст из файла (страница 25)
we haveAð1Þ ¼x þ wðxÞ¼ q x; wðxÞ ¼ a11 a12 x þ wðxÞ þ ;xwðxÞwhere x ¼ F y1 ; wðy2 Þ , and qðx; yÞ is the power series uniquely determined by the equation F ðx; yÞ ¼ x þ y xyqðx; yÞ. In particular,sinceFx;wðxÞ¼ 0 by definition of thepower series wðxÞ, we have x þ wðxÞ xwðxÞq x; wðxÞ ¼ 0 which justifies the second equality. For the last equality, we used computation of the first few terms of F ðx; yÞ and wðxÞfrom [14], 2.5. Here a11 , a12 etc. denote the coe‰cients of the universal formal group law,that is,F ðx; yÞ ¼ x þ y þ a11 xy þ a12 xy 2 þ :The coe‰cients aij are the elements of the Lazard ring L , e.g. a11 ¼ ½P 1 ,a12 ¼ a21 ¼ ½P 1 2 P 2 (see [14], 2.5).
We also haveAðy1 Þ ¼ y2 Að1Þ þF ðx; y2 Þ y2¼ y2 q x; wðxÞ y2 qðx; y2 Þ þ 1 ¼ 1 þ a12 y1 y2 þ :xThe pull-back p : W ðX Þ ! W ðY Þ gives W ðY Þ the structure of an W ðX Þ-module. Recallthat by the projective bundle formula we have an isomorphism of W ðX Þ-modulesW ðY Þ G p W ðX Þ l xp W ðX Þ;where x ¼ c1 OE ð1Þ . Since the push-forward is a homomorphism of W ðX Þ-modules, it isenough to determine the action of p on 1Y and on x. The following result is a special caseof [21], Theorem 5.30, which gives an explicit formula for the push-forward p for vectorbundles of arbitrary rank.Bereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:46Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordism65Proposition 2.1 ([21], Theorem 5.30).
Let x1 and x2 be the Chern roots of E, that is,formal variables satisfying the conditions x1 þ x2 ¼ c1 ðEÞ and x1 x2 ¼ c2 ðEÞ. Then the pushforward acts on 1Y and x as follows:p ð1Y Þ ¼ ½Að1Þðx1 ; x2 Þ;p ðxÞ ¼ ½Aðy1 Þðx1 ; x2 Þ;where Að1Þ and Aðy1 Þ are the formal power series in two variables defined above.Since Að1Þ and Aðy1 Þ are symmetric in y1 and y2 , they can be written as power series iny1 þ y2 and y1 y2 .
Hence, the right-hand sides are power series in c1 ðEÞ and c2 ðEÞ and evenpolynomials (as all terms of degree greater than dim X will vanish by [15]). So the right-handsides indeed define elements in W ðX Þ.For the Chow ring and K0 , analogous statements were proved in [10], Propositions2.3 and 2.6, for certain morphisms Y ! X . Note that for both of these theories, the formula for p ðxÞ reduces to p ðxÞ ¼ 1 since the corresponding formal group laws do notcontain terms of degree greater than two. As Vishik showed (see [21], Theorem 5.35), hisformula is equivalent to Quillen’s formula [18] for complex cobordism, as also proved byShinder in the algebraic setting [19].If we identify W ðY Þ with the polynomial ring W ðX Þ½x= x 2 c1 ðEÞx þ c2 ðEÞ bythe projective bundle formula, we can reformulate Proposition 2.1 as follows: p f ðxÞ ¼ A f ðy1 Þ ðx1 ; x2 Þfor any polynomial f with coe‰cients in W ðX Þ (where f ðy1 Þ in the right-hand side is regarded as an element in W ðX Þ½½y1 ; y2 ).
In this form, Proposition 2.1 is consistent with theclassical formula for the push-forward in the case of Chow ring (cf. [16], Remark 3.5.4).Indeed, since the formal group law for Chow ring is additive, we haveAð1Þ ¼11y1y2þ¼ 0 and Aðy1 Þ ¼þ¼ 1:y1 y2 y2 y1y1 y2 y2 y1Definition 2.2. We define an W ðX Þ-linear operator Ap on W ðY Þ as follows. Wehave an isomorphismW ðX Þ½½y1 ; y2 = y1 þ y2 c1 ðEÞ; y1 y2 c2 ðEÞ G W ðY Þgiven by f ðy1 ; y2 Þ 7! f x; c1 ðEÞ xÞ . Then the operator A on W ðX Þ½½y1 ; y2 descendsto an operator Ap on W ðY Þ, which can be described using the above isomorphism asfollows: Ap : f x; c1 ðEÞ x ! A f ðy1 ; y2 Þ x; c1 ðEÞ x :We also define an W ðX Þ-linear endomorphism sp of W ðY Þ by the formulasp : f x; c1 ðEÞ x ¼ f c1 ðEÞ x; x :Bereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:4666Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordismThe operator Ap is well-defined since A preserves the idealy1 þ y2 c1 ðEÞ; y1 y2 c2 ðEÞ :Indeed, for any power series f ðy1 ; y2 Þ symmetric in y1 and y2 (in particular, fory1 þ y2 c1 ðEÞ and y1 y2 c2 ðEÞ) and any power series gðy1 ; y2 Þ, we have Að fgÞ ¼ fAðgÞ.The operator Ap decreases degrees by one, and its image is contained inp W ðX Þ H W ðY Þ;which can be identified using the above isomorphism for W ðX Þ with the subring ofsymmetric polynomials in y1 and y2 .
Proposition 2.1 tells us that the push-forwardp : W ðY Þ ! W ðX Þ is the composition of Ap with the isomorphism p W ðX Þ G W ðX Þ,which sends (underthe aboveidentifications) a symmetric polynomial f ðy1 ; y2 Þ into thepolynomial g c1 ðEÞ; c2 ðEÞ such that gðy1 þ y2 ; y1 y2 Þ ¼ f ðy1 ; y2 Þ. Hence, we get the following corollary, which we will use in the sequel.Corollary 2.3. The composition p p : W ðY Þ ! W ðY Þ is equal to the operator Ap :p p ¼ Ap :In the special case Y ¼ G=B and X ¼ G=Pi (and this is the main application we have,see Section 3.2), the topological analogue of this formula appeared in [3], Corollary–Definition 1.9, for a di¤erent definition of Ap .2.2.
Algebraic cobordism groups of cellular varieties. We start with the definitionof a cellular variety. The following definition is taken from [11], Example 1.9.1, althoughother authors sometimes consider slight variations.Definition 2.4. We say that a smooth variety X over k is ‘‘cellular’’ or ‘‘admits acellular decomposition’’ if X has a filtration j ¼ X1 H X0 H X1 H H Xn ¼ X by closedsubvarieties such that the Xi Xi1 are isomorphic to a disjoint union of a‰ne spaces A difor all i ¼ 0; . . . ; n, which are called the ‘‘cells’’ of X .Examples of cellular varieties include projective spaces and more general Grassmannians, and complete flag varieties G=B where G is a split reductive group and B is a Borelsubgroup.The following theorem is a corollary of [22], Corollary 2.9. We thank Sascha Vishikfor explaining to us how it can be deduced using the projective bundle formula.
Themain point is that for d ¼ dim X and i an arbitrary integer, one has for A ¼ W thatiW ðX Þ ¼: Wdi ðX Þ isisomorphic to Hom Aðd iÞ½2ðd iÞ; MðX Þ using that in the notation of loc. cit. Hom Aðd iÞ½2d 2i; MðX Þ is a direct summand indiLHom MðP di Þ; MðX Þ ¼ Adi ðP di X Þ ¼Adij ðX Þ;j¼0and it is not di‰cult to see that it corresponds to the summand with j ¼ 0.Bereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:46Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordism67Theorem 2.5. Let X be a variety with a cellular decomposition as in the definitionabove.
Then we have an isomorphism of graded abelian groups (and even of L-modules)W ðX Þ GLL½di iwhere the sum is taken over the cells of X . There is a basis in W ðX Þ given by resolutions ofcell closures (choose one resolution for each cell).The second statement of this theorem follows from the first one if we show that thecobordism classes of resolutions of the cell closures generate W ðX Þ. This can be deducedfrom the analogous statement for the Chow ring using [14], Theorem 1.2.19, Remark 4.5.6.For complex cobordism of topological complex cellular spaces, the corresponding theoremsimply follows from an iterated use of the long exact localization sequence which alwayssplits as everything in sight has MU -groups concentrated in even degrees only. Notealso that in the topological case, the Atiyah–Hirzebruch spectral sequence degenerates forthese spaces, which allows it to transport information from singular cohomology to complex cobordism.
As Morel points out, the analogous motivic spectral sequence invented byHopkins–Morel (unpublished) converging to algebraic cobordism does not in general degenerate even for the point SpecðkÞ, because the one converging to algebraic K-theorydoes not.We now turn to the ring structure.
First, we note that if k ¼ C, then there is a map ofgraded rings and even of L-algebras W ðX Þ ! MU 2 X ðCÞ an by universality of algebraiccobordism [14], Example 1.2.10. Using the geometric description of push-forwards both forW and MU and the fact that the above morphism respects push-forwards [14] as well as[15], we may describe this mapby mapping an element ½Y ! X of W ðX Þ to explicitlyananan 2½Y ðCÞ ! X ðCÞ in MU X ðCÞ . As both product structures are defined by takingcartesian products of the geometric representatives and pulling them back along the diagonal of X resp. X ðCÞ an , we see that this map does indeed preserve the graded L-algebrastructure.
Also, for any embedding k ! C we obtain a ring homomorphism from algebraiccobordism over k to algebraic cobordism over C.For the flag variety of GLn , this is an isomorphism by Theorem 2.6 below which isalso valid for MU , as both base change from k to C and complex topological realizationrespect products and first Chern classes.
For general cellular varieties, it is still an isomorphism. This is probably known to the experts, we provide a proof in the appendix.For some varieties X , the ring structure of W ðX Þ can be completely determined usingthe projective bundle formula [14], Section 1.1. This is the case for the variety of completeflags for G ¼ GLn (see Theorem 2.6 below) and also for Bott–Samelson resolutions ofSchubert cycles in a complete flag variety for any reductive group G (see Section 3).2.3. Borel presentation for the flag variety of GLn .
We now turn to the case of thecomplete flag variety X for G ¼ GLn ðkÞ. The points of X are identified with complete flagsin k n . A complete flag is a strictly increasing sequence of subspacesF ¼ ff0g ¼ F 0 H F 1 H F 2 H H F n ¼ k n gBereitgestellt von | Universitätsbibliothek Regensburg (Universitätsbibliothek Regensburg)Angemeldet | 172.16.1.226Heruntergeladen am | 01.03.12 17:4668Hornbostel and Kiritchenko, Schubert calculus for algebraic cobordismwith dimðF k Þ ¼ k.
The group G acts transitively on the set of all flags, and the stabilizer ofa point is isomorphic to a Borel subgroup B H G, which makes X ¼ G=B into a homogeneous space under G. By this definition, X has the structure of an algebraic variety.Note that over C, one may equivalently define the flag variety X to be the homogeneous space K=T under the maximal compact subgroup K H G, where T is a maximalcompact torus in K (that is, the product of several copies of S 1 ) [2].
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
