Диссертация (1136188), страница 45
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To do this, we only have to observe from the projection formula forthe map pG/B : G/B → pt that pG/B ∗ ρ · p∗G/B (x) = pG/B ∗ (ρ) · x = x, whereρ ∈ M U ∗ (G/B) is the class of a point. This gives a splitting of the map p∗G/B andhence a splitting of 1 ⊗ id = p∗G/B ⊗ id.To prove the surjectivity, we follow the proof of the analogous result for theChow groups in [39, Thm. 1.3].
Since the Atiyah-Hirzebruch spectral sequence degenerates over the rationals and since the analogue of our proposition is known forthe singular cohomology groups by [39, Thm. 1.3(2)], we see that the propositionholds over the rationals (cf. [22, Thm. 8.8]).We now let α : M U ∗ (G/B) → L be the map α(y) = pG/B ∗ (ρ · y) and setβ = α ⊗ id : M U ∗ (G/B) ⊗L M U ∗ (BG ) → M U ∗ (BG ). Set f ∗ = p∗G/B ⊗ id andf∗ = pG/B ∗ ⊗ id. The projection formula as above implies that f ∗ βf ∗ (x) = f ∗ (x)for all x ∈ M U ∗ (BG ). Thus f ∗ β(y) = y for all y in the image of 1 ⊗ id. We∼=identify S −→ M U ∗ (BT ) with M U ∗ (G/B) ⊗L M U ∗ (BG ) over R as in Lemma 4.3and consider the commutative diagramSβf∗// M U ∗ (BG )gSQ// Sgβ// M U ∗ (BG )Qf∗(4.7)// SQwhere g : S → SQ is the natural change of coefficients map.WLet us fix an element x ∈ S W .
Since g S W ⊆ (SQ ) , it follows from our resultover rationals thatg (f ∗ β(x)) = f ∗ β (g(x)) = g(x).That is, g (x − f ∗ β(x)) = 0. Since S is torsion-free, we must have x = f ∗ β(x) onthe top row of (4.7). Since x is an arbitrary element of S W , we conclude thatS W ⊆ Image(f ∗ ) over R. Combining Lemma 4.3 and Proposition 4.4, we immediately get:Corollary 4.5.
Let X be a smooth scheme with an action of G such that HT∗ (X, R)is torsion-free. Then, the map'∗Ψtop→ M UT∗ (X)X : S ⊗S W M UG (X) −is an isomorphism of R-algebras. In particular, M UT∗ (G/B) is isomorphic toS ⊗S W S.This extends to cobordism a well-known result for cohomology (see, e.g., [4,Prop. 1(iii)] or [16, Prop.
2.2(ii)]).404VALENTINA KIRITCHENKO, AMALENDU KRISHNA5. Equivariant algebraic cobordism of G/BLet k be any field of characteristic zero. Let G be a connected and reductivegroup over k. We assume that G has a split maximal torus T contained in a Borelsubgroup B. In this section, we prove our main result on the Borel presentation ofΩ∗T (G/B). We demonstrate how this presentation can be used to define Demazureoperators and establish Schubert calculus in Ω∗T (G/B).5.1. Borel presentation of Ω∗T (G/B)Using the natural restriction map rTG : Ω∗G (G/B) → Ω∗T (G/B) ([22, Subsect.
4.1])and the isomorphisms ([22, Props. 5.4, 8.1])S∼= Ω∗T (k) ∼= Ω∗B (k) ∼= Ω∗G (G/B),∗we get the characteristic ring homomorphism c eqG/B : S → ΩT (G/B). On theother hand, the structure map G/B → Spec(k) gives another L-algebra map S →Ω∗T (G/B).Theorem 5.1. The natural map of S-algebras∗ΨalgG/B : S ⊗S W S → ΩT (G/B),eqΨalgG/B (a ⊗ b) = a · cG/B (b)is an isomorphism over R.Proof. Since G contains a split maximal torus, it is a split reductive group over kand hence it is uniquely described by a root system.
In particular, there is a splitσreductive group GQ over the prime field Q ,→ k such that G ∼= GQ ⊗Q k. It follows∼∼==∗∗∗from [23, Thm. 4.7] that σ : ΩTQ (GQ /BQ ) −→ ΩT (G/B). Since Ω∗TQ (Q) −→ Ω∗T (k),it is enough to prove the theorem when k = Q.
By using the same argument forthe inclusion Q ,→ C, we are reduced to proving the theorem when the base fieldis C.In this case, we observe that ceqG/B is simply the change of group homomorphism,π∗∗and hence it is the algebraic analogue of the restriction map M UG(G/B) −−X→M UT∗ (G/B) in (4.2).
Furthermore, the map S → Ω∗T (G/B) is the algebraicanalogue of the map p∗T,G/B in (4.2). Using Corollary 4.5, we get a diagramΨalgG/B// Ω∗ (G/B)S ⊗S W SNTNNNNNNΦtopNNNG/BΨtopN''G/BM UT∗ (G/B)(5.1)which commutes by the above comparison of the various algebraic and topological maps. The right vertical map is an isomorphism by Corollary 3.8 and thediagonal map is an isomorphism by Corollary 4.5. We conclude that ΨalgG/B is anisomorphism too. EQUIVARIANT COBORDISM OF FLAG AND SYMMETRIC VARIETIES405The following consequence of Theorem 5.1 improves [23, Thm. 8.1], which wasproven with the rational coefficients. It also improves the computation of the nonequivariant cobordism ring of G/B in [8, Thm.
13.12] (see also [8, Remark 2.21]),where a presentation of Ω∗ (G/B) was obtained in terms of the completion of Swith respect to its augmentation ideal.Corollary 5.2. There is an R-algebra isomorphism∼=S ⊗S W L −→ Ω∗ (G/B).Proof. This follows immediately from Theorem 5.1 and [23, Thm. 3.4].The following result extends Proposition 4.4 to the algebraic setting and givesthe isomorphism S(G) ' M U ∗ (BG ) over R for all reductive groups. The isomorphism S(G) ' M U ∗ (BG ) over the integers was earlier proven in [10] for classicalgroups.Proposition 5.3. The natural map S(G) → S W is an isomorphism of R-algebras.Proof. We follow the proof of an analogous result for the Chow groups in [39,Thm. 1.3].
For every good pair (Vj , Uj ) approximating BT = BB , consider thefiber bundleiπG/B →− Uj /B −→ Uj /G.By Lemma 4.2, there exists an element a ∈ Ωd (BT ) such that i∗ (a) is the classof a point in Ωd (G/B). We now show that the push-forward map π∗ : Ω∗ (BT ) →Ω∗−d (BG ) sends a to an invertible element of Ω0 (BG ) (for Chow rings, this isstraightforward since π ∗ (a) = 1). The projection formula yieldsπ∗ (aπ ∗ ([pt])) = π∗ (a)[pt](here [pt] is the class of a point in Ω∗ (BG )). Since π ∗ ([pt]) = i∗ (1), we have thataπ ∗ ([pt]) = i∗ (i∗ (a)) is the class of a point in Ω∗ (BT ). Hence, [pt] = π∗ (a)[pt],that is, π∗ (a) is equal to 1 modulo F 1 Ω∗ (BG ) by [29, Thm. 1.2.19, Remark 4.5.6].Put b = 1 − π∗ (a). Since b ∈ F 1 Ω∗ (BG ), the power series 1 + b + b2 + · · · gives awell-defined element of S(G).
Indeed, for every variety Uj /G, this series terminates(since b is nilpotent in Ω∗ (Uj /G)). It follows that π∗ (a) is invertible.The rest of the proof is completely analogous to the proof [39, Thm. 1.3]. Weprovide the details below for the reader’s convenience.We now show the injectivity of π ∗ . The projection formula impliesπ∗ (π ∗ (x)a) = xπ∗ (a)for all x ∈ M U ∗ (BT ). In particular, π ∗ (x) = 0 implies x = 0, since π∗ (a) isinvertible.To prove the surjectivity of π ∗ , we use that the analogue of our proposition holdsover the rationals (cf.
[22, Thm. 8.6]). Next, observe that for any x ∈ M U ∗ (BG )and y = π ∗ (x), we have π ∗ (π∗ (ay)) = π ∗ (π∗ (a)x) = π ∗ (π∗ (a))y. Since M U ∗ (BT )is torsion-free, it follows thatπ ∗ (π∗ (ay)) = π ∗ (π∗ (a))yfor all y ∈ M U ∗ (BT )W , that is, y is the image of π∗ (ay)π∗ (a)−1 under π ∗ .406VALENTINA KIRITCHENKO, AMALENDU KRISHNA5.2. Divided difference operatorsVarious definitions of generalized divided difference (or Demazure) operators weregiven in [3] for complex cobordism and in [18], [8] for algebraic cobordism in orderto establish Schubert calculus in M U ∗ (G/B) and Ω∗ (G/B). Corollary 5.2 allowsus to compare these definitions. We also outline Schubert calculus in equivariantcobordism using Theorem 5.1.Let xχ ∈ S denote the first T -equivariant Chern class cT1 (Lχ ) of the T -equivariant line bundle Lχ on Spec(k) associated with the character χ of T . Recall thatthe isomorphism S ' Ω∗T (k) sends χ to xχ .
The Weyl group WG acts on S: anelement w ∈ WG sends xχ to xwχ . For each simple root α, define an L-linearoperator ∂α on the ring S:f∂α : f 7→ (1 + sα ),x−αwhere sα ∈ W is the reflection corresponding to the root α. One can show that∂α is indeed well-defined using arguments of [18, Sect. 5] (in [18] the ring of allpower series is considered, but it is easy to check that ∂α (f ) is homogeneous if fis homogeneous). It is also easy to check that ∂α is S W -linear. In particular, ∂αdescends to S ⊗S W L.The comparison result below follows directly from definitions and Corollary 5.2.(1) Under the isomorphism M U ∗ (BT ) ' S, the operator Cα considered in [3,Prop. 3] coincides with the operator ∂α .(2) Under the isomorphism of S ⊗S W L ' Ω∗ (G/B), the operator ∂α descendsto the operator Aα defined in [18, Sect. 3].(3) The operator ∂α coincides with the restriction of the operator Cα given in[8, Def.
3.11] from the ring of all power series to S.Note that most of the operators considered above also have geometric meaning(see [3], [18], [8] for details). In particular, they were used to compute the Bott–Samelson classes in cobordism.We now define an equivariant generalized Demazure operator ∂αT on S ⊗S W S:∂αT : f ⊗ g 7→ f ⊗ ∂α (g).It is well-defined since ∂α is S W -linear. It follows immediately from Theorem 5.1that ∂αT defines an S-linear operator on Ω∗T (G/B). Similarly to the ordinarycobordism, these operators can be used to compute the equivariant Bott–Samelsonclasses. We outline the main steps but omit those details that are the same as forthe ordinary cobordism. We use notation and definitions of [18].Recall that to each sequence I = {α1 , .
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