Rost motives, affine varieties, and classifying spaces

Petrov, Viktor; Semenov, Nikita

doi:10.1112/jlms.12040

Cited by 7 publications

(7 citation statements)

References 47 publications

(84 reference statements)

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“…If is a standard generic torsor (see [PS17, § 3]), then for every free theory (cf. [PS17, Lemma 3.1]). Therefore, in this case .…”

Section: General Hopf-theoretic Statementsmentioning

confidence: 99%

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Hopf-theoretic approach to motives of twisted flag varieties

Petrov¹,

Semenov²

2021

Compositio Math.

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Let $G$ be a split semisimple algebraic group over a field and let $A^*$ be an oriented cohomology theory in the Levine–Morel sense. We provide a uniform approach to the $A^*$ -motives of geometrically cellular smooth projective $G$ -varieties based on the Hopf algebra structure of $A^*(G)$ . Using this approach, we provide various applications to the structure of motives of twisted flag varieties.

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“…If is a standard generic torsor (see [PS17, § 3]), then for every free theory (cf. [PS17, Lemma 3.1]). Therefore, in this case .…”

Section: General Hopf-theoretic Statementsmentioning

confidence: 99%

“…Due to nilpotency results [CNZ21, § 5] (cf. [PS17, NPSZ18, Theorem 5.5]) one can lift motivic decompositions of the -motives of twisted flag varieties to a motivic decomposition of the -equivariant -motive of .…”

Section: Applications To Other Cohomology Theoriesmentioning

confidence: 99%

Hopf-theoretic approach to motives of twisted flag varieties

Petrov¹,

Semenov²

2021

Compositio Math.

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“…Conversely, if a generic flag variety

X = E / P

is generically split, then for any field

L / F

and any

G

‐torsor

E^{'}

over

L

, the flag variety

E^{'} / P

is also generically split. It follows by [, § 5] that

P

is special. Below is a direct proof of both implications due to Merkurjev.…”

Section: Chern Subring For Generic Generically Split Flag Varietiesmentioning

confidence: 99%

On generically split generic flag varieties

Karpenko

2018

Bull. London Math. Soc.

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Let G be a split semisimple algebraic group over an arbitrary field F , let E be a G-torsor over F , and let P be a parabolic subgroup of G. The quotient variety X := E/P , known as a flag variety, is generically split, if the parabolic subgroup P is special. It is generic, provided that the G-torsor E over F is a standard generic G k -torsor for a subfield k ⊂ F and a split semisimple algebraic group G k over k with (G k )F = G.For any generically split generic flag variety X, we show that the Chow ring CH X is generated by Chern classes (of vector bundles over X). This implies that the topological filtration on the Grothendieck ring of X coincides with the computable gamma filtration. The results were already known in some cases including the case where P is a Borel subgroup.We also provide a complete classification of generically split generic flag varieties and, equivalently, of special parabolic subgroups for split simple groups.Contents 497 any P -torsor over any extension field of the base field is trivial. This is equivalent to the fact that the generic flag variety X := E/P is generically split, see Lemma 7.1.In the present paper, we show that a generically split generic flag variety X = E/P has the same property as in the above particular case with P = B: the topological filtration on the Grothendieck ring K(X) coincides with the gamma filtration. This confirms that the introduced by Grothendieck gamma filtration is a good approximation of the topological filtration. And not only in the well-known sense that they both coincide after tensoring by Q.The reason for this coincidence (with Z-coefficients) is our main result (Theorem 7.3) saying that the ring CH X coincides with its Chern subring. In its turn, Theorem 7.3 is a consequence of the similar statement for the Chow ring of the classifying space of P , see Proposition 5.5.The assumption that P is special is essential. For instance, taking for G a spinor group Spin(n), we may take a parabolic subgroup P ⊂ G such that the variety X = E/P is a projective quadric. Then the gamma filtration on K(X) is different from the topological filtration provided that n 9. Indeed, as shown in [9], the component of codimension 2 of the graded ring associated with the gamma filtration contains an element of order 2; on the other hand, by [8, Theorem 6.1 and § 3.1], the component of codimension 2 of the graded ring associated with the topological filtration is torsion-free for such n.For an overview of sections of the paper we refer to the table of contents: the titles of sections are self-explaining. Note that the paper ends with a classification of special parabolic subgroups (or, equivalently, of generically split generic flag varieties).

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“…The invariant subring h T (X) W is related to the G-equivariant cohomology via the forgetful map h G (X) → h T (X) W which has been recently studied in the context of motivic decompositions of (generic) flag varieties [CM06], [PS16], [NPSZ], [CNZ].…”

Section: Introductionmentioning

confidence: 99%

“…As for the latter, consider the category of G-equivariant h-motives (see [GV18,§2] and [PS16] for definitions and basic properties). Let M G be its full additive subcategory generated by motives of G/P for all standard parabolic subgroups P of G. The morphisms in M G are given by classes in h G (X) and their composition is given by the so called correspondence product.…”

Section: Introductionmentioning

confidence: 99%

Oriented cohomology sheaves on double moment graphs

Devyatov¹,

Lanini²,

Zainoulline³

2017

Preprint

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In the present paper we extend the theory of sheaves on moment graphs due to Braden-MacPherson and Fiebig to the context of an arbitrary oriented equivariant cohomology h (e.g. to algebraic cobordism). We introduce and investigate structure h-sheaves on double moment graphs to describe equivariant oriented cohomology of products of flag varieties. We show that in the case of a total flag variety X of Dynkin type A the space of global sections of the double structure h-sheaf also describes the endomorphism ring of the equivariant h-motive of X. Contents 1. Introduction 1 2. Preliminaries 3 3. Equivariant oriented cohomology of the products 7 4. Double moment graphs 11 5. Closed double moment graphs 15 6. Global sections of structure sheaves 20 7. Correspondence product and equivariant motives 25 8. Appendix (type B) 28 References 36

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Rost motives, affine varieties, and classifying spaces

Cited by 7 publications

References 47 publications

Hopf-theoretic approach to motives of twisted flag varieties

Hopf-theoretic approach to motives of twisted flag varieties

On generically split generic flag varieties

Oriented cohomology sheaves on double moment graphs

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