Integral motives, relative Krull–Schmidt principle, and Maranda-type theorems

Semenov, Nikita; Zhykhovich, Maksim

doi:10.1007/s00208-014-1158-8

Cited by 7 publications

(5 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Only very few facts are known concerning the subcategory [E/B] of Chow motives with integer coefficients. An integer version of the motive R was introduced and discussed in [31]; in [5], [10], [27] it was shown that [E/B] is not Krull-Schmidt (the uniqueness of a direct sum decomposition fails).…”

Section: Introductionmentioning

confidence: 99%

Motivic decompositions of twisted flag varieties and representations of Hecke-type algebras

Neshitov

Petrov

Semenov

et al. 2018

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

Let G be a split semisimple linear algebraic group over a field k 0 . Let E be a G-torsor over a field extension k of k 0 . Let h be an algebraic oriented cohomology theory in the sense of Levine-Morel. Consider a twisted form E/B of the variety of Borel subgroups G/B over k.Following the Kostant-Kumar results on equivariant cohomology of flag varieties we establish an isomorphism between the Grothendieck groups of the h-motivic subcategory generated by E/B and the category of finitely generated projective modules of certain Hecke-type algebra H which depends on the root datum of G, on the torsor E and on the formal group law of the theory h.In particular, taking h to be the Chow groups with finite coefficients Fp and E to be a generic G-torsor we prove that all indecomposable submodules of an affine nil-Hecke algebra H of G with coefficients in Fp are isomorphic to each other and correspond to the (non-graded) generalized Rost-Voevodsky motive for (G, p).

show abstract

Section: Introductionmentioning

confidence: 99%

Motivic decompositions of twisted flag varieties and representations of Hecke-type algebras

Neshitov

Petrov

Semenov

et al. 2018

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Motivic decompositions which we considered in this paper were usually with modulo coefficients. Nevertheless, there is a standard technique to lift motivic isomorphisms and motivic decompositions from - to - or -coefficients (see, for example, [SZ15]).…”

Section: Applications To Other Cohomology Theoriesmentioning

confidence: 99%

Hopf-theoretic approach to motives of twisted flag varieties

Petrov¹,

Semenov²

2021

Compositio Math.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…Thus,

{CH}_{G} false(G / P^{'} false) \to {CH}_{T} false(G / P^{'} false)

also splits. Remark Since we are mainly interested in motives with

double-struckZ

‐ or

Z / p

‐coefficients we could avoid the construction of a splitting of homomorphism using . Namely, it follows from [, Theorem 4.3] that all decompositions of ordinary motives which we consider below hold with

Z_{p}

‐coefficients in the same way as with

Z / p

‐coefficients.…”

Section: G‐equivariant Motivesmentioning

confidence: 99%

“…Remark Since we are mainly interested in motives with

double-struckZ

‐ or

Z / p

Z_{p}

‐coefficients in the same way as with

Z / p

‐coefficients. But since

Z_{p}

is flat over

double-struckZ

, homomorphism remains injective after tensoring with

Z_{p}

and this is its only property needed in the proofs below.…”

Section: G‐equivariant Motivesmentioning

confidence: 99%

Rost motives, affine varieties, and classifying spaces

Petrov

Semenov

2017

J. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

In the present article we investigate ordinary and equivariant Rost motives. We provide an equivariant motivic decomposition of the variety X of full flags of a split semisimple algebraic group over a smooth base scheme, study torsion subgroup of the Chow group of twisted forms of X, define some equivariant Rost motives over a field and ordinary Rost motives over a general base scheme, and relate equivariant Rost motives with classifying spaces of some algebraic groups.Comment: 21 page

show abstract

Integral motives, relative Krull–Schmidt principle, and Maranda-type theorems

Cited by 7 publications

References 24 publications

Motivic decompositions of twisted flag varieties and representations of Hecke-type algebras

Motivic decompositions of twisted flag varieties and representations of Hecke-type algebras

Hopf-theoretic approach to motives of twisted flag varieties

Rost motives, affine varieties, and classifying spaces

Contact Info

Product

Resources

About