Differential graded algebras over some reductive groups

Cao, Jin

doi:10.4310/hha.2019.v21.n2.a9

Cited by 1 publication

(3 citation statements)

References 9 publications

(24 reference statements)

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“…Without loss of generality, we assume that M is a cell module. Using Remark 6.5 in [7], we obtain that:…”

Section: Dg Modules and Motives For An Elliptic Curvementioning

confidence: 96%

“…Our strategy for the CM case is extending the cycles algebra and representations over K rather than Q and using the isomorphism Res K/Q G m ⊗ K ∼ = G m,K × G m,K . Studying the cdga object in the category of GL 2 -representations is one of the main motivation for us to develop a theory of cdgas over some reductive group in [7], which generalize Kriz and May's theory of Adams cdgas. Compared with mixed Tate motives, the left things are constructing a reasonable elliptic type cdga and further connecting with DM gm (k, Q) (resp.…”

Section: Introductionmentioning

confidence: 99%

“…Then we can construct a functor from the derived category of dg E * ell -modules to Voevodsky's big category of motives DM(k, Q), whose restriction on the compact objects leads to a functor to DM gm (k, Q). In Section 8, we provide such construction and show that this functor induces an equivalence between the compact objects in the derived category of dg E * ell -modules and the idempotent complete rigid tensor triangulated subcategory generated by the motives of E. As a corollary of Theorem 8.3 in [7], if E * ell is cohomological connected, i.e., Conjecture 2.6 and Conjecture 8.16 hold for E, then there exists an abelian category of mixed motives for E. In the last section, we show that the embedding of the triangulated category of mixed Tate motives into motives for E can be understood as a map between the derived categories of dg modules, which is induced by the inclusion of a sub-algebra N of E * ell to E * ell itself.…”

Section: Introductionmentioning

confidence: 99%

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Motives for an elliptic curve

Cao

2018

Math. Ann.

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In this paper we describe the rigid tensor triangulated subcategory of Voevodsky's triangulated category of motives generated by the motive of an elliptic curve as a derived category of dg modules over a commutative differential graded algebra in the category of representations over some reductive group.1 For the CM case, we only consider the complex multiplication is defined over k.Furthermore, the piecewhere V (n,0) ⊗ V ⊗i ⊗V ⊗j are pairwise non-isomorphic irreducible representations over a K-algebra End Res K/Q Gm⊗K ((F K ) ⊗n ) and V i,j are pairwise non-isomorphic irreducible representation over Res K/Q G m ⊗ K = T K . For simplicity, we delete V i,j and one may think that both V andV are endowed with the G m -action. In fact, End Res K/Q Gm⊗K ((F K ) ⊗n ) is a special case defined in the Section 3.9 of [1], which is called B n,K . Ancona's main result -Theorem 4.1 in [1] implies that the decomposition like Lemma 2.1 is holding for the CM elliptic motives:3 Here λ t is the transpose (or conjugate) of λ, which is defined by interchanging rows and columns in the Young diagram associated to λ. 4 We view cn as an idempotent in End((F K ) ⊗n ), which lies in End Res K/Q Gm⊗K ((F K ) ⊗n ).

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“…Without loss of generality, we assume that M is a cell module. Using Remark 6.5 in [7], we obtain that:…”

Section: Dg Modules and Motives For An Elliptic Curvementioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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