Smooth duality in natural characteristic

Kohlhaase, Jan

doi:10.1016/j.aim.2017.06.038

Cited by 30 publications

(43 citation statements)

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“…Kohlhaase takes this a step further and shows that also S 2 (π V ) = 0 by checking that S 1 (T ) is bijective ([Koh17, p. 45]), which is also a key ingredient in his long verification that S 1 (π V ) ≃ πV . The latter is another of his main results, [Koh17, Thm. 5.13].…”

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confidence: 76%

“…Remark 11.2. For GL 2 (Q p ) Theorem 1.1 amounts to [Koh17, Thm. 5.11], which is one of the main results of that paper.…”

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confidence: 99%

“…Moreover, the smooth dual functor S 0 is not exact. In [Koh17] Kohlhaase extended S 0 to a δ-functor consisting of certain higher smooth duality functors…”

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confidence: 99%

“…and N, N ′ ⊳ K. As N varies we get a direct system, and passing to the limit gives the stable cohomology groups of M , cf [Koh17,. p. 18].…”

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A vanishing result for higher smooth duals

Sorensen

2019

Alg. Number Th.

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In this paper we prove a general vanishing result for Kohlhaase's higher smooth duality functors S i . If G is any unramified connected reductive p-adic group, K is a hyperspecial subgroup, and V is a Serre weight, we show that S i (ind G K V ) = 0 for i > dim(G/B) where B is a Borel subgroup. (Here and throughout the paper dim refers to the dimension over Qp.) This is due to Kohlhaase for GL 2 (Qp) in which case it has applications to the calculation of S i for supersingular representations. Our proof avoids explicit matrix computations by making use of Lazard theory, and we deduce our result from an analogous statement for graded algebras via a spectral sequence argument. The graded case essentially follows from Koszul duality between symmetric and exterior algebras.whether the bound dim(G/B) is sharp for π = ind G I 1 in the sense that S dim(G/B) (ind G I 1) is nonzeroeven in the case of GL 2 (Q p ).We now state the main result of this paper which we alluded to above. Let F/Q p be a finite extension, and G /F a connected reductive group with a Borel subgroup B. We assume that G is unramified 2 and choose a hyperspecial maximal compact subgroup K ⊂ G along with a finite-dimensional smooth representation K → GL(V ) with coefficients in some algebraic extension E/F p . Let ind G K V be the compact induction. Then we have the following vanishing result for its higher smooth duals S i .). What we actually prove is a slightly stronger result on the vanishing of the transition maps. Namely, if N ⊳ K has an Iwahori factorization and acts trivially on V , then the restriction mapas long as m is greater than some constant depending only on i and V, N (and an auxiliary filtration on N ). Here (−) ∨ denotes Pontryagin duality, and Λ(N ) = E[[N ]] is the completed group algebra over E. Note that the set of p m -powers N p m is a group for m large enough for any p-valuable group N , cf. [Sch11, Rem. 26.9].We have been unable to show that S dim(G/B) (ind G K V ) = 0 but we believe this should be true under a suitable regularity condition on the weight V , cf. section 12. We hope to address this in future work, and to say more about the action of Hecke operators on S i (ind G K V ) for all i. Let us add that the bound dim(G/B) is not sharp for all V . Indeed S i (ind G K 1) = 0 for all i > 0, cf. Remark 11.2. For GL 2 (Q p ) Theorem 1.1 amounts to [Koh17, Thm. 5.11], which is one of the main results of that paper. There is a small difference coming from the center Z ≃ Q × p though. He assumes V is an irreducible representation of K = GL 2 (Z p ) which factors through GL 2 (F p ), extends the central character of V to Z by sending p → 1, and considers ind G KZ V instead. The latter carries a natural Hecke operator T = T V whose cokernel π V is an irreducible supersingular representation. As V varies this gives all supersingular representations of GL 2 (Q p ) (with p acting trivially), cf. [BL94, Prop. 4] and [Bre03, Thm. 1.1]. The shortgives rise to a long exact sequence of higher smooth dualsSince S i (ind G KZ V ) = 0 for i ...

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confidence: 76%

“…Remark 11.2. For GL 2 (Q p ) Theorem 1.1 amounts to [Koh17, Thm. 5.11], which is one of the main results of that paper.…”

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confidence: 99%

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A vanishing result for higher smooth duals

Sorensen

2019

Alg. Number Th.

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“…It is well-known that this sets up a duality between the two categories. See [Sch95,ST02,Eme10] and the more recent [Koh17]. We summarize this below.…”

Section: Pontryagin Duality For Profinite Groupsmentioning

confidence: 99%

Koszul duality for Iwasawa algebras modulo 𝑝

Sorensen

2020

Represent. Theory

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In this article we establish a version of Koszul duality for filtered rings arising from p-adic Lie groups. Our precise setup is the following. We let G be a uniform pro-p group and consider its completed group algebra Ω = k[[G]] with coefficients in a finite field k of characteristic p. It is known that Ω carries a natural filtration and grΩ = S(g) where g is the (abelian) Lie algebra of G over k. One of our main results in this paper is that the Koszul dual grΩ ! = g ∨ can be promoted to an A∞-algebra in such a way that the derived category of pseudocompact Ω-modules D(Ω) becomes equivalent to the derived category of strictly unital A∞-modules D∞( g ∨ ). In the case where G is an abelian group we prove that the A∞-structure is trivial and deduce an equivalence between D(Ω) and the derived category of differential graded modules over g ∨ which generalizes a result of Schneider for Zp.but hereD is not the derived category, it is rather squeezed in between the homotopy category and the derived category. We should emphasize that the filtered rings U considered in [Pos93] are of a completely different nature than the Iwasawa algebras we will consider in this paper. Positselski requires U to be a non-homogeneous quadratic algebra U = T (V )/(P ) for some vector space V and ideal of relations (P ) in the tensor algebra generated by a subspace P ⊂ k ⊕ V ⊕ V ⊗2 with P ∩ (k ⊕ V ) = 0. He associates the quadratic algebra A = T (V )/(Q) where Q ⊂ V ⊗2 is the image of P under projection and decrees that the natural map A → grU is an isomorphism.Analogously we will consider completed group rings Ω = Ω(G) modulo p of certain compact p-adic Lie groups G for which grΩ ≃ S(g) where g = Lie(G) is the Lie algebra, and promote the Koszul dual algebra grΩ ! ≃ g ∨ to a so-called A ∞ -algebra in such a way that D(Ω) ∼ −→ D ∞ ( g ∨ ). The notation and terminology will be explained in more detail below.

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A quotient of the Lubin–Tate tower II

Ludwig

2020

Math. Ann.

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In this article we construct the quotient $$\mathcal {M}_\mathbf {1}/P(K)$$ M 1 / P ( K ) of the infinite-level Lubin–Tate space $$\mathcal {M}_\mathbf {1}$$ M 1 by the parabolic subgroup $$P(K) \subset \mathrm {GL} _n(K)$$ P ( K ) ⊂ GL n ( K ) of block form $$(n-1,1)$$ ( n - 1 , 1 ) as a perfectoid space, generalizing the results of Ludwig (Forum Math Sigma 5:e17, 41, 2017) to arbitrary n and $$K/{\mathbb {Q}} _p$$ K / Q p finite. For this we prove some perfectoidness results for certain Harris–Taylor Shimura varieties at infinite level. As an application of the quotient construction we show a vanishing theorem for Scholze’s candidate for the mod p Jacquet–Langlands and mod p local Langlands correspondence. An appendix by David Hansen gives a local proof of perfectoidness of $$\mathcal {M}_\mathbf {1}/P(K)$$ M 1 / P ( K ) when $$n=2$$ n = 2 , and shows that $$\mathcal {M}_\mathbf {1}/Q(K)$$ M 1 / Q ( K ) is not perfectoid for maximal parabolics Q not conjugate to P.

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Smooth duality in natural characteristic

Cited by 30 publications

References 18 publications

A vanishing result for higher smooth duals

A vanishing result for higher smooth duals

Koszul duality for Iwasawa algebras modulo 𝑝

A quotient of the Lubin–Tate tower II

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