A Variational Tate Conjecture in crystalline cohomology

Morrow, Matthew

doi:10.4171/jems/907

Cited by 24 publications

(34 citation statements)

References 38 publications

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“…Let us first assume that is algebraically closed. By [Mor14, Proposition 3.2] the inclusions and induce an isomorphism and a surjection The claim follows. In general, we argue as in [Mor14, Theorem 1.4]: the claim for algebraically closed shows that lifts to after making the base change .…”

Section: Morrow’s Variational Tate Conjecture For Divisorsmentioning

confidence: 90%

“…By [Mor14, Proposition 3.2] the inclusions and induce an isomorphism and a surjection The claim follows. In general, we argue as in [Mor14, Theorem 1.4]: the claim for algebraically closed shows that lifts to after making the base change . Let denote the strict Henselisation of inside ; by Néron–Popescu desingularisation there exists some smooth local -algebra such that lifts to after making the base change .…”

Section: Morrow’s Variational Tate Conjecture For Divisorsmentioning

confidence: 90%

“…The goal of this section is to offer a simpler proof of a special case of [Mor14, Theorem 3.5] for smooth and proper schemes over the power series ring . This result essentially states that a line bundle on the special fibre of lifts if and only if its first Chern class in does, and should be viewed as an equicharacteristic analogue of Berthelot and Ogus’s theorem [BO83, Theorem 3.8] stating that a line bundle on the special fibre of a smooth proper scheme over a DVR in mixed characteristic lifts if and only if its Chern class lies in the first piece of the Hodge filtration.…”

Section: Morrow’s Variational Tate Conjecture For Divisorsmentioning

confidence: 99%

“…It seems entirely plausible that the methods of this section can be easily adapted to give a proof of [Mor14, Theorem 3.5] in general, that is, over rather than just .…”

Section: Morrow’s Variational Tate Conjecture For Divisorsmentioning

confidence: 99%

“…Applying the principle that deformation problems in characteristic should be studied using -adic cohomology, Morrow in [Mor14] formulated a crystalline variational Tate conjecture for smooth and proper families of varieties in characteristic , and proved the conjecture for divisors, at least when is projective. The key step of the proof is a version of this result over , with a perfect field of characteristic .…”

Section: Introductionmentioning

confidence: 99%

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Cycle classes in overconvergent rigid cohomology and a semistable Lefschetz theorem

Lazda¹,

Pál²

2019

Compositio Math.

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In this article we prove a semistable version of the variational Tate conjecture for divisors in crystalline cohomology, showing that for k a perfect field of characteristic p, a rational (logarithmic) line bundle on the special fibre of a semistable scheme over k t lifts to the total space if and only if its first Chern class does.The proof is elementary, using standard properties of the logarithmic de Rham-Witt complex. As a corollary, we deduce similar algebraicity lifting results for cohomology classes on varieties over global function fields. (1, 1) Morrow's proof of the local statement uses some fairly heavy machinery from motivic homotopy theory, in particular a 'continuity' result for topological cyclic homology. In this article we provide a new proof of the local crystalline variational Tate conjecture for divisors, at least over the base S = Spec (k t ), which only uses some fairly basic properties of the de Rham-Witt complex, and is close in spirit to the approach taken in [Mor15]. The point of giving this proof is that it adapts essentially verbatim to the case of semistable reduction, once the corresponding basic properties of the logarithmic de Rham-Witt complex are in place. Cycle classes and LefschetzSo let X be a semistable, projective scheme over k t , with special fibre X 0 and generic fibre X. Write K = W (k)[1/p] and let R denote the Robba ring over K. Then there is an isomorphism

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Section: Morrow’s Variational Tate Conjecture For Divisorsmentioning

confidence: 90%

Section: Morrow’s Variational Tate Conjecture For Divisorsmentioning

confidence: 90%

Section: Morrow’s Variational Tate Conjecture For Divisorsmentioning

confidence: 99%

“…It seems entirely plausible that the methods of this section can be easily adapted to give a proof of [Mor14, Theorem 3.5] in general, that is, over rather than just .…”

Section: Morrow’s Variational Tate Conjecture For Divisorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Cycle classes in overconvergent rigid cohomology and a semistable Lefschetz theorem

Lazda¹,

Pál²

2019

Compositio Math.

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Unramified Cohomology, Algebraic Cycles and Rationality

Schreieder

2021

Rationality of Varieties

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We show that the Tate conjecture for divisors over a finite field F is equivalent to an explicit algebraic problem about the third Milnor K-group of the function field F(x, y, z) in three variables over F.Theorem 1.2. Let F = F q be a finite field with q = p m elements and let ℓ be a prime invertible in F. The following hold true:(1)(3) The operator Frob q −q • id is injective on the p-adic completion K M 3 ( F(P 3 )) ⊗Q p . (4) There is a Galois-equivariant surjectionand the action of Frob q −q • id on (( F(P 3 ) * ) ⊗3 ) ⊗Q ℓ is injective. (5) If we setthen S q ⊂ {±1, ±q 1/2 , ±q} and {±1, ±q 1/2 } ⊂ S q for q = p 4m . Some comments are in order. Item (1) shows that the a priori very complicated Z ℓ -module K M 3 ( Fq (P 3 )) ⊗Z ℓ that is the key player in Theorem 1.1 is in fact quite simple: it is the ℓ-adic completion of a free G F -module M of countable rank. We will see in the proof (see Theorem 4.2 below) that M is a G F -submodule of a countable direct sum N = P i of permutation G F -modules P i ≃ Z n i . The Frobenius eigenvalues on N ⊗Q ℓ are roots of unity, see Lemma 5.2 below. Item ( 5) is thus somewhat

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Supersingular Irreducible Symplectic Varieties

2021

Rationality of Varieties

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A Variational Tate Conjecture in crystalline cohomology

Cited by 24 publications

References 38 publications

Cycle classes in overconvergent rigid cohomology and a semistable Lefschetz theorem

Cycle classes in overconvergent rigid cohomology and a semistable Lefschetz theorem

Unramified Cohomology, Algebraic Cycles and Rationality

Supersingular Irreducible Symplectic Varieties

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