2.3 Overconvergent sheaves and homotopy types
2.3.2 A comparison theorem
In this section, I prove the following comparison result.
Theorem 2.3.6. There is an isomorphism
RΓTh(Ω∗(OX/K† )) ∼= RΓ(OX/K† ) (2.55) in Ho(dgaK).
The idea is that after using simplicial methods to (essentially) reduce to the case where there exists an embedding of X into a smooth formal V-scheme, it only really needs observ-ing that Le Stum’s comparison of rigid cohomology and cohomology of the overconvergent site respects multiplicative structures.
So suppose that U• = (U•, U•,U•) is a framing system for X, with U• → X a Zariski hyper-covering. Define the category dga(O†U
•/K) of dga’s on the simplicial ringed topos (U•/K)An† in the standard way. As before, consider the functor of Thom-Whitney global
2.3. Overconvergent sheaves and homotopy types
Proposition 2.3.7. This natural transformation is an isomorphism when evaluated on O†X/K.
Proof. As usual, it suffices to show that it induces an isomorphism on cohomology. But this just follows from cohomological descent for overconvergent cohomology, see Section 3.6 of [49].
I now want to extend Le Stum’s overconvergent version of ‘linearisation of differential operators’ to deal both with dga’s and with simplicial Berkovich spaces. To start with, consider the following diagram of simplicial ringed topoi
(U•, ]U•[U•)An† and ϕU• is the ‘realisation map’. For more details, see §1.4 and §2.1 of loc. cit.
The induced maps jU−1•(OU†
•/K) → O†U
•/]U•[U• and ϕ−1U•(K) → O†U
•/]U•[U•/K are both flat, and so there derived linearisation of the overconvergent de Rham dga
RL(i−1U•Ω∗(U•)
K) := RjU•∗ϕ∗U•(i−1U•Ω∗(U•)
K) (2.59)
as in Chapter 3 of loc. cit., exists as an object of Ho(dga(OU†
•/K)).
Proposition 2.3.8. There is an isomorphism
Proof. To define the morphism, it suffices to define a morphism OU† func-torially in n, such that the composite map O†U
n/K → jUn∗ϕ∗Un(i−1UnΩ1(U
n)K) is zero. But ex-actly as in Proposition 3.3.10 of [49], since O†U
n/K is a crystal, jU−1
K) is a quasi-isomorphism, I may forget the algebra structure, and prove that it is a quasi-isomorphism of complexes in each simplicial degree.
But by definition, the map O†U
n/K → RL(i−1U
•Ω∗(U
•)K)n is exactly the augmentation map that Le Stum constructs. That this is a quasi-isomorphism is then Proposition 3.5.4. of loc. cit.
Proposition 2.3.9. There is an isomorphism
RΓ(RL(i−1U•Ω∗(U•)K)) ∼= RΓ(i−1U•Ω∗(U•)K)) (2.62) in Ho(dgaK)∆.
Proof. Just note that the proof of Proposition 3.3.9 of loc. cit. carries over mutatis mutandis to the simplicial/dga situation.
Combining these two results, in order to complete the proof of Theorem 2.3.6, I just need to verify that there is a canonical isomorphism
RΓ(j†Ω∗V
0(U)•) ∼= RΓ(i−1U•Ω∗(U•)
K) (2.63)
in Ho(dgaK)∆. I may work level-wise, where there is a natural map RΓ(j†Ω∗V
iso-2.3. Overconvergent sheaves and homotopy types
morphism is I may forget the algebra structure, and invoke Le Stum’s results from §3 of loc. cit.
2.3.3 Frobenius structures
I am now in a position to prove that the rigid rational homotopy type is functorial. Indeed, it is clear that the definition in terms of the overconvergent site is functorial, and it is also not too difficult to see by functoriality of the comparison morphism that the map f∗ : RΓ(O†Y /K) → RΓ(O†X/K) induced by any morphism f : X → Y is the same as that induced by any lift of f to a map f between framing systems for X and Y . In particular, this latter map is independent of the lift f.
In order to put Frobenius structures on the dga’s obtained from the overconvergent site, Le Stum’s base change morphism 1.4.6 of [49] will need to be examined slightly more closely. So suppose that α : K → K0 is a finite extension of complete, discretely valued fields, and let V → V0 (resp. k → k0) be the induced finite extension of rings of integers (resp. residue fields). Then there is a morphism of sites
α : An†(V0) → An†(V) (2.65)
which is induced by (X ⊂P ← V ) 7→ (Xk0 ⊂PV0 ← VK0). This base extension functor has an adjoint, which considers an overconvergent variety (Y, W ) over K0 as one over K -note that this holds only if the extension K → K0 is finite. Hence the pull-back morphism α−1 on presheaves has a simple description - namely (α−1F )(Y, W ) = F (Y, W ) where on the LHS (Y, W ) is considered as an overconvergent variety over K0, and on the RHS as one over K. In particular, α−1(O†V) = O†V0, and α extends to a morphism of ringed sites.
Now suppose that X is a k-variety, and consider the sheaf (X/K) on An†(V), which is the sheafification of the presheaf (C, O) 7→ Mork(C, X). By the above comments, α−1(X/K) is the sheafification of the presheaf (C0, O0) 7→ Mork(C0, X) = Mork0(C0, Xk0).
Thus α−1(X/K) = (Xk0/K0), and hence there is a morphism of ringed topoi
(Xk0/K0)An† → (X/K)An†. (2.66) More generally, exploiting functoriality of (Y /K0)An† in Y as a k0-variety, for any k0 variety Y and any commutative square
Y f //
X
Spec(k0) //Spec(k)
(2.67)
there is an induced morphism of ringed topoi
f : (Y /K0)An† → (X/K)An† (2.68) such that that f−1(O†X/K) = O†Y /K0. The situation I am interested in is when σ : K → K is a lifting of the absolute Frobenius on k, and FX : X → X is the absolute Frobenius on X. Then there is a morphism
FX : (X/K)An† → (X/K)An† (2.69)
of ringed topoi, and if f : X → Y is a morphism of k-varieties, then there is a commutative square
(X/K)An†
FX //
f
(X/K)An†
f
(Y /K)An† FY //(Y /K)An†.
(2.70)
Hence there is a base change map
ΦX/Y : FY−1Rf∗(OX/K† ) → Rf∗(O†X/K) (2.71) in Ho(dga(OY /K† )).
Proposition 2.3.10. If Y = Spec(k) is a point, then ΦX/k is a quasi-isomorphism.
Proof. This is a straightforward application of the comparison theorem. It is not too difficult to check that this is compatible base change, and hence the morphism induced by ΦX/k on cohomology is the usual Frobenius on rigid cohomology, which is an isomorphism.
Remark 2.3.11. Similarly to the problem of functoriality, the map ΦX/k is the same as the map induced by a lift of the absolute Frobenius to a framing system for X. Again, this implies that the latter is independent of the choice of this lift.
Remark 2.3.12. I do not know whether or not ΦX/Y is a quasi-isomorphism in general. It would follow, for example, if it Frobenius was known to be bijective on relative overcon-vergent cohomology.