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Finiteness theorems for the cohomology of an overconvergent isocrystal on a curve. (English) Zbl 0943.14008

Consider a smooth scheme \(X\) over a perfect field \(k\) of characteristic \(p>0\). Berthelot’s theory of overconvergent isocrystals on \(X\) and the corresponding rigid cohomology of such isocrystals often lead to infinite dimensional cohomology spaces. Here the case of a smooth curve \(X/k\) is treated in detail. A fairly general sufficient condition for an overconvergent isocrystal on \(X\) to have finite dimensional cohomology is given. In such cases one can also give a proof of Poincaré duality. The paper consists of an introduction to set the stage, then three parts treating in detail the necessary functional analysis over discretely valuated fields, local duality, global duality and finiteness with applications to quasi-unipotent isocrystals on smooth curves.
The introduction gives some philosophy on quasi-unipotent isocrystals and their relation to Grothendieck’s local monodromy theorem. It seems reasonable to expect that overconvergent isocrystals “of geometric origin” are quasi-unipotent.
Part I may be considered as a very wellcome and readible overview of non-archimedean functional analysis. Starting from general definitions, notions such as local convexity, balance, boundedness and linear compactness are discussed. Also Banach, Fréchet, dual-of-Fréchet and Montel spaces enter the stage, as well as \(LF\)-spaces, strictness and duality, in particular, a \(LF\)-space is bornological and barreled, the Banach-Steinhaus theorem holds and a map from an \(LF\)-space to a locally convex space is continuous if and only if it is bounded. A locally convex space which is Fréchet and dual-of-Fréchet is necessarily finite dimensional.
In part II local duality is treated. Let \(K\) be a discretely valued field. A topological \(K\)-algebra isomorphic to the algebra \(A\) of Laurent series convergent in some annulus \(r<|x|<1\) is called a local algebra. Let \(\Omega_A^1\) be the free topological \(A\)-module of rank one with basis \(dx/x\) and let \(M\) be a finite free \(A\)-module. Then one has a pairing \(M\times(M^\vee\otimes\Omega^1_A)\rightarrow K\) given by \((m,m^\vee\otimes dx/x)\mapsto\text{Res} m^\vee(m)\otimes dx/x\), where \(\text{Res}\) denotes the residue at \(x=0\). As a first result one may state: Any free \(A\)-module \(M\) of finite type is a Montel space for which the map \(M\rightarrow(M^\vee\otimes\Omega^1_A)^\prime_s\) induced by the pairing above, is a topological isomorphism. Here \((\ldots)^\prime_s\) denotes the strong dual. For an \(A\)-module \(M\) one has the usual notion of connection \(\nabla:M\rightarrow M\otimes\Omega^1_A\), and for an \(A\)-module with connection \((M,\nabla)\) one defines the cohomology \(H^0(M):=\text{Ker} \nabla\) and \(H^1(M):=\text{Coker} \nabla\). The pairing above induces for any free \(A\)-module of finite type with connection \((M,\nabla)\) a pairing in cohomology \(H^i(M)\times H^{1-i}(M^\vee)\rightarrow K\). A connection \(\nabla\) on \(M\) induces a dual connection \(\nabla^\vee\) on \(M^\vee\). One calls an \(A\)-module \(M\) strict if the connection \(M\rightarrow M\otimes\Omega^1_A\) is a strict map of topological vector spaces. Similarly for \(M^\vee\). One has the result:
For any free \(A\)-module of finite type with connection \((M,\nabla)\) the following are equivalent:
(i) \(M\) is strict;
(ii) \(M^\vee\) is strict;
(iii) \(H^1(M)\) is finite dimensional and separated.
Furthermore for strict \(M\), the cohomogical pairing above is a perfect pairing between finite dimensional \(K\)-vector spaces. A connection \(\nabla\) on a finite free \(A\)-module \(M\) is called unipotent if \((M,\nabla)\) is a successive extension of trivial rank one objects \((A,d)\). Then an \(A\)-module with unipotent connection is strict.
Let \(X/k\) be a smooth affine \(k\)-curve and let \(X\hookrightarrow\overline X\) be a smooth projective embedding. Let \(X\hookrightarrow\overline X\) lift to a morphism \(\mathfrak X\hookrightarrow\overline{\mathfrak X}\) of formally smooth \(R\)-schemes, where \(R\) is the discrete valuation ring of \(K\). For a strict neighborhood \(V\) of the tube \(]X[\) of \(X\) let \(A_V:=\Gamma(V,\mathcal O_V)\). For a cofinal set of strict neighborhoods \(V\) of \(]X[\), put \(\displaystyle{A^\dagger_X:=\lim_{\longrightarrow\atop V}A_V}\). For \(a\in D=\overline{X}-X\) define \(\displaystyle{A(a):=\lim_{\longrightarrow\atop V}\Gamma(V\cap]a[,\mathcal O_{\mathfrak X}^{\text{an}})}\) and let \(\displaystyle{A_X^{\text{loc}}:=\bigoplus_{a\in D}A(a)}\). Define \(A_X^{\text{qu}}\) by the exactness of the sequence \(0\rightarrow A^\dagger_X\rightarrow A_X^{\text{loc}}\rightarrow A_X^{\text{qu}}\rightarrow 0\). Define \(\displaystyle{\Omega_{A^\dagger}^1:=\lim_{\longrightarrow\atop V}\Gamma(V,\Omega^1_V)}\). Then, for a locally free \(A^\dagger\)-module \(M\) one has the global pairing \((M\otimes A^{\text{loc}}_X)\times(M^\vee\otimes A^{\text{loc}}_X)\rightarrow K\), with \(\displaystyle{((m_a),(\omega_a))\mapsto\langle(m_a),(\omega_a) \rangle_X:=\sum_{a\in D}\langle m_a,\omega_a\rangle_a}\), where \(\langle , \rangle_a\) denotes the local pairing as above. The following result summarizes the global duality: With notations as above, for a coherent locally free \(A^\dagger\)-module \(M\), the natural topology on \(M\) is dual-of-Fréchet, \(M\otimes_{A^{\dagger}}A^{\text{qu}}\) is Fréchet, and the maps \(M\rightarrow(M^\vee\otimes\Omega^{\text{qu}})^\prime_s\) and \(M\otimes\Omega^{\text{qu}}\rightarrow(M^\vee)^\prime_s\) are topological isomorphisms. The above theory may now be worked out for overconvergent isocrystals on a curve \(X/k\). One may define cohomology with compact support \(H^1_c(X,M)\), \(i=0,1,2\), where \((M,\nabla)\) is an overconvergent isocrystal on \(X/K\) with connection, and relate it to de Rham cohomology via an exact sequence. One also has parabolic cohomology \(H^i_p(X,M)\). The main result of the paper can now be stated:
For a smooth affine curve \(X/k\) and a strict overconvergent isocrystal \(M\) on \(X/K\), the \(K\)-vector spaces \(H^i(X,M)\), \(H^i_c(X,M)\) and \(H^i_p(X,M)\) are finite dimensional and there are perfect pairings \(H^i_c(X,M)\times H^{2-i}(X,M^{\vee})\rightarrow K\) and \(H^1_p(X,M)\times H^1_p(X,M^{\vee})\rightarrow K\) which can be given explicitly. In the final section the case of quasi-unipotent isocrystals is studied as an analogue of Deligne’s Weil II article [P. Deligne, Publ. Math., Inst. Hautes Étud. Sci. 52, 137-252 (1980; Zbl 0456.14014)] for \(\ell\)-adic étale cohomology related to Grothendieck’s local monodromy theorem. Notions of weight and pureness can be introduced. Several interesting analogues can be formulated. As a final result we mention (as a first step to the Riemann hypothesis):
For a quasi-unipotent isocrystal \((M,\Phi)\) on the curve \(X/K\), \(\imath\)-pure of weight \(\beta\), the weights of Frobenius on \(H^1_c(X,M)\) are strictly less than \(\beta+2\).

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
32B05 Analytic algebras and generalizations, preparation theorems

Citations:

Zbl 0456.14014
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