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Dualité locale et holonomie pour les \({\mathcal D}\)-modules arithmétiques. (Local duality and holonomy for arithmetic \({\mathcal D}\)-modules). (French) Zbl 0955.14015
This paper deals with the local duality functor on Berthelot’s \(\mathcal D^ {\dagger}\)-modules for algebraic varieties defined over a field of characteristic \(p>0\). Let \(X\) be an algebraic variety over a field \(k\) of positive characteristic \(p\). Assume that \(X\) can be lifted to a smooth formal scheme \(\mathcal X\), of relative dimension \(d=d_{\mathcal X}\), over the formal spectrum \(\mathcal S\) of a discrete valuation ring \(\mathcal V\) of unequal characteristics \((0,p)\), residue field \(k\), and maximal ideal \(\mathfrak m\), \(\mathcal V\) being complete with respect to the \(\mathfrak m\)-adic topology. In his theory of \(\mathcal D^{\dagger}\)-modules, D. Berthelot [“\({\mathcal D}\)-modules arithmétiques. II: Descente de Frobenius”, Mém. Soc. Math. Fr., Nouv. Sér. 81 (2000; Zbl 0948.14017)] studied as a central object a coherent sheaf of rings, of finite cohomological dimension, \(\mathcal D^{\dagger}_{\mathcal X,\mathbb Q}\). \(\mathcal D^{\dagger}_{\mathcal X,\mathbb Q}=\mathcal D^{\dagger}_{\mathcal X}\otimes\mathbb Q\), where \(\mathcal D^{\dagger}_{\mathcal X}\) is the inductive limit (over \(m\)) of the \(\mathfrak m\)-adic completions of a family of sheaves \(\mathcal D^{(m)}_{\mathcal X}\) of rings of differential operators, locally generated by a finite number of differential operators. One is interested in the differential operators of \(\mathcal D^{\dagger}_{\mathcal X,\mathbb Q}\) with coefficients in an \(\mathcal O_{\mathcal X}\)-algebra \(\mathcal B\), not necessarily of finite cohomological dimension, endowed with a compatible structure of left \(\mathcal D^{\dagger}_{\mathcal X,\mathbb Q}\)-module. The main motivation for studying such operators comes from rigid cohomology where one deals with sheaves of differential operators with coefficients in algebras of functions with overconvergent singularities. Passing to derived categories one is led to consider perfect complexes of sheaves.
To deal with algebraic and formal aspects of the theory at the same time, write \(\mathbb Z_{(p)}\) for the localization of \(\mathbb Z\) with respect to the ideal \((p)\), and let \(S\) be a \(\mathbb Z_{(p)}\)-scheme or a formal \(\mathcal V\)-scheme (with notations as above). \(X\) will denote a smooth \(S\)-scheme, respectively a smooth formal \(S\)-scheme. Using divided powers one may define sheaves of differential operators of level \(m\), \(\mathcal D^{(m)}_X\), on \(X\) \(\dots\), etc. One can also take completions and pass to the limit, etc. One assumes the existence of a ‘Frobenius’ \(F:X\rightarrow X''\) for smooth (formal) \(S\)-schemes and Frobenius descent à la Berthelot. For a left \(\mathcal D_{X''}^{(m)}\)-module one has an induced left \(\mathcal D^{(m+1)}_X\)-module \(F^*\mathcal E\), and as a matter of fact, \(F^*\) induces an equivalence of categories between the left \(\mathcal B^{\prime (m)}\otimes_{\mathcal O_{X''}}\mathcal D^{(m)}_{X''}\)-modules and the left \(\mathcal B^{(m+1)}\otimes_{\mathcal O_{X}}\mathcal D^{(m+1)}_{X}\)-modules, where the \(\mathcal B^{\prime(m)}\) are \(\mathcal O_{X''}\)-algebras with left \(\mathcal D^{(m)}_X\)-module structure. Concerning the cohomological dimension of \(\mathcal D^{\dagger}_{\mathcal X,\mathbb Q}\), one has Berthelot’s result, at least in the formal case if the relative dimension \(d_{\mathcal X}\) of \(\mathcal X\) over \(\mathcal S\) is constant, that \(\text{dimcoh}(\mathcal D^{\dagger}_{\mathcal X,\mathbb Q})\leq 2d_{\mathcal X}+1\). This contrasts with the classical result for \(\mathcal D\)-modules over, say, the Witt vectors, à la Malgrange.
Let \(\mathcal R\) be a sheaf of commutative rings and let \(\mathcal A\) and \(\mathcal B\) be sheaves of (not necessarily commutative) flat \(\mathcal R\)-algebras on \(X\). Assume one has a sheaf of commutative rings \(\mathcal O_{\mathcal A}\) on \(X\) with a ring morphism \(\mathcal O_{\mathcal A}\rightarrow\mathcal A\). Also assume one has an \(\mathcal O_{\mathcal A}\)-module \(\omega_{\mathcal A}\), locally free of rank \(1\), to make the (right) \((\mathcal O_{\mathcal A}^d,\mathcal A^d)\)-bimodule \(\omega_{\mathcal A} \otimes_{\mathcal O_{\mathcal A}}\mathcal A\). Assume there is another right \(\mathcal A\)-module structure on \(\omega_{\mathcal A}\otimes_{\mathcal O_{\mathcal A}}\mathcal A\) inducing an \(\mathcal O_{\mathcal A}\)-structure defined by left multiplication and such that \(\omega_{\mathcal A}\otimes_{\mathcal O_{\mathcal A}}\mathcal A\) with the previous right \(\mathcal A\)-module structure, is a right \(\mathcal A\)-bimodule. Finally one assumes the existence of a right \(\mathcal A\)-bimodule involution on \(\omega_{\mathcal A}\otimes_{\mathcal O_{\mathcal A}}\mathcal A\) which interchanges both right \(\mathcal A\)-module structures. Similarly one assumes such properties for \(\mathcal B\). Then, with \(\omega_{\mathcal A}^{-1}:= \operatorname{Hom}_{\mathcal O_{\mathcal A}}(\omega_{\mathcal A},\mathcal O_{\mathcal A})\), the natural isomorphisms \(\mathcal A\otimes_{\mathcal O_{\mathcal A}}\omega_{\mathcal A}^{-1}{\buildrel \sim\over\rightarrow}\operatorname{Hom}_{\mathcal O_{\mathcal A}}(\omega_{\mathcal A},\mathcal A) {\buildrel\sim\over\rightarrow}\operatorname{Hom}_{\mathcal A^d}(\omega_{\mathcal A}\otimes _{\mathcal O_{\mathcal A}}\mathcal A,\mathcal A)\) endow \(\mathcal A\otimes_{\mathcal O_{\mathcal A}} \omega_{\mathcal A}^{-1}\) with the structure of a left \(\mathcal A\)-bimodule. The existence of such an \(\omega_{\mathcal A}\) implies that the functor \(\omega_ {\mathcal A}\otimes_{\mathcal O_{\mathcal A}}\) induces an equivalence of categories between \(D_{\text{parf}}({}^g \mathcal A)\) and \(D_{\text{parf}} (\mathcal A^d)\), where \(D_{\text{parf}}({}^g \mathcal A)\) denotes the derived category of perfect complexes of left \(\mathcal A\)-modules. Similarly for \(D_{\text{parf}}(\mathcal A^d)\) for right \(\mathcal A\)-modules. The functor \(\otimes_{\mathcal O_{\mathcal A}}\omega_{\mathcal A}^{-1}\) is a quasi-inverse of \(\omega_{\mathcal A}\otimes_{\mathcal O_{\mathcal A}}\). These results also hold for the categories \(D_{\text{parf}}^b\) of perfect complexes with bounded cohomology. Now for \(\mathcal E\in D^b_{\text{parf}}({}^g \mathcal A)\), one defines its dual \(\mathbb D(\mathcal E)\) by \(\mathbb D(\mathcal E)=\mathbb R\operatorname{Hom}_{{}^g \mathcal A}(\mathcal E,\mathcal A[d_X])\otimes_{\mathcal O_{\mathcal A}} \omega_{\mathcal A}^{-1}\), where \(d_X\) is the relative dimension of \(X/S\). Similarly, for \(\mathcal F\in D^b_{\text{parf}}(\mathcal A^d)\), one defines the dual \(\mathbb D''(\mathcal F)\) by \(\mathbb D''(\mathcal F)=\omega_ {\mathcal A}\otimes_{\mathcal O_{\mathcal A}}\mathbb R\mathcal Hom_{\mathcal A^d}(\mathcal F,\mathcal A[d_X])\). One has the bidualities:
There exist canonical isomorphisms \[ \imath:\mathcal E{\buildrel\sim\over\rightarrow}\mathbb D \circ\mathbb D(\mathcal E)\quad\text{and}\qquad\imath'':\mathcal F{\buildrel \sim\over\rightarrow}\mathbb D''\circ\mathbb D''(\mathcal F). \] Also, taking duals commutes with extension of scalars:
If there are ring morphisms \(v:\mathcal A\rightarrow\mathcal B\) and \(w:\mathcal O_{\mathcal A}\rightarrow \mathcal O_{\mathcal B}\) such that \(\omega_{\mathcal B}\cong\omega_{\mathcal A}\otimes_ {\mathcal O_{\mathcal A}}\mathcal O_{\mathcal B}\), then for \(\mathcal F\in D^b_{\text{parf}} (\mathcal A^d)\), there exists in \(D^b_{\text{parf}}(\mathcal B^d)\) a canonical isomorphism \(\mathbb D''(\mathcal F)\otimes^{\mathbb L}_{\mathcal A}\mathcal B{\buildrel \sim\over\longrightarrow}\mathbb D''(\mathcal F\otimes^{\mathbb L}_{\mathcal A} \mathcal B)\). Similarly for \(\mathcal E\in D^b_{\text{parf}}({}^g \mathcal A)\).
The theory may be applied to the geometric situation with differential operators by taking \(\omega=\omega_{\mathcal X}=\wedge^{d_{\mathcal X}}\Omega^1_{\mathcal X/\mathcal S}\). It leads to an expression for \(\mathbb D(\mathcal E)\) where \(\mathcal D^{\dagger}_{\mathcal X,\mathbb Q}\) shows up.
Back to the geometric situation, let \(Y\) be a locally noetherian scheme and let \(f:X\rightarrow Y\) be a finite morphism. In case \(f\) is a homeomorphism and \(\mathcal M\in D^b(Y)\), one may define \(f^{\flat}\mathcal M= \mathbb R\mathcal Hom_{\mathcal O_Y}(\mathcal O_X,\mathcal M)\in D^b(X)\). In particular, this applies to Frobenius \(F:X\rightarrow X''\) as described above. One has functors \(F^*\) and \(F^{\flat}\) à la Grothendieck-Hartshorne [see R. Hartshorne, “Residues and duality”, Lect. Notes Math. 20 (1966; Zbl 0212.26101)]. One has \(\omega_X\simeq F^{\flat}\omega_{X''}\).
From now on assume \(X\) and \(X''\) are smooth formal \(\mathcal V\)-schemes and write them as \(\mathcal X\) and \(\mathcal X''\). One considers inductive systems of \(\mathcal O_{\mathcal X}\)- and \(\mathcal O_{\mathcal X^ {\prime}}\)-algebras \(\mathcal B\) and \(\mathcal B''\) and differential operators \(\mathcal D\) and \(\mathcal D''\) with coefficients in \(\mathcal B\) and \(\mathcal B''\). Then the functor \(F^*\) induces an equivalence of categories \(D_{\text{parf}}({}^g \mathcal D''){\buildrel\sim \over\rightarrow} D_{\text{parf}}({}^g \mathcal D)\). \(F^{\flat}\) induces an equivalence \(D_{\text{parf}}(\mathcal D^{\prime d}){\buildrel\sim\over\rightarrow} D_{\text{parf}}(\mathcal D^d)\). One obtains the result:
Let \(\mathcal E\in D_{\text{parf}}({}^g \mathcal D'')\) and \(\mathcal F\in D_{\text{parf}}(\mathcal D^{\prime d})\). Then there are canonical isomorphisms:
(i) \(\eta_{\mathcal E}:F^{\flat}(\omega_{\mathcal X''}\otimes_{\mathcal O_{\mathcal X''}}\mathcal E){\buildrel\sim\over\rightarrow}\omega_{\mathcal X}\otimes_{\mathcal O_{\mathcal X}}F^*\mathcal E\),
(ii) \(\eta_{\mathcal F}:F^{\flat}\mathcal F\otimes_{\mathcal O_{\mathcal X}}\omega_{\mathcal O_{\mathcal X}}^{-1}{\buildrel\sim\over\rightarrow}F^*(\mathcal F\otimes_{\mathcal O_{\mathcal X''}}\omega_{\mathcal X}^{-1})\).
Applying the dualities \(\mathbb D\) and \(\mathbb D''\) one finds the result:
(i) Let \(\mathcal E\in D_{\text{parf}}({}^g \mathcal D'')\). Then there is a canonical isomorphism \[ \rho:\mathbb D(F^*\mathcal E){\buildrel\sim\over\longrightarrow}F^*\mathbb D(\mathcal E). \] (ii) Let \(\mathcal F\in D_{\text{parf}}(\mathcal D^{\prime d})\). Then there is a canonical isomorphism \[ \rho'':\mathbb D''(F^{\flat}\mathcal F){\buildrel\sim\over\rightarrow}F^{\flat}\mathbb D''(\mathcal F). \] Furthermore the afore mentioned bidualities are compatible with \(F^*\) and \(F^{\flat}\).
Let \(\mathcal V=W(k)\), where \(k\) is a perfect field of characteristic \(p>0\). Write \(\mathcal S=\text{Spf} \mathcal V\), and let \(\mathcal X\) be a smooth formal \(\mathcal S\)-scheme and let \(\mathcal X''\) be its inverse image under Frobenius acting on \(\mathcal S\). Suppose there is an \(\mathcal S\)-morphism \(F:\mathcal X\rightarrow\mathcal X''\) that lifts the relative Frobenius of \(X=\mathcal X\times_{\mathcal S}\text{Spec}(k)\). A \(\mathcal D^{\dagger}_{\mathcal X, \mathbb Q}\)-module \(\mathcal E\) is called a \(F\)-\(\mathcal D^{\dagger}_{\mathcal X,\mathbb Q}\)-module if \(\mathcal E\cong F^*\mathcal E\). One can associate to such modules a characteristic variety in a canonical way, thanks to Berthelot’s theorem (loc. cit.) on Frobenius descent. For a coherent \(F\)-\(\mathcal D^{\dagger}_{\mathcal X,\mathbb Q}\) -module \(\mathcal E\) one defines its dimension and codimension as the dimension and codimension of its characteristic variety in the cotangent bundle \(T^*_X\). Coherent \(F\)-\(\mathcal D^{\dagger}_{\mathcal X,\mathbb Q}\)-modules behave very much as in characteristic zero. Berthelot proved Bernstein’s inequality for coherent \(F\)-\(\mathcal D^{\dagger} _{\mathcal X,\mathbb Q}\)-modules \(\mathcal E\): \(\dim(\mathcal E)\geq\dim(\mathcal X)\). Then the following result is proved:
Let \(\mathcal E\) be a coherent left \(F\)-\(\mathcal D^{\dagger}_{\mathcal X,\mathbb Q}\)-module. Then:
(i) \(\text{codim}(\mathcal Ext^i_{\mathcal D^{\dagger}_{\mathcal X,\mathbb Q}}(\mathcal E,\mathcal D^{\dagger}_{\mathcal X,\mathbb Q}))\geq i\), for every \(i\geq 0\).
(ii) \(\mathcal Ext^i_{\mathcal D^{\dagger}_{\mathcal X,\mathbb Q}}(\mathcal E,\mathcal D^{\dagger}_{\mathcal X,\mathbb Q})=0\) for every \(i<\text{codim}(\mathcal E)\).
Finally, a coherent left \(F\)-\(\mathcal D^{\dagger}_{\mathcal X,\mathbb Q}\)-module \(\mathcal E\) is called holonomous if \(\mathcal E=0\) or \(\dim(\mathcal E)=d_{\mathcal X}\). Then:
Theorem. \(\mathcal E\) is holonomous \(\iff {\mathcal E}xt_{{\mathcal D}_{{\mathcal X},\mathbb Q}^\dagger}^i ({\mathcal E}, {\mathcal D}^{\dagger}_{\mathcal X,\mathbb Q})=0\) for \(i\neq d_{\mathcal X}\).
The dual \(\mathcal E^*\) of a holonomous \(F\)-\(\mathcal D^{\dagger}_{\mathcal X,\mathbb Q}\)-module \(\mathcal E\) is defined as \(\mathcal E^*=\mathcal Ext^{d_{\mathcal X}}_ {\mathcal D^{\dagger}_{\mathcal X,\mathbb Q}}(\mathcal E,\mathcal D^{\dagger}_{\mathcal X,\mathbb Q})\otimes_{\mathcal O_{\mathcal X,\mathbb Q}}\omega^{-1}_{\mathcal X,\mathbb Q}\). Then \({}^*\) defines an exact duality functor on the category of holonomous \(F\)-\(\mathcal D^{\dagger}_{\mathcal X,\mathbb Q}\)-modules, and one has the biduality \(\mathcal E{\buildrel\sim\over\rightarrow}(\mathcal E^*)^*\).

14F30 \(p\)-adic cohomology, crystalline cohomology
14G22 Rigid analytic geometry
14G20 Local ground fields in algebraic geometry
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
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