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$${\mathcal D}^\dagger$$-affinity of projective space. – With an appendix by P. Berthelot. ($${\mathcal D}^\dagger$$-affinité de l’espace projectif. (Avec un appendice de P. Berthelot).) (French) Zbl 0956.14010
The Beilinson-Bernstein theorem [A. Beilinson and J. Bernstein, C. R. Acad. Sci., Paris, Sér. I 292, 15-18 (1981; Zbl 0476.14019)] asserts that for a reductive group over a field of characacteristic 0 the flag variety $$X$$ is $${\mathcal D}$$-affine in the following sense: Every $${\mathcal D}_X$$-module $${\mathcal E}$$ is generated by its global sections and $$H^i (X;{\mathcal E})=0$$ for $$i>0$$. It follows that the global section functor induces an equivalence of categories between the category of coherent $${\mathcal D}_X$$-modules and the category of coherent $$\Gamma(X; {\mathcal D}_X)$$-modules.
In non-zero characteristic the usual sheaf of differential operators is not finitely generated. To overcome this difficulty P. Berthelot [Ann. Sci. Éc. Norm. Supér (4) 29, No. 2, 185-272 (1996; Zbl 0886.14004)] introduced sheaves of differential operators of level $$m$$, $${\mathcal D}_X^{(m)}$$, using a notion of partial divided power. The sheaves $${\mathcal D}_X^{(m)}$$ are finitely generated and satisfy $${\mathcal D}_X=\varinjlim {\mathcal D}_X^{(m)}$$; $${\mathcal D}^†_{\mathcal X}$$ is then defined as $$\varinjlim\widehat {\mathcal D}_{\mathcal X}^{(m) }$$, where $$\widehat{\mathcal D}_{\mathcal X}^{(m)}$$ is the $$p$$-adic completion of $${\mathcal D}^{(m)}_X$$, and $${\mathcal D}^†_{{\mathcal X}, \mathbb{Q}}={\mathcal D}^†_{\mathcal X}\otimes \mathbb{Q}$$.
The aim of the paper under review is to prove a Beilinson-Bernstein theorem for the projective space in the framework of arithmetic $${\mathcal D}$$-modules. Let $$V$$ be a discrete valuation ring of characteristics $$(0,p)$$, $${\mathcal V}$$ its completion, and $${\mathcal X}$$ the formal projective space of dimension $$N$$ over $$\text{Spf} {\mathcal V}$$. The author shows that $$\Gamma({\mathcal X}; {\mathcal D}^†_{{\mathcal X}, \mathbb{Q}})$$ is a coherent $${\mathcal V}$$-algebra, that coherent $${\mathcal D}^†_{{\mathcal X},\mathbb{Q}}$$-modules are acyclic and that taking global sections induces equivalences between the categories of coherent $${\mathcal D}^†_{{\mathcal X}, \mathbb{Q}}$$-modules and of coherent $$\Gamma({\mathcal X};{\mathcal D}^†_{{\mathcal X}, \mathbb{Q}})$$-modules.
In characteristic 0 it is well known that the associated graded ring of $${\mathcal D}_X$$ is isomorphic to the symmetric algebra of the tangent bundle of $$X$$. In the first part the author constructs the “symmetric algebra of level $$m$$” of an $${\mathcal O}_X$$-module and shows that the associated graded ring of $${\mathcal D}^{(m)}_X$$ is isomorphic to the symmetric algebra of level $$m$$ of the tangent bundle. Using this and the fact that the tangent bundle of the projective space is ample, she proves in the second part that, for $$n\geq 1$$ and for any coherent $${\mathcal D}_X^{(m)}$$-module $${\mathcal E}$$, $$H^n(X;{\mathcal E})$$ is of finite type and torsion.
She considers the behavior of cohomology when taking projective limits and obtains the vanishing of cohomology in degree greater than 1 for the $${\mathcal D}^†_{{\mathcal X}, \mathbb{Q}}$$-modules. In the last two parts she gives finiteness results for the global sections and proves the equivalence of categories.
It should be noted that these results are given in a more general framework where the sheaf $${\mathcal D}_X$$ is tensored by a commutative algebra with a structure of $${\mathcal D}$$-module. In particular, for a suitable algebra associated to a divisor $$Z$$, introduced by Berthelot, this gives the sheaf of differential operators with overconvergent singularities along $$Z$$.
Vanishing results for the cohomology of this algebra are given in an appendix by P. Berthelot.

##### MSC:
 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 14F30 $$p$$-adic cohomology, crystalline cohomology 14F17 Vanishing theorems in algebraic geometry
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