\({\mathcal D}^\dagger\)-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 0956.14010The 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.

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.

Reviewer: Stephane Guillermou (MR 99c:14031)

##### 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 |