\(D^\dagger (\infty)\)-affinité des schémas projectifs. (\(D^\dagger (\infty)\)-affinity of projective schemes.) (Q1271446)

Suppose \(X\) is a smooth projective \(p\)-adic formal scheme, \(U=X-D\) the complement of an ample divisor. Then Berthelot has defined a sheaf (on \(X)\) of overconvergent differential operators. Here it is shown that coherent modules over this sheaf satisfy the usual ``Theorem A'' and ``Theorem B''. Previously this was known for affine space (with its canonical compactification). The proof uses various filtrations and clever choices of integral models. At the end it is also shown that the category of coherent overconvergent \(D\)-modules is invariant under maps of compactifications \(f:(U,X) \to(U',X')\) with \(U=U'\).