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Finiteness of the homological dimension of weakly complete algebras of differential operator with overconvergent coefficients. (Finitude de la dimension homologique d’algèbres d’opérateurs différentiels faiblement complètes et à coefficients surconvergents.) (French. English summary) Zbl 1111.14006
Summary: We prove that the sheaf of arithmetic differential operators with overconvergent coefficients, introduced by P. Berthelot [Ann. Sci. Éc. Norm. Supér., IV. Sér. 29, 185–272 (1996; Zbl 0886.14004) and Mém. Soc. Math. Fr., Nouv. Sér. 81 (2000; Zbl 0948.14017)], has finite cohomological dimension. A similar geometrical proof shows that the weak \(p\)-adic completion of the Weyl algebra has also finite cohomological dimension. Moreover, this algebra can be naturally endowed with a filtration which is compatible with the Frobenius.

MSC:
14F30 \(p\)-adic cohomology, crystalline cohomology
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
13N10 Commutative rings of differential operators and their modules
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