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Let A be a nite-dimensional simple Lie algebra over the complex numbers. It is shown that a module is complete (or relatively complete) in the sense of Enright if and only if it is injectively copresented by certain injective modules in the BGG-category O.L etA be the nite-dimensional algebra associated to a block ofO. Then the corresponding block of the category of complete modules is equivalent to the category of eAe-modules for a suitable choice of the idempotente. Using this equivalence, a very easy proof is given for Deodhar's theorem (also proved by Bouaziz) that completion functors satisfy braid relations. The algebra eAe is left properly and standardly stratied. It satises a double centralizer property similar to Soergel's \combinatorial description" ofO. Its simple objects, their characters and their multiplicities in projective or standard objects are determined.

Enright’s completions and injectively copresented modules