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In the present paper, the concept of asymptotic closure f Y M of a filtration f relative to a module M is introduced and investigated. The methods used by the authors in previous notes on integral and pru È ferian closure operations of a filtration have proved to be efficient here, despite the complexity of the asymptotic closure operation comparatively to the integral and pru È ferian closure operation. Our main result gives a complete description of the asymptotic closure "