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We find conditions on the local cohomology modules of multi-Rees algebras ofadmissible filtrations which enable us to predict joint reduction numbers. As aconsequence we are able to prove a generalisation of a result ofReid-Roberts-Vitulli in the setting of analytically unramified local rings forcompleteness of power products of complete ideals.

Local cohomology of multi-Rees algebras, Joint reduction numbers and product of complete ideals