Abstract
Let n and t be non-negative integers, R a commutative Noetherian ring with an ideal of R, M and N finite R-modules, and X an arbitrary R-module. We prove that if is an R-module for all then is an -cofinite R-module for all i < t, is an R-module, and is a finite set for all It shows that is -cofinite and is finite for all i. In particular, is -cofinite for all i whenever Also, is finite for all i when R is semi-local with
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