!THE $\aleph_1$-PRODUCT OF DG-INJECTIVE COMPLEXES

Abstract

Given a left Noetherian ring $R$, we give a necessary and sufficient condition in order that a complex of $R$-modules be DG-injective. Using this result we prove that if $(K_i)_{i\in I}$ is a family of DG-injective complexes of left $R$-modules and $K$ is the $\aleph_1$-product of $(K_i)_{i\in I}$ (i.e. $K\subset\prod_{i\in I}K_i$ is such that, for each $n$, $K^n\subset\prod_{i\in I}K_i^n$ consists of all $(x_i)_{i\in I}$ such that $\{i\mid x_i\neq0\}$ is at most countable), then $K$ is DG-injective.