Let $A$ be a noetherian ring, and let $\{I\_i \mid i \in J\}$ be a collection of K-injective complexes over $A$.


Is the direct sum
$$
\bigoplus\_{i \in J} I\_i
$$
also a K-injective complex over $A$?


Recall that a complex $I$ is called K-injective if for any acyclic complex $M$, the complex $\operatorname{Hom}\_A(M,I)$ is also acyclic.



Direct sum of K-injectives over a noetherian ring
No.


The $K$-injective complexes form a triangulated subcategory of the homotopy category of complexes, and so if they were also closed under coproducts, the homotopy colimit of a sequence of $K$-injective complexes would also be $K$-injective.


Let $A=\mathbb{C}[x]/(x^2)$ and let $X$ be the complex
$$\dots\stackrel{x}{\to}A\stackrel{x}{\to}A\stackrel{x}{\to}A\stackrel{x}{\to}\dots.$$


Then $X$ is acyclic but not contractible, so can't be $K$-injective, but it is the homotopy colimit of its truncations to the left, which are all bounded below complexes of injectives and therefore $K$-injective.


