**Question:** Let $\mathbb{F}$ be an algebraically closed field of characteristic zero and let $\mathrm{Ch}\_{\mathbb{F}}$ be the category whose objects are chain complexes (of $\mathbb{F}$-modules) and whose morphisms are chain maps. Is it true that every object in $\mathrm{Ch}\_{\mathbb{F}}$ is injective? If so, does it remain true if $\mathbb{F}$ is not algebraically closed? What if $\mathbb{F}$ has positive characteristic?


On one hand this seems to good to be true, but on the other hand things are often exceedingly nice when dealing with vector spaces.


**Edit:** removed a flawed proof. Thanks darij for spotting the error.



Are chain complexes over a field always injective?
No.


Let $X$ be the chain complex with $\mathbb{k}$ in degree $n$ (and all differentials zero) and let $Y$ be the chain complex with $\mathbb{k}$ in degrees $n$ and $n+1$ with the identity as differential $Y\_{n+1}\to Y\_n$ (I choose for my convention that the differential lowers degree).


Then $X$ is not injective because there is no chain map $Y\to X$ which factors the identity on $X$ through the evident inclusion of $X$ into $Y$.


To be injective over a field it is necessary and sufficient to have vanishing homology.


A similar argument shows that not every chain complex is projective, and that to be projective over a field it is necessary and sufficient to have vanishing homology.


