One of the properties of the category $\mathcal{O}$(w.r.t a finite dimensional semisimple Lie algebra $\mathfrak{g}$) is:
If $0\to M_1\to M_2\to M_3\to 0$ is an exact sequence of objects of $\mathcal{O}$ and $L$ is a finite dimensional $\mathfrak{g}$-module,
then $0\to L\otimes M_1\to L\otimes M_2\to L\otimes M_3\to 0$ is also an exact sequence of objects of $\mathcal{O}$.
But in the proof Humphreys' book, it is only proved that $L\otimes M_i$ is an object of $\mathcal{O}$ and I don't see why the exactness should be preserved.
$L$ should be decomposed into a finite direct sum of finite dimensional irreducible $\mathfrak{g}$-modules and each of these is generated by a maximal vector as a $U(\mathfrak{g})$-module. However, this doesn't mean that they are free modules.
Can anyone explain why the exactness is preserved?
Best Answer
I think I figured it out. Like I commented, $L\otimes M$ usually means $L\otimes_k M$ but not $L\otimes_{U(\mathfrak{g})}M$. Then my question is obvious.