Sorry for my bad English.
In Hartshorne, III.Lemma 12.3, we prove $h^i(L^{\cdot}\otimes M)\cong h^i(C^{\cdot}\otimes M)$ as follows.
But, Im trouble that why he had to reduct to “$M$ is finitely generated”.
In other words, where we use “$M$ is finitely generated”.
Help me,thanks.