The Stacks project

Remark 63.4.8 (Alternative construction). Lemma 63.4.7 gives an alternative construction of the functor $f_!$ for locally quasi-finite morphisms $f$. Namely, given a locally quasi-finite morphism $f : X \to Y$ of schemes we can choose an open covering $X = \bigcup _{i \in I} X_ i$ such that each $f_ i : X_ i \to Y$ is separated. For example choose an affine open covering of $X$. Then we can define $f_!\mathcal{F}$ as the cokernel of the penultimate map of the complex of the lemma, i.e.,

\[ f_!\mathcal{F} = \mathop{\mathrm{Coker}}\left( \bigoplus \nolimits _{i_0, i_1} f_{i_0i_1, !} \mathcal{F}|_{X_{i_0i_1}} \to \bigoplus \nolimits _{i_0} f_{i_0, !} \mathcal{F}|_{X_{i_0}} \right) \]

where we can use the construction of $f_{i_0, !}$ and $f_{i_0i_1, !}$ in Section 63.3 because the morphisms $f_{i_0}$ and $f_{i_0 i_1}$ are separated. One can then compute the stalks of $f_!$ (using the separated case, namely Lemma 63.3.17) and obtain the result of Lemma 63.4.5. Having done so all the other results of this section can be deduced from this as well.

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