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This is in Section III.12. Algebraic Geometry by Hartshorne. Assume $X\to\mathrm{Spec}(A)$ is a projective morphism of Noetherian schemes. Let $\mathscr{F}$ be coherent over $X$, flat over $A$.

Proposition 12.2 says that there exists a bounded complex of finite free $A$-modules $\mathscr{L}^\bullet$ such that $$H^i(X,\mathscr{F}\otimes_AM)\cong H^i(\mathscr{L}^\bullet\otimes_AM)$$ for every $A$-module $M$.

Here are my questions:

  • In what sense is $\mathscr{L}^\bullet$ unique? Is $\mathscr{L}^\bullet$ unique up to chain homotopy or up to quasi-isomorphisms? Now I only know these two notions of uniqueness.
  • Which category should $\mathscr{L}^\bullet$ situate? Does the association $\mathscr{F}\mapsto\mathscr{L}^\bullet$ define a functor? I expect some derived category, and the functor needs answer to the question above.
  • What are generalizations of this proposition? This is vague. Actually I want to know the modern formulation of the proposition.
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    $\begingroup$ $\mathscr{L}^{\bullet}=Rf_*\mathscr{F}$, an object of the (bounded) derived category $D(Y)$. The nontrivial point is that it is a perfect object in $D(Y)$, commuting with base change. You can have a look at chapter 36 in the Stacks project. $\endgroup$
    – abx
    Commented Jun 29, 2023 at 5:51
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    $\begingroup$ In particular, see [Tag 0B91 (2)] for existence of the 'Grothendieck complex' $\mathscr L^{\bullet}$ (using [Tags 08EB and 0657] for translation to the case where the base $Y$ is affine). $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jun 29, 2023 at 9:32
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    $\begingroup$ The discussion of this in Chapter 2 of Mumford's "Abelian varieties" is particularly short and clear. I usually tell students to skip that section of Hartshorne's book and read it instead in Mumford's book. $\endgroup$
    – Jason Starr
    Commented Jun 29, 2023 at 10:26

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On your questions:

In what sense is $\mathscr{L}^\bullet$ unique? Is $\mathscr{L}^\bullet$ unique up to chain homotopy or up to quasi-isomorphisms? Now I only know these two notions of uniqueness.

It is unique up to quasi-isomorphism. Of course, if $Y$ is affine this uniqueness can be upgraded to uniqueness up to homotopy because in this case the complex $\mathscr{L}^\bullet$ corresponds to a complex of finitely generated projective modules over the global sections of the structure sheaf of $Y$.

Which category should $\mathscr{L}^\bullet$ situate? Does the association $\mathscr{F} \mapsto \mathscr{L}^\bullet$ define a functor? I expect some derived category, and the functor needs answer to the question above.

As it is remarked by @abx for $f \colon X \to Y$ projective and $\mathscr{F}$ flat over $Y$, the functor takes coherent flat complexes to perfect complexes, where we consider the category of perfect complexes $\mathrm{Perf}(Y)$ as a subcategory of the derived category.

What are generalizations of this proposition? This is vague. Actually I want to know the modern formulation of the proposition.

Recently the theorem has been generalized as follows, for $f \colon X \to Y$ proper and pseudo-coherent the functor $\mathbf{R}f_*$ restricts to a functor $$ \mathbf{R}f_* \colon \mathrm{Perf}(f) \longrightarrow \mathrm{Perf}(Y) $$ where $\mathrm{Perf}(f)$ denotes the relative perfect complexes on $X$ over $Y$, that, whenever $Y$ is regular, are all bounded pseudo-coherent complexes. Notice that the Noetherian hypothesis is not needed.

For more details, see this paper, especially section 3. As a bonus, in section 5 the semicontinuity is discussed.

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