So far I have studied fundamental part of derived category theory, for example, the existence of derived functors, the "composition of derived functors", and so on.
Now I came up with some questions about derived functors in the sense of derived category theory.
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(1) Are the derived categorical derived functors universal in the classical sense?
Let $\mathscr{A, B}$ be two abelian categories, $f : \mathscr{A} \to \mathscr{B}$ a left exact functor, and assume that $\mathscr{A}$ has enough injective.
Then there exists "the" right derived functor $\mathbb{R}^+f: D^+(\mathscr{A}) \to D(\mathscr{B})$ of $f$,
and its $i$-th cohomology of an object $X$ in $\mathscr{A}$ (considering as the complex which has $X$ at $0$-th degree) is the classical $R^if(X)$.
So $\{ H^i(\mathbb{R}^+f(X)) \}_i$ is universal, in the sense of Hartshorne's AG, chapter III, i.e., for every $\delta$-functors $\{ g^i \}_i$ from $\mathscr{A}$ to $\mathscr{B}$ (i.e., a collection of additive functors satisfying the following condition: for every short exact sequence $0 \to X \to Y \to Z \to 0$ in $\mathscr{A}$, there exists the "connection map" $g^iZ \to g^{i+1}X$, making the long sequence exact.), if we have a natural transformation $f \to g^0$, there exists a unique natural transformations $R^if \to g^i$.
Now is there a derived categorical interruption of this phenomena?
I.e., for such $\{g^i\}$ and $f \to g^0$, does there exist a $\delta$-functor $g: D^+(\mathscr{A}) \to D(\mathscr{B})$ (such that for every $X \in \mathscr{A}$, $H^i g(X) = g^i(X)$) and $\mathscr{Q}f \to g \mathscr{Q}$?
($\mathscr{Q}$ is the localizing functor.)
If so, then by the universality (in derived categorical sense), we have $\mathbb{R}^+f \to g$.
(I read this post, but it seems to be a bit different from my question.)
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(2) Can we show the "Grothendieck's Tohoku" easily using derived category?
This is related to (1).
This theorem says that, in particular, for a $\delta$-functor $\{g^i\}$, if $g^i(I) = 0$ for every $i \gt 0$ and every injective object $I$, then this is universal, i.e., $g^i \cong R^i g^0$.
If (1) is true, then I think that this theorem can be translated into the following form:
Let $\{ g^i \}$ be a $\delta$-functor and $g : D^+(\mathscr{A}) \to D(\mathscr{B})$ be "the morphism" as in (1). If $g^i(I) = 0$ for every $i$ and injective $I \in \mathscr{A}$, then $g$ is the right derived functor of $g^0 : K^+(\mathscr{A}) \to K(\mathscr{B})$.
Is this true?
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(3) How can we use derived categorical derived functors in order to show propositions around spectral sequences?
I'm studying derived categories by Hartshorne's "Residues and duality".
In this text, the author says "What used to be a spectral sequence is now simply a composition of functors. And of course one can recover the old spectral sequence..." (see the remarks after the Proposition 5.4.)
I think the author means that we can show "all" propositions which are used to be shown by the spectral sequence argument using derived categories.
For example:
a) Let $\mathscr{A,B,C}$ be abelian categories, $f : \mathscr{A} \to \mathscr{B}, g: \mathscr{B} \to \mathscr{C}$ left exact functors, and suppose that $\mathscr{A,B}$ has enough injectives and that $f$ takes every injective object of $\mathscr{A}$ to a $g$-acycic object. Let $X \in \mathscr{A}$. If $R^if(X) = 0$ for every $i \gt 0$, then $R^n(gf)(X) \cong (R^ng)(f(X))$.
(This is obvious. I could show it.)b) More generally, in the situation of (a), if $R^ifX = 0$ for $0 \lt i \lt q$, then there exists an exact sequence $0 \to R^qg(f(X)) \to R^q(gf)X \to g R^q f X$.
c) Let $f : X \to Y$ be a morphism of proper schemes over a field, $\mathscr{F}$ a coherent sheaf on $X$. Then $\chi(\mathscr{F}) = \sum_p (-1)^p \chi(R^p f_* \mathscr{F})$.
d) Let $f : X \to Y$ be a morpshim of schemes, and $\mathscr{F}$ a sheaf of modules on $X$. Assume that for all $q$, $\dim \operatorname{supp}R^qf_*\mathscr{F} = 0$. Then $H^0(Y, R^nf_*\mathscr{F}) \cong H^n(X, \mathscr{F})$. (This proposition is used in Mumford's Abelian varieties, $\S$8, theorem1.)
(All propositions are trivial if we use the Grothendieck spectral sequence.)
I think there are a lot of such proposition related the Grothendieck spectral sequence, but If I understand these (a)~(d), it seems to be similar to show any other such type propositions.
Related post: this (a) and this(c).
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(4) Grothendieck group of a derived category
This is a related question to (3).
This post shows (3) (c).
As we can see from this post, it seems to be very important to understand Grothendieck groups of derived categories.
But I don't know any references of Grothendieck groups of derived categories, especially of algebraic geometrical objects.
I've found some fundamental properties.
(e.g., this post and this pdf.)
Are there any other important propositions of Grothendieck groups?
And would you give me some references?
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Lastly,
(5) Easier proof of classical homological propositions
I've heard that we can show the Kunneth's formula (of schemes) more easily using derived categories.
(see here and for singular cohomology here.)
I want to know such propositions.
Would you give me references?
I also want references of derived categorical proof of cohomology and base change theorems(see III.12 of Hartshorne's AG or here.
The later statement is too abstract for me.
I prefer Hartshorne-like concrete statement.)
Any help will be much appreciated!!
