# Derived categories and classical theorems in homological algebra

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!!

• You might want to ask these as separate questions. – Ben McKay Nov 6 '19 at 18:32

Your question might be compacted to someting like: Do I need derived categories to study cohomology of sheaves? Of course, this depends on your particular interests. Let me anyeway give you some starting points to help you to make up your mid.

(1) Are the derived categorical derived functors universal in the classical sense?

Let $$f : \mathcal{A} \to \mathcal{B}$$ a left exact functor between abelian categories and denote also by $$f$$ its extension to the corresponding homotopy categories. $$\mathbf{K}(\mathcal{A})$$ denotes the category of complexes with maps up to homotopy. The derived functor $$\mathbf{R}f : \mathbf{D}(\mathcal{A}) \to \mathbf{D}(\mathcal{B})$$ satisfies a universal property that implies that the collection $$\{\mathbf{R}^if\}_{i \in \mathbb{N}}$$ is a universal $$\delta$$-functor such that $$\mathbf{R}^0f = f$$. Where $$\mathbf{R}^if: = \mathrm{H}^i\mathbf{R}f$$. See [L, $$\S2.1$$].

(2) Can we show the "Grothendieck's Tohoku" easily using derived category?

This is related to the existence of acyclic objects for a certain functor. For details, I suggest you to look at [L, $$\S$$ 2.2].

(3) How can we use derived categorical derived functors in order to show propositions around spectral sequences?

As long as you refer to the so called Grothendieck spectral sequence on the composition of functors, it is replaced by the following theorem. Let $$f : \mathcal{A} \to \mathcal{B}$$ and $$g : \mathcal{B} \to \mathcal{C}$$ left exact functors between abelian categories. If $$f$$ takes $$f$$-acyclic objects into $$g$$-acyclic objects we have a natural isomorphism $$\mathbf{R}gf \cong \mathbf{R}g\mathbf{R}f$$

(there is an analogous theorem for left derived functors).

The every time $$\mathbf{R}f$$ reduces to $$F$$ you obtain similar formulas than the ones you obtain by the collapse of the spectral sequence. The advantage is that the argument is simpler and you don't have limitations on finiteness or boundedness of the complex involved.

Also, arguments involving three or more functors are seamless, something that would require multi-graded spectral sequences.

On the other hand, derived categories will never help you in computing delicate properties for certain spectral sequences like the Adams spectral sequence or similar ones.

(4) Grothendieck group of a derived category

In favorable cases $$\mathrm{K}_0(\mathbf{D}(\mathcal{A}))$$ and $$\mathrm{K}_0(\mathcal{A})$$ agree. A small advantage would be that the oposite of a class of an object $$X$$ is not a virtual object but $$-[X] = [X[1]]$$. A discussion of $$\mathrm{K}_0$$ in the geometric context is in SGA6, exposé IV, to begin with.

(5) Easier proof of classical homological propositions

I won't assert they are easier but they are in my opinion clearer and broader. For base-change and Künneth I suggest you to look at [L, Theorem (3.10.3)]. The Künneth formula is more general than any other I've seen in the literature. Its expression via spectral sequences, if possible, it would look extremely complicated.

Besides, if you really want to understand Grothendieck-Serre duality beyond Cohen-Macaulay maps and schemes, then derived categories are indispensable.

Final remarks

For me, [L] is a very good introduction to the use of derived categories in Algebraic Geometry. Unfortunately it is not self-contained, so you need at least the first chapter in [KS1] and looking at Spaltenstein paper on unbounded resolutions. Alternatively you have all the needed prerequisites in [KS2].

Bibliography

[KS1] Kashiwara, Masaki; Schapira, Pierre: Sheaves on manifolds. Grundlehren der Mathematischen Wissenschaften, 292. Springer-Verlag, Berlin, 1994.

[KS2] Kashiwara, Masaki; Schapira, Pierre: Categories and sheaves. Grundlehren der Mathematischen Wissenschaften, 332. Springer-Verlag, Berlin, 2006.

[L] Lipman, Joseph: Notes on derived functors and Grothendieck duality. Foundations of Grothendieck duality for diagrams of schemes, 1–259, Lecture Notes in Math., 1960, Springer, Berlin, 2009.

[SGA6] Berthelot, Pierre; Alexandre Grothendieck; Luc Illusie, eds. Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225). Berlin; New York: Springer-Verlag, 1971.