So far I have studied a fundamental part of the theory of derived categories, for example, the existence of derived functors, the "composition of derived functors", etc.

Now I have some questions about derived functors in the sense of derived category theory.

(one) **Are categorical derivative functors universal derivatives in the classical sense?**

Leave $ mathscr {A, B} $ be two abelian categories, $ f: mathscr {A} to mathscr {B} $ an exact left functor, and suppose that $ mathscr {A} $ You have enough injection.

Then there is "the" right derived functor $ mathbb {R} ^ + f: D ^ + ( mathscr {A}) to D ( mathscr {B}) $ from $ f $,

And it is $ i $-th cohomology of an object $ X $ in $ mathscr {A} $ (considering as the complex that has $ X $ to $ 0 $-th grade) is the classic $ R ^ yes (X) $.

So $ {H ^ i ( mathbb {R} ^ + f (X)) } _ i $ it is universal, in the sense of Hartshorne's AG, chapter III, that is, for each $ delta $-functors $ {g ^ i } _ i $ from $ mathscr {A} $ to $ mathscr {B} $ (i.e., a collection of additive functors that satisfy the following condition: for each exact sequence it cuts $ 0 a X a Y a Z a 0 $ in $ mathscr {A} $, there is the "connection map" $ g ^ iZ to g i + 1} X $, doing the exact long sequence.), if we have a natural transformation $ f to g ^ 0 $, there is a unique natural transformation $ R ^ if to g ^ i $.

Is there now a categorical interruption derived from this phenomenon?

That is, for such $ {g ^ i } $ Y $ f to g ^ 0 $there is $ delta $-functor $ g: D ^ + ( mathscr {A}) a D ( mathscr {B}) $ (such that for each $ X in mathscr {A} $, $ H ^ i g (X) = g ^ i (X) $) Y $ mathscr {Q} f to g mathscr {Q} $?

($ mathscr {Q} $ is the location functor).

If so, then by universality (in the derived categorical sense), we have $ mathbb {R} ^ + f to g $.

(I read this post, but it seems to be a little different from my question).

(2) **Can we display the "Grothendieck Tohoku" easily using the derived category?**

This theorem says that, in particular, for a $ delta $-functor $ {g ^ i } $Yeah $ g ^ i (I) = 0 $ for each $ i gt 0 $ and each injectable object $ I $, so this is universal, that is, $ g ^ i cong R ^ i g ^ 0 $.

If (1) is true, then I believe that this theorem can be translated as follows:

Leave $ {g ^ i } $ be a $ delta $-functor and $ g: D ^ + ( mathscr {A}) a D ( mathscr {B}) $ be "morphism" as in (1).

Yes $ g ^ i (I) = 0 $ for each $ i $ and injective $ I in mathscr {A} $, then $ g $ is the right derived functor of $ g ^ 0: K ^ + ( mathscr {A}) a K ( mathscr {B}) $.

Is this true?

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

I am studying categories derived from "Waste and duality" of Hartshorne.

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 retrieve the old spectral sequence …" (see observations after Proposition 5.4.)

I think the author means that we can show "all" the propositions that the spectral sequence argument used to show using derived categories.

For example:

leave $ mathscr {A, B, C} $ be abelian categories, $ f: mathscr {A} a mathscr {B}, g: mathscr {B} a mathscr {C} $ left exact functors, and suppose that $ mathscr {A, B} $ has enough injectables and that $ f $ take each injectable object of $ mathscr {A} $ yet $ g $– acyclic object.

Leave $ X in mathscr {A} $.

Yes $ R ^ if (X) = 0 $ for each $ 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 is an exact sequence $ 0 a R ^ qg (f (X)) a R ^ q (gf) X a g R ^ q f X $.

c) Leave $ f: X a Y $ be a morphism of appropriate schemes on a field, $ mathscr {F} $ a coherent sheaf in $ X $.

So $ chi ( mathscr {F}) = sum_p (-1) ^ p chi (R ^ p f_ * mathscr {F}) $.

d) Leave $ f: X a Y $ be a morpshim of schemes, and $ mathscr {F} $ a bundle of modules in $ X $.

Suppose for everyone $ q $, $ dim operatorname {supp} R ^ qf _ * mathscr {F} = 0 $.

So $ 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 many of these propositions related to the Grothendieck spectral sequence, but if I understand these (a) ~ (d), it seems similar to show any other type proposition.

Related publication: this

Finally,

(4)**Easier test of classical homological propositions**

I have heard that we can show Kunneth's formula more easily using derived categories.

I want to know such propositions.

Would you give me references?

I also want references to categorical evidence derived from cohomology and base change theorems (see III.12 of Hartshorne & # 39; s AG or here.

The subsequent statement is too abstract for me.

I prefer Hartshorne's concrete statement).

Thank you!