Derived functor

In the mathematical subfield of category theory and homological algebra , a derived functor (also: derived functor ) of a left or right exact functor is a measure of how far it deviates from the exactness. The name comes from the fact that analogously the derivatives of a function measure how much it deviates from a constant function.

For the remainder of this article, let and be Abelian categories and a covariant left exact functor . In the case of a contravariant and / or right-hand exact functor, the same applies, whereby arrows may have to be turned around and injective objects replaced by projective objects . ${\displaystyle C}$ ${\displaystyle D}$ ${\displaystyle F\colon C\to D}$

motivation

Is

${\displaystyle 0\to A'\to A\to A''\to 0}$

exact, so is the corresponding sequence

${\displaystyle 0\to F(A')\to F(A)\to F(A'')}$

exactly, but generally not the continuation through . ${\displaystyle \to 0}$

In principle, one could indeed continue the sequence - this is how the coke is finally defined - by exactly continuing, but this continuation would then depend on the homomorphism . One would like to have a dependency only on the objects. ${\displaystyle \to \operatorname {coker} (F(A)\to F(A''))\to 0}$ ${\displaystyle A\to A''}$

The fact that one of the objects involved can already severely limit the deviation from the accuracy can be seen, for example, in the case that it is an injective object . It then follows that the original sequence cleaves and is isomorphic . This is carried over to the image sequence, which in this case is also a short, exact sequence. ${\displaystyle A'}$ ${\displaystyle A}$ ${\displaystyle A'\oplus A''}$

In this respect, it is reasonable to assume that (at least under suitable additional conditions) one generally has an exact sequence

${\displaystyle 0\to F(A')\to F(A)\to F(A'')\to R^{1}F(A')}$

can be found, wherein the object functorial of dependent. In addition, there should be an object as “simple” as possible among all candidates; so should apply when is injective. ${\displaystyle R^{1}F(A')}$ ${\displaystyle A'}$ ${\displaystyle R^{1}F(A')}$ ${\displaystyle R^{1}F(A')=0}$ ${\displaystyle A'}$

definition

A sequence of functors for all hot δ- functors if there is any short exact sequence ${\displaystyle G^{*}}$ ${\displaystyle G^{n}\colon C\to D}$ ${\displaystyle n\geq 0}$

${\displaystyle 0\to A'\to A\to A''\to 0}$

natural homomorphisms there, so the long sequence ${\displaystyle \delta ^{n}\colon G^{n}(A'')\to G^{n+1}(A')}$

${\displaystyle 0\to G^{0}(A')\to G^{0}(A)\to G^{0}(A'')\to G^{1}(A')\to G^{1}(A)\to G^{1}(A'')\to G^{2}(A')\to \ldots }$

is exact. Strictly speaking, one should even count those with the data of a δ-functor, which results in a functor from the category of short exact sequences to the category of long exact sequences. ${\displaystyle \delta ^{n}}$

Let be universal among the δ-functors with natural transformation , i.e. H. there is a natural transformation and for everyone who in turn has a natural transformation , uniquely certain natural transformations for all , so that the corresponding long exact sequences are compatible. Then the -th (right-) derived functor of is called . ${\displaystyle R^{*}F}$ ${\displaystyle G^{*}}$ ${\displaystyle F\to G^{0}}$ ${\displaystyle F\to R^{0}F}$ ${\displaystyle G^{*}}$ ${\displaystyle F\to G^{0}}$ ${\displaystyle R^{n}F\to G^{n}}$ ${\displaystyle n}$ ${\displaystyle R^{n}F}$ ${\displaystyle n}$ ${\displaystyle F}$

Existence and calculation

The following applies: If there are enough injective objects, the derived functors exist . ${\displaystyle C}$ ${\displaystyle R^{n}F}$

Sufficient number of injective objects means that there is an injective object and a monomorphism for each object . For each one such is chosen and it applies for the sake of simplicity if it is already injective. ${\displaystyle A\in \operatorname {Ob} (C)}$ ${\displaystyle I_{A}}$ ${\displaystyle A\to I_{A}}$ ${\displaystyle A}$ ${\displaystyle I_{A}}$ ${\displaystyle I_{A}=A}$ ${\displaystyle A}$

Then we can set as well as (see above) for and injectives and then get from the short exact sequence ${\displaystyle R^{0}:=F}$ ${\displaystyle R^{n}F(I):=0}$ ${\displaystyle n>0}$ ${\displaystyle I}$

${\displaystyle 0\to A\to I_{A}\to I_{A}/A\to 0}$

the long exact sequence to be formed

${\displaystyle 0\to F(A)\to F(I_{A})\to F(I_{A}/A)\to R^{1}F(A)\to 0\to R^{1}F(I_{A}/A)\to R^{2}F(A)\to 0\ldots }$,

Which

${\displaystyle R^{1}F(A):=\operatorname {coker} (F(I_{A})\to F(I_{A}/A))}$

such as

${\displaystyle R^{n+1}F(A)\,:=\,R^{n}F(I_{A}/A)}$

suggests.

In order to make all of them functors, one still has to investigate the effect on homomorphisms, where it suffices to consider. If there is a homomorphism, this can be continued (in an ambiguous way!), So that a commutative diagram is obtained ${\displaystyle R^{n}F}$ ${\displaystyle R^{1}F}$ ${\displaystyle f\colon A\to B}$

${\displaystyle {\begin{matrix}0\to &A&\to &I_{A}&\to &I_{A}/A&\to &0\\&\downarrow &&\downarrow &&\downarrow \\0\to &B&\to &I_{B}&\to &I_{B}/B&\to &0\end{matrix}}}$

gets a chart

${\displaystyle {\begin{matrix}0\to &F(A)&\to &F(I_{A})&\to &F(I_{A}/A)&\to &R^{1}F(A)&\to &0\\&\downarrow &&\downarrow &&\downarrow &&\downarrow \\0\to &F(B)&\to &F(I_{B})&\to &F(I_{B}/B)&\to &R^{1}F(B)&\to &0\end{matrix}}}$

induced. The fact that at least the right vertical arrow is unambiguous (and thus indeed defines a functor) can be demonstrated by a hunt for a diagram . Because if the null homomorphism is factored over , i.e. H. You can add a diagonal commutative to the original diagram , and consequently also the second diagram , which in turn results in the zero homomorphism on the right. ${\displaystyle R^{1}F}$ ${\displaystyle f}$ ${\displaystyle I_{A}/A\to I_{B}/B}$ ${\displaystyle I_{B}\to I_{B}/B}$ ${\displaystyle I_{A}/A\to I_{B}}$ ${\displaystyle F(I_{A}/A)\to F(I_{B})}$

Alternatively, one forms an injective resolution of , i. H. an exact sequence ${\displaystyle A}$

${\displaystyle \ldots \to 0\to A\to I^{0}\to I^{1}\to I^{2}\to \ldots }$

with injective objects (e.g. , etc.). One then wins all at once as the -th cohomology of the complex ${\displaystyle I^{n}}$ ${\displaystyle I^{0}:=I_{A}}$ ${\displaystyle I^{1}:=I_{I^{0}/A}}$ ${\displaystyle R^{n}F(A)}$ ${\displaystyle n}$

${\displaystyle F(I^{*})=(\ldots \to 0\to F(I^{0})\to F(I^{1})\to F(I^{2})\to \ldots )}$

with at the -th place, which is why this is probably the most common method in literature. ${\displaystyle F(I^{n})}$ ${\displaystyle n}$

With the snake lemma and the horseshoe lemma one then shows that there is indeed a δ-functor. By further chart hunts one proves that the has universal property. Therefore, the result in particular "essentially" does not depend on the choice of the injective resolution. For the concrete calculation, instead of an injective one, you can even use a resolution using -acyclic objects (i.e. for is already known). It then applies . ${\displaystyle R^{*}F}$ ${\displaystyle R^{*}F}$ ${\displaystyle F}$ ${\displaystyle M^{i}}$ ${\displaystyle R^{n}F(M^{i})=0}$ ${\displaystyle n=1,2,\ldots }$ ${\displaystyle H^{i}(F(M^{*}))\cong R^{i}F(A)}$

Correspondingly, one can calculate left derivatives of right exact functors for categories with a sufficient number of projective objects (ie for each there is a projective and an epimorphism ) via projective resolutions . ${\displaystyle A\in \operatorname {Ob} (C)}$ ${\displaystyle P}$ ${\displaystyle P\to A}$

properties

• More general and only naturally equivalent functors; Equality is a peculiarity of the first construction given above. ${\displaystyle R^{0}F}$ ${\displaystyle F}$
• Is injective, so is for . ${\displaystyle A}$ ${\displaystyle R^{n}F(A)=0}$ ${\displaystyle n\geq 1}$
• If is an exact functor, then the zero functor is for . ${\displaystyle F}$ ${\displaystyle R^{n}F}$ ${\displaystyle n\geq 1}$

Examples

• Ext is the right derivative of the Hom functor .
• Tor is the left derivative of the tensor product .
• Sheaf cohomology is the right derivative of the functor global cuts .
• Group cohomology is the right derivative of the functor invariants .

Individual evidence

1. ↑ Peter Hilton : Lectures in Homological Algebra , American Mathematical Society (1971), ISBN 0821816578 , chap. 3: Properties of derived functors
2. ^ Saunders Mac Lane : Homology , Springer Grundlehren der Mathematischen Wissenschaften Volume 114 (1967), Chapter XII: Derived Functors