Derived functor

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

For the rest of this article and are abelian categories and a covariant left exact functor. In the case of a contravariant and / or right exact functor applies the corresponding, where appropriate, to turn arrows are injective and be replaced by projective objects.

Motivation

Is

Although precise, as is the corresponding sequence

Exactly, but generally not the continuation by.

In principle, one could, although the sequence - so the cokernel is finally defined - continue through exactly, but this sequel would then depend on homomorphism. One would like to have a dependence only on the objects.

The fact that one of the objects involved can already severely limit the deviation of the precision, can be seen for example in the case that is an injective object. Then arises, namely, that the original sequence splits and is isomorphic. This carries over to the image sequence, which therefore also is a short exact sequence in this case.

In this respect, it seems likely that ( at least under suitable additional conditions ) is generally an exact sequence

Can see, the object being functorial depends on. In addition, among all the candidates should be a possible " simple" object; so should be approximately valid if is injective.

Definition

A sequence of functors for all hot δ - functor, if for every short exact sequence

Are natural homomorphism, so that a long sequence

Is exact. Strictly speaking, one should even count to the data of a δ - functor, which results in a total functor from the category of short exact sequences in the category of long exact sequences.

Be among the universal δ - functors with natural transformation, ie there is a natural transformation, and to each, in turn, has a natural transformation, clearly certain natural transformations for all, so that the corresponding long exact sequences are compatible. Then called the -th (right ) derived functor of.

Existence and calculation

It is: Do enough many injective objects, then there are the derived functors.

Here, sufficient means many injective objects that, for every object is an injective object and a monomorphism. It should at all any such set chosen and suppose for simplicity, if already is injective.

Then we can set as well (see above) and for injective and then get out of the short exact sequence

To forming long exact sequence

Which

As well as

Suggests.

To make all to functors, one the effect on homomorphisms must still investigate, where it is sufficient to look at. Is a homomorphism, so can this continue ( not in a unique way! ), So that one is a commutative diagram

Receives, which chart a

Induced. The fact that in this case at least, the right vertical arrow is clearly (and thus a functor defined in the Act), we can prove by diagram chase. Because if the Nullhomomorphismus is factored over, ie you, the original diagram by a diagonal complement commutative, consequently, also to the second chart, which in turn results in the right Nullhomomorphismus.

Alternatively, one forms an injective resolution, that is, an exact sequence

Injective with objects (for example, etc.). One then recovers all in one fell swoop as the - th cohomology of the complex

With th of the place, which is why this is probably the most widely used method in the literature.

One can now prove by further chart hunts, that in fact is a δ - functor and that it has the universal property. Therefore, the result is particularly "substantially" does not depend on the choice of injective resolution. For the concrete calculation you can even use a resolution by - acyclic objects instead of injective (that is, for it is already known ). It then applies.

Accordingly, one can leftmost derivations quite exact functors for categories with sufficiently many projective objects (ie, for every there exists a projective and an epimorphism ) via projective resolutions calculated.

Properties

• Are more general and only naturally equivalent functors; Equality is a feature of the first above-mentioned construction.
• Is injective, then for.
• Is an exact functor, so is the Nullfunktor for.

Examples

• Ext is the right derivative of the Hom - functor.
• Gate is the left derivative of the tensor product.
• Garbenkohomologie is the right derivative of the functor global sections.
• Gruppenkohomologie is the right derivative of the functor invariants.