Definition:Right Derived Functor

Definition

Let $\mathbf A$ be an abelian category with enough injectives.

Let $\mathbf B$ be an abelian category.

Let $F: \mathbf A \to \mathbf B$ be a left exact functor.

Let $X$ and $Y$ be objects of $\mathbf A$.

Let $f: X \to Y$ be a morphism of $\mathbf A$.

Let $I$ be an arbitrary injective resolution of $X$.

Let $J$ be an arbitrary injective resolution of $Y$.

Let $\tilde f : I \to J$ be a morphism of cochain complexes induced by $f$.

Let $\map F I$ denote the cochain complex defined by applying the functor on cochains induced by $F$ to $I$.

Let $i \in \Z_{\ge 0}$ be a non-negative integer.

Let $\map {H^i} {\map F I}$ denote the $i$-th cohomology of $\map F I$.

The $i$-th right derived functor $\mathrm R^i F : \mathbf A \to \mathbf B$ of $F$ is defined on objects as:

$\mathrm R^i \map F X := \map {H^i} {\map F I}$

This article, or a section of it, needs explaining. In particular: If $\mathrm R^i \map F X$ is just defined the same as $\map {H^i} {\map F I}$, then why define it at all? This article defines a sequence of functors $\mathrm R^i F$ attached to $F$. The definition of the right derived functors of a functor is a central definition in homological algebra and should not be omitted. --Wandynsky (talk) 11:00, 28 July 2021 (UTC) You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

This article, or a section of it, needs explaining. In particular: It is not clear what exactly is being defined here. Do the following lines contribute to the definition? Can't figure out exactly what is what. As has been done here in the above rewrite, the best approach to defining something (and standard $\mathsf{Pr} \infty \mathsf{fWiki}$ style) is: a) Write at the start all the objects that contribute to the definition: "Let... let... let..." b) State the definition in terms of all those objects. Do not use the word "any", it is ambiguous and loose. Tried to fix it. Does it look better now? It's a bit tricky in this case. --Wandynsky (talk) 08:22, 28 July 2021 (UTC) Definite improvement, but some way to go. Further explain templates have been added. Once I understand what this page says, I will be able to try to put it into a form that others on my level (I failed my CSE mathematics) can get to grips with. How straightforward would it be to go to a source work and present the material as presented there? In extremis I may reconcile it with my copy of Freyd, but "derived functor" is in an exercise right at the end, and I'd need to work through the book to understand it, and I've barely cracked it open. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

This article, or a section of it, needs explaining. In particular: Are there in fact two different definitions being set up here? If that is the case, we need two different pages for them. Perhaps transclude one inside the other. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

The $i$-th right derived functor $\mathrm R^i F$ of $F$ is defined on morphisms as follows:

Define $\mathrm R^i \map F f: \mathrm R^i \map F X \to \mathrm R^i \map F Y$ by the induced map $\map {H^i} {\map F {\tilde f} } : \map {H^i} {\map F I} \to \map {H^i} {\map F J}$.

Also see

• Injective Resolution Exists Iff Enough Injectives
• Induced Functor of Cochain Complex is Cochain Complex
• Cochain Morphism Induced by Morphism
• Right Derived Functor is Unique up to Unique Isomorphism
• Definition:Left Derived Functor

Sources

There are no source works cited for this page. Source citations are highly desirable, and mandatory for all definition pages. Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted. To discuss this page in more detail, feel free to use the talk page.