Editing Derived functor (section)

== Examples ==

* If <math>A</math> is an abelian category, then its category of morphisms <math>A^{\{\ast\to\ast\}}</math> is also abelian. The functor <math>\ker: A^{\{\ast\to\ast\}}\to A</math> which maps each morphism to its kernel is left exact. Its right derived functors are 
::<math>R^i(\ker)(f) = \begin{cases} \ker(f) & i=0 \\ \operatorname{coker}(f) & i=1 \\ 0 & i>1\end{cases}</math> 
:Dually the functor <math>\operatorname{coker}</math> is right exact and its left derived functors are 
::<math>L_i(\operatorname{coker})(f)=\begin{cases} \operatorname{coker}(f) & i=0 \\ \ker(f) & i=1 \\ 0 & i>1\end{cases}</math> 
:This is a manifestation of the [[snake lemma]].

=== Homology and cohomology ===

====[[Sheaf cohomology]]====
If <math>X</math> is a [[topological space]], then the category <math>Sh(X)</math> of all [[sheaf (mathematics)|sheaves]] of [[abelian group]]s on <math>X</math> is an abelian category with enough injectives. The functor <math>\Gamma: Sh(X)\to Ab</math> which assigns to each such sheaf <math>\mathcal{F}</math> the group <math>\Gamma(\mathcal{F}) := \mathcal{F}(X)</math> of global sections is left exact, and the right derived functors are the [[sheaf cohomology]] functors, usually written as <math>H^i(X,\mathcal{F})</math>. Slightly more generally: if <math>(X,\mathcal{O}_X)</math> is a [[ringed space]], then the category of all sheaves of <math>\mathcal{O}_X</math>-modules is an abelian category with enough injectives, and we can again construct sheaf cohomology as the right derived functors of the global section functor.

There are various notions of cohomology which are a special case of this:

* '''[[De Rham cohomology]]''' is the sheaf cohomology of the sheaf of [[Locally constant function|locally constant]] <math>\R</math>-valued functions on a [[manifold]]. The De Rham complex is a resolution of this sheaf not by injective sheaves, but by [[fine sheaf|fine sheaves]].
* '''[[Étale cohomology]]''' is another cohomology theory for sheaves over a scheme. It is the right derived functor of the global sections of abelian sheaves on the [[Étale topology|étale site]].

====[[Ext functor]]s====
If <math>R</math> is a [[ring (mathematics)|ring]], then the category of all left [[module (mathematics)|<math>R</math>-modules]] is an abelian category with enough injectives. If <math>A</math> is a fixed left <math>R</math>-module, then the functor <math>\operatorname{Hom}(A,-): R\text{-Mod} \to \mathfrak{Ab}</math> is left exact, and its right derived functors are the [[Ext functor]]s <math>\operatorname{Ext}_R^i(A,-)</math>. Alternatively <math>\operatorname{Ext}_R^i(-,B)</math> can also be obtained as the left derived functor of the right exact functor <math>\operatorname{Hom}_R(-,B): R\text{-Mod} \to \mathfrak{Ab}^{op}</math>.

Various notions of cohomology are special cases of Ext functors and therefore also derived functors.

* '''[[Group cohomology]]''' is the right derived functor of the invariants functor <math>(-)^G : k[G]\text{-Mod}\to k[G]\text{-Mod}</math> which is the same as <math>\operatorname{Hom}_{k[G]}(k,-)</math> (where <math>k</math> is the trivial <math>k[G]</math>-module) and therefore <math>H^i(G,M) = \operatorname{Ext}_{k[G]}^i(k,M)</math>.
* '''[[Lie algebra cohomology]]''' of a [[Lie algebra]] <math>\mathfrak{g}</math> over some commutative ring <math>k</math> is the right derived functor of the invariants functor <math>(-)^{\mathfrak{g}}: \mathfrak{g}\text{-Mod}\to k\text{-Mod}</math> which is the same as <math>\operatorname{Hom}_{U(\mathfrak{g})}(k,-)</math> (where <math>k</math> is again the trivial <math>\mathfrak{g}</math>-module and <math>U(\mathfrak{g})</math> is the [[universal enveloping algebra]] of <math>\mathfrak{g}</math>). Therefore <math>H^i(\mathfrak{g},M) = \operatorname{Ext}_{U(\mathfrak{g})}^i(k,M)</math>.
*'''[[Hochschild cohomology]]''' of some [[associative algebra|<math>k</math>-algebra]] <math>A</math> is the right derived functor of invariants <math>(-)^A: (A,A)\text{-Bimod}\to k\text{-Mod}</math> mapping a [[bimodule]] <math>M</math> to its [[center of a bimodule|center]], also called its set of invariants <math>M^A := Z(M) := \{m\in M \mid \forall a\in A: am=ma\}</math> which is the same as <math>\operatorname{Hom}_{A^e}(A,M)</math> (where <math>A^e:=A\otimes_k A^{op}</math> is the enveloping algebra of <math>A</math> and <math>A</math> is considered an <math>(A,A)</math>-bimodule via the usual left and right multiplication). Therefore <math>HH^i(A,M) = \operatorname{Ext}_{A^e}^i(A,M)</math>:

====[[Tor functor]]s====
The category of left <math>R</math>-modules also has enough projectives. If <math>A</math> is a fixed right <math>R</math>-module, then the [[tensor product]] with <math>A</math> gives a right exact covariant functor <math>A\otimes_R - : R\text{-Mod} \to Ab</math>; The category of modules has enough projectives so that left derived functors always exists. The left derived functors of the tensor functor are the [[Tor functor]]s <math>\operatorname{Tor}_i^R(A,-)</math>. Equivalently <math>\operatorname{Tor}_i^R(-,B)</math> can be defined symmetrically as the left derived functors of <math>-\otimes B</math>. In fact one can combine both definitions and define <math>\operatorname{Tor}_i^R(-,-)</math> as the left derived of <math>-\otimes-: \text{Mod-}R \times R\text{-Mod} \to Ab</math>.

This includes several notions of homology as special cases. This often mirrors the situation with Ext functors and cohomology.

* '''[[Group homology]]''' is the left derived functor of taking coinvariants <math>(-)_G: k[G]\text{-Mod}\to k\text{-Mod}</math> which is the same as <math>k\otimes_{k[G]}-</math>.
* '''[[Lie algebra homology]]''' is the left derived functor of taking coinvariants <math>\mathfrak{g}\text{-Mod}\to k\text{-Mod}, M\mapsto M/[\mathfrak{g},M]</math> which is the same as <math>k\otimes_{U(\mathfrak{g})}-</math>.
* '''[[Hochschild homology]]''' is the left derived functor of taking coinvariants <math>(A,A)\text{-Bimod}\to k\text{-Mod}, M\mapsto M/[A,M]</math> which is the same as <math>A \otimes_{A^e} -</math>.

Instead of taking individual left derived functors one can also take the total derived functor of the tensor functor. This gives rise to the [[derived tensor product]] <math>-\otimes^L-: D(\text{Mod-}R) \times D(R\text{-Mod}) \to D(Ab)</math> where <math>D</math> is the [[derived category]].