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  "slug": "derived-functor",
  "title": "Derived functor",
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  "lead": "In mathematics, specifically category theory, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.",
  "body": "In mathematics, specifically category theory, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. From this we see that F is an exact functor if and only if R^1F=0 ; so in a sense the right derived functors of F measure \"how far\" F is from being exact. If the object A in the above short exact sequence is injective, then the sequence splits. Applying any additive functor to a split sequence results in a split sequence, so in particular R^1F(A)=0 . Right derived functors (for i>0 ) are zero on injectives: this is the motivation for the construction given below. is exact, so R 0 F(X) = F(X), so we only get something interesting for i>0. (Technically, to produce well-defined derivatives of F, we would have to fix an injective resolution for every object of A. This choice of injective resolutions then yields functors R i F. Different choices of resolutions yield naturally isomorphic functors, so in the end the choice doesn't really matter.) The above-mentioned property of turning short exact sequences into long exact sequences is a consequence of the snake lemma. This tells us that the collection of derived functors is a δ-functor. If X is itself injective, then we can choose the injective resolution 0 → X → X → 0, and we obtain that R i F(X) = 0 for all i ≥ 1. In practice, this fact, together with the long exact sequence property, is often used to compute the values of right derived functors. An equivalent way to compute R i F(X) is the following: take an injective resolution of X as above, and let K i be the image of the map I i-1 →I i (for i=0, define I i-1 =0), which is the same as the kernel of I i →I i+1 . Let φ i  : I i-1 →K i be the corresponding surjective map. Then R i F(X) is the cokernel of F(φ i ). where the P_i are projective. We apply G to this sequence, chop off the last term, and compute homology to get L_iG(X) . As before, L_0G(X)=G(X) . In this case, the long exact sequence will grow \"to the left\" rather than to the right: 0\\to A \\to B \\to C \\to 0 is turned into \\cdots\\to L_2G(C) \\to L_1G(A) \\to L_1G(B)\\to L_1G(C)\\to G(A)\\to G(B)\\to G(C)\\to 0 . Left derived functors are zero on all projective objects. One may also start with a contravariant left-exact functor F ; the resulting right-derived functors are then also contravariant. The short exact sequence 0\\to A \\to B \\to C \\to 0 is turned into the long exact sequence 0\\to F(C)\\to F(B)\\to F(A)\\to R^1F(C) \\to R^1F(B) \\to R^1F(A)\\to R^2F(C)\\to \\cdots These right derived functors are zero on projectives and are therefore computed via projective resolutions. R^i(\\ker)(f) = \\begin{cases} \\ker(f) & i=0 \\\\ \\operatorname{coker}(f) & i=1 \\\\ 0 & i>1\\end{cases} Dually the functor \\operatorname{coker} is right exact and its left derived functors are L_i(\\operatorname{coker})(f)=\\begin{cases} \\operatorname{coker}(f) & i=0 \\\\ \\ker(f) & i=1 \\\\ 0 & i>1\\end{cases} This is a manifestation of the snake lemma. ====Sheaf cohomology==== If X is a topological space, then the category Sh(X) of all sheaves of abelian groups on X is an abelian category with enough injectives. The functor \\Gamma: Sh(X)\\to Ab which assigns to each such sheaf \\mathcal{F} the group \\Gamma(\\mathcal{F}) := \\mathcal{F}(X) of global sections is left exact, and the right derived functors are the sheaf cohomology functors, usually written as H^i(X,\\mathcal{F}) . Slightly more generally: if (X,\\mathcal{O}_X) is a ringed space, then the category of all sheaves of \\mathcal{O}_X -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 \\R -valued functions on a manifold. The De Rham complex is a resolution of this sheaf not by injective sheaves, but by 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 site. ====Ext functors==== If R is a ring, then the category of all left R -modules is an abelian category with enough injectives. If A is a fixed left R -module, then the functor \\operatorname{Hom}(A,-): R\\text{-Mod} \\to \\mathfrak{Ab} is left exact, and its right derived functors are the Ext functors \\operatorname{Ext}_R^i(A,-) . Alternatively \\operatorname{Ext}_R^i(-,B) can also be obtained as the left derived functor of the right exact functor \\operatorname{Hom}_R(-,B): R\\text{-Mod} \\to \\mathfrak{Ab}^{op} . 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 (-)^G : k[G]\\text{-Mod}\\to k[G]\\text{-Mod} which is the same as \\operatorname{Hom}_{k[G]}(k,-) (where k is the trivial k[G] -module) and therefore H^i(G,M) = \\operatorname{Ext}_{k[G]}^i(k,M) . Lie algebra cohomology of a Lie algebra \\mathfrak{g} over some commutative ring k is the right derived functor of the invariants functor (-)^{\\mathfrak{g}}: \\mathfrak{g}\\text{-Mod}\\to k\\text{-Mod} which is the same as \\operatorname{Hom}_{U(\\mathfrak{g})}(k,-) (where k is again the trivial \\mathfrak{g} -module and U(\\mathfrak{g}) is the universal enveloping algebra of \\mathfrak{g} ). Therefore H^i(\\mathfrak{g},M) = \\operatorname{Ext}_{U(\\mathfrak{g})}^i(k,M) . Hochschild cohomology of some k -algebra A is the right derived functor of invariants (-)^A: (A,A)\\text{-Bimod}\\to k\\text{-Mod} mapping a bimodule M to its center, also called its set of invariants M^A := Z(M) := \\{m\\in M \\mid \\forall a\\in A: am=ma\\} which is the same as \\operatorname{Hom}_{A^e}(A,M) (where A^e:=A\\otimes_k A^{op} is the enveloping algebra of A and A is considered an (A,A) -bimodule via the usual left and right multiplication). Therefore HH^i(A,M) = \\operatorname{Ext}_{A^e}^i(A,M) : ====Tor functors==== The category of left R -modules also has enough projectives. If A is a fixed right R -module, then the tensor product with A gives a right exact covariant functor A\\otimes_R - : R\\text{-Mod} \\to Ab . The left derived functors of the tensor functor are the Tor functors \\operatorname{Tor}_i^R(A,-) . Equivalently \\operatorname{Tor}_i^R(-,B) can be defined symmetrically as the left derived functors of -\\otimes B . In fact one can combine both definitions and define \\operatorname{Tor}_i^R(-,-) as the left derived of -\\otimes-: \\text{Mod-}R \\times R\\text{-Mod} \\to Ab . 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 (-)_G: k[G]\\text{-Mod}\\to k\\text{-Mod} which is the same as k\\otimes_{k[G]}- . Lie algebra homology is the left derived functor of taking coinvariants \\mathfrak{g}\\text{-Mod}\\to k\\text{-Mod}, M\\mapsto M/[\\mathfrak{g},M] which is the same as k\\otimes_{U(\\mathfrak{g})}- . Hochschild homology is the left derived functor of taking coinvariants (A,A)\\text{-Bimod}\\to k\\text{-Mod}, M\\mapsto M/[A,M] which is the same as A \\otimes_{A^e} - . 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 -\\otimes^L-: D(\\text{Mod-}R) \\times D(R\\text{-Mod}) \\to D(Ab) where D is the derived category. 0\\to A\\xrightarrow{f} B\\xrightarrow{g} C\\to 0 is a short exact sequence, then a commutative diagram is induced. Both of these naturalities follow from the naturality of the sequence provided by the snake lemma. Conversely, the following characterization of derived functors holds: given a family of functors R i : A → B, satisfying the above, i.e. mapping short exact sequences to long exact sequences, such that for every injective object I of A, R i (I)=0 for every positive i, then these functors are the right derived functors of R 0 .",
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    "title": "Derived functor",
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