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| In [[mathematics]], certain [[functor]]s 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.
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| == Motivation ==
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| It was noted in various quite different settings that a [[short exact sequence]] often gives rise to a "long exact sequence". The concept of derived functors explains and clarifies many of these observations.
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| Suppose we are given a covariant [[left exact functor]] ''F'' : '''A''' → '''B''' between two [[abelian category|abelian categories]] '''A''' and '''B'''. If 0 → ''A'' → ''B'' → ''C'' → 0 is a short exact sequence in '''A''', then applying ''F'' yields the exact sequence 0 → ''F''(''A'') → ''F''(''B'') → ''F''(''C'') and one could ask how to continue this sequence to the right to form a long exact sequence. Strictly speaking, this question is ill-posed, since there are always numerous different ways to continue a given exact sequence to the right. But it turns out that (if '''A''' is "nice" enough) there is one [[canonical form|canonical]] way of doing so, given by the right derived functors of ''F''. For every ''i''≥1, there is a functor ''R<sup>i</sup>F'': '''A''' → '''B''', and the above sequence continues like so: 0 → ''F''(''A'') → ''F''(''B'') → ''F''(''C'') → ''R''<sup>1</sup>''F''(''A'') → ''R''<sup>1</sup>''F''(''B'') → ''R''<sup>1</sup>''F''(''C'') → ''R''<sup>2</sup>''F''(''A'') → ''R''<sup>2</sup>''F''(''B'') → ... . From this we see that ''F'' is an exact functor if and only if ''R''<sup>1</sup>''F'' = 0; so in a sense the right derived functors of ''F'' measure "how far" ''F'' is from being exact.
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| If the object ''A'' in the above short exact sequence is [[injective object|injective]], then the sequence [[Splitting lemma|splits]]. Applying any additive functor to a split sequence results in a split sequence, so in particular ''R''<sup>1</sup>''F''(''A'') = 0. Right derived functors are zero on injectives: this is the motivation for the construction given below.
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| == Construction and first properties ==
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| The crucial assumption we need to make about our abelian category '''A''' is that it has ''enough injectives'', meaning that for every object ''A'' in '''A''' there exists a [[monomorphism]] ''A'' → ''I'' where ''I'' is an [[injective object]] in '''A'''.
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| The right derived functors of the covariant left-exact functor ''F'' : '''A''' → '''B''' are then defined as follows. Start with an object ''X'' of '''A'''. Because there are enough injectives, we can construct a long exact sequence of the form
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| :<math>0\to X\to I^0\to I^1\to I^2\to\cdots</math>
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| where the ''I''<sup> ''i''</sup> are all injective (this is known as an ''injective resolution'' of ''X''). Applying the functor ''F'' to this sequence, and chopping off the first term, we obtain the [[chain complex]]
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| :<math>0\to F(I^0)\to F(I^1) \to F(I^2) \to\cdots</math>
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| Note: this is in general ''not'' an exact sequence anymore. But we can compute its [[homology (mathematics)|homology]] at the ''i''-th spot (the kernel of the map from ''F''(''I''<sup>''i''</sup>) modulo the image of the map to ''F''(''I''<sup>''i''</sup>)); we call the result ''R<sup>i</sup>F''(''X''). Of course, various things have to be checked: the end result does not depend on the given injective resolution of ''X'', and any morphism ''X'' → ''Y'' naturally yields a morphism ''R<sup>i</sup>F''(''X'') → ''R<sup>i</sup>F''(''Y''), so that we indeed obtain a functor. Note that left exactness means that
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| 0 →''F''(''X'') → ''F''(''I''<sup>0</sup>) → ''F''(''I''<sup>1</sup>)
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| is exact, so ''R''<sup>0</sup>''F''(''X'') = ''F''(''X''), so we only get something interesting for ''i''>0.
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| (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<sup>i</sup>F''. Different choices of resolutions yield [[naturally isomorphic]] functors, so in the end the choice doesn't really matter.)
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| The above-mentioned property of turning short exact sequences into long exact sequences is a consequence of the [[snake lemma]]. This tell us that the collection of derived functors is a [[Delta-functor|δ-functor]].
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| If ''X'' is itself injective, then we can choose the injective resolution 0 → ''X'' → ''X'' → 0, and we obtain that ''R<sup>i</sup>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.
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| An equivalent way to compute ''R<sup>i</sup>F''(''X'') is the following: take an injective resolution of ''X'' as above, and let ''K''<sup>''i''</sup> be the image of the map ''I''<sup>''i''-1</sup>→''I<sup>i</sup>'' (for ''i''=0, define ''I''<sup>''i''-1</sup>=0), which is the same as the kernel of ''I''<sup>''i''</sup>→''I''<sup>''i''+1</sup>. Let φ<sub>''i''</sub> : ''I''<sup>''i''-1</sup>→''K''<sup>''i''</sup> be the corresponding surjective map. Then ''R<sup>i</sup>F''(''X'') is the cokernel of ''F''(φ<sub>''i''</sub>).
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| == Variations ==
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| If one starts with a covariant ''right-exact'' functor ''G'', and the category '''A''' has enough projectives (i.e. for every object ''A'' of '''A''' there exists an epimorphism ''P'' → ''A'' where ''P'' is a [[projective module|projective object]]), then one can define analogously the left-derived functors ''L<sub>i</sub>G''. For an object ''X'' of '''A''' we first construct a projective resolution of the form
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| :<math>\cdots\to P_2\to P_1\to P_0 \to X \to 0</math>
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| where the ''P''<sub>''i''</sub> are projective. We apply ''G'' to this sequence, chop off the last term, and compute homology to get ''L<sub>i</sub>G''(''X''). As before, ''L''<sub>0</sub>''G''(''X'') = ''G''(''X'').
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| In this case, the long exact sequence will grow "to the left" rather than to the right:
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| :<math>0\to A \to B \to C \to 0</math>
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| is turned into
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| :<math>\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</math>.
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| Left derived functors are zero on all projective objects.
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| One may also start with a ''contravariant'' left-exact functor ''F''; the resulting right-derived functors are then also contravariant. The short exact sequence
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| :<math>0\to A \to B \to C \to 0</math>
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| is turned into the long exact sequence
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| :<math>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</math>
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| These right derived functors are zero on projectives and are therefore computed via projective resolutions.
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| == Applications ==
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| '''Sheaf cohomology.''' If ''X'' is a [[topological space]], then the category of all [[sheaf (mathematics)|sheaves]] of [[abelian group]]s on ''X'' is an abelian category with enough injectives. The functor which assigns to each such sheaf ''L'' the group ''L''(''X'') of global sections is left exact, and the right derived functors are the [[sheaf cohomology]] functors, usually written as ''H''<sup> ''i''</sup>(''X'',''L''). Slightly more generally: if (''X'', O<sub>''X''</sub>) is a [[ringed space]], then the category of all sheaves of O<sub>''X''</sub>-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.
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| '''[[Étale cohomology]]''' is another cohomology theory for sheaves over a scheme.
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| '''Ext functors.''' If ''R'' is a [[ring (mathematics)|ring]], then the category of all left [[module (mathematics)|''R''-modules]] is an abelian category with enough injectives. If ''A'' is a fixed left ''R''-module, then the functor Hom(''A'',-) is left exact, and its right derived functors are the [[Ext functor]]s Ext<sub>''R''</sub><sup>''i''</sup>(''A'',-).
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| '''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 on the category of left ''R''-modules; its left derivatives are the [[Tor functor]]s Tor<sup>''R''</sup><sub>''i''</sub>(''A'',-).
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| '''Group cohomology.''' Let ''G'' be a [[group (mathematics)|group]]. A [[G-module|''G''-module]] ''M'' is an [[abelian group]] ''M'' together with a [[group action]] of ''G'' on ''M'' as a group of automorphisms. This is the same as a [[module (mathematics)|module]] over the [[group ring]] '''Z'''''G''. The ''G''-modules form an abelian category with enough injectives. We write ''M''<sup>''G''</sup> for the subgroup of ''M'' consisting of all elements of ''M'' that are held fixed by ''G''. This is a left-exact functor, and its right derived functors are the [[group cohomology]] functors, typically written as H<sup> ''i''</sup>(''G'',''M'').
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| == Naturality ==
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| Derived functors and the long exact sequences are "natural" in several technical senses.
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| First, given a [[commutative diagram]] of the form
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| <math>\begin{array}{ccccccccc} 0&\xrightarrow{}&A_1&\xrightarrow{f_1}&B_1&\xrightarrow{g_1}&C_1&\xrightarrow{}&0\\ &&\alpha\downarrow\quad&&\beta\downarrow\quad&&\gamma\downarrow\quad&&\\ 0&\xrightarrow{}&A_2&\xrightarrow{f_2}&B_2&\xrightarrow{g_2}&C_2&\xrightarrow{}&0 \end{array}</math>
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| (where the rows are exact), the two resulting long exact sequences are related by commuting squares:
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| [[Image:two long exact sequences.png]]
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| Second, suppose η : ''F'' → ''G'' is a [[natural transformation]] from the left exact functor ''F'' to the left exact functor ''G''. Then natural transformations ''R<sup>i</sup>''η : ''R<sup>i</sup>F'' → ''R<sup>i</sup>G'' are induced, and indeed ''R<sup>i</sup>'' becomes a functor from the [[functor category]] of all left exact functors from '''A''' to '''B''' to the full functor category of all functors from '''A''' to '''B'''. Furthermore, this functor is compatible with the long exact sequences in the following sense: if
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| :<math>0\xrightarrow{}A\xrightarrow{f}B\xrightarrow{g}C\xrightarrow{} 0</math>
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| is a short exact sequence, then a commutative diagram
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| [[Image:two long exact sequences2.png]]
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| is induced.
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| Both of these naturalities follow from the naturality of the sequence provided by the [[snake lemma]].
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| Conversely, the following characterization of derived functors holds: given a family of functors ''R''<sup>''i''</sup>: '''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''<sup>''i''</sup>(''I'')=0 for every positive ''i'', then these functors are the right derived functors of ''R''<sup>0</sup>.
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| == Generalization ==
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| The more modern (and more general) approach to derived functors uses the language of [[derived category|derived categories]].
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| == References==
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| * {{Citation | last1=Manin | first1=Yuri Ivanovich | author1-link= Yuri Ivanovich Manin | last2=Gelfand | first2=Sergei I. | title=Methods of Homological Algebra | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-43583-9 | year=2003}}
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| * {{Weibel IHA}}
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| {{Functors}}
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| [[Category:Homological algebra]]
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| [[Category:Functors]]
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