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| In [[mathematics]], particularly in [[abstract algebra]] and [[homological algebra]], a '''resolution''' (or '''left resolution'''; dually a '''coresolution''' or '''right resolution'''<ref>{{harvnb|Jacobson|2009|loc=§6.5}} uses ''coresolution'', though ''right resolution'' is more common, as in {{harvnb|Weibel|1994|loc=Chap. 2}}</ref>) is an [[exact sequence]] of [[module (mathematics)|module]]s (or, more generally, of [[Object (category theory)|object]]s in an [[abelian category]]), which is used to describe the structure of a specific module or object of this category. In particular, projective and injective resolutions induce a [[quasi-isomorphism]] between the exact sequence and the module, which may be regarded as a [[Weak equivalence (homotopy theory)|weak equivalence]], with the resolution having nicer properties [[Quasi-isomorphism#Applications|as a space]].<ref>{{nlab|id=projective+resolution|title=projective resolution}}, {{nlab|id=resolution}}</ref>
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| Generally, the objects in the sequence are restricted to have some property ''P'' (for example to be free). Thus one speaks of a ''P resolution'': for example, a '''flat resolution''', a '''free resolution''', an '''injective resolution''', a '''projective resolution'''. The sequence is supposed to be infinite to the left (to the right for a coresolution). However, a '''finite resolution''' is one where only finitely many of the objects in the sequence are [[Zero object|non-zero]].
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| ==Resolutions of modules==
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| ===Definitions===
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| Given a module ''M'' over a ring ''R'', a '''left resolution''' (or simply '''resolution''') of ''M'' is an [[exact sequence]] (possibly infinite) of ''R''-modules
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| :<math>\cdots\overset{d_{n+1}}{\longrightarrow}E_n\overset{d_n}{\longrightarrow}\cdots\overset{d_3}{\longrightarrow}E_2\overset{d_2}{\longrightarrow}E_1\overset{d_1}{\longrightarrow}E_0\overset{\epsilon}{\longrightarrow}M\longrightarrow0,</math>
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| with all the ''E''<sub>''i''</sub> modules over ''R''. The homomorphisms ''d<sub>i</sub>'' 's are called boundary maps. The map ε is called an '''augmentation map'''. For succinctness, the resolution above can be written as
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| :<math>E_\bullet\overset{\epsilon}{\longrightarrow}M\longrightarrow0.</math>
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| The [[dual (category theory)|dual notion]] is that of a '''right resolution''' (or '''coresolution''', or simply '''resolution'''). Specifically, given a module ''M'' over a ring ''R'', a right resolution is a possibly infinite exact sequence of ''R''-modules
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| :<math>0\longrightarrow M\overset{\epsilon}{\longrightarrow}C^0\overset{d^0}{\longrightarrow}C^1\overset{d^1}{\longrightarrow}C^2\overset{d^2}{\longrightarrow}\cdots\overset{d^{n-1}}{\longrightarrow}C^n\overset{d^n}{\longrightarrow}\cdots,</math>
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| where each ''C<sup>i</sup>'' is an ''R''-module (it is common to use superscripts on the objects in the resolution and the maps between them to indicate the dual nature of such a resolution). For succinctness, the resolution above can be written as
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| :<math>0\longrightarrow M\overset{\epsilon}{\longrightarrow}C^\bullet.</math>
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| <br>
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| A (co)resolution is said to be '''finite''' if only finitely many of the modules involved are non-zero. The '''length''' of a finite resolution is the maximum index ''n'' labeling a nonzero module in the finite resolution. | |
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| ===Free, projective, injective, and flat resolutions===
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| In many circumstances conditions are imposed on the modules ''E''<sub>''i''</sub> resolving the given module ''M''. For example, a ''free resolution'' of a module ''M'' is a left resolution in which all the modules ''E''<sub>''i''</sub> are free ''R''-modules. Likewise, ''projective'' and ''flat'' resolutions are left resolutions such that all the ''E''<sub>''i''</sub> are [[projective module|projective]] and [[flat module|flat]] ''R''-modules, respectively. Injective resolutions are ''right'' resolutions whose ''C''<sup>''i''</sup> are all [[injective module]]s. | |
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| Every ''R''-module possesses a free left resolution.<ref>{{harvnb|Jacobson|2009|loc=§6.5}}</ref> [[A fortiori]], every module also admits projective and flat resolutions. The proof idea is to define ''E''<sub>0</sub> to be the free ''R''-module generated by the elements of ''M'', and then ''E''<sub>1</sub> to be the free ''R''-module generated by the elements of the kernel of the natural map ''E''<sub>0</sub> → ''M'' etc. Dually, every ''R''-module possesses an injective resolution. Flat resolutions can be used to compute [[Tor functor]]s.
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| Projective resolution of a module ''M'' is unique up to a [[chain homotopy]], i.e., given two projective resolution ''P''<sub>0</sub> → ''M'' and ''P''<sub>1</sub> → ''M'' of ''M'' there exists a chain homotopy between them.
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| Resolutions are used to define [[homological dimension]]s. The minimal length of a finite projective resolution of a module ''M'' is called its ''[[projective dimension]]'' and denoted pd(''M''). For example, a module has projective dimension zero if and only if it is a projective module. If ''M'' does not admit a finite projective resolution then the projective dimension is infinite. For example, for a commutative [[local ring]] ''R'', the projective dimension is finite if and only if ''R'' is [[regular local ring|regular]] and in this case it coincides with the [[Krull dimension]] of ''R''. Analogously, the [[injective dimension]] id(''M'') and [[flat dimension]] fd(''M'') are defined for modules also.
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| The injective and projective dimensions are used on the category of right ''R'' modules to define a homological dimension for ''R'' called the right [[global dimension]] of ''R''. Similarly, flat dimension is used to define [[weak global dimension]]. The behavior of these dimensions reflects characteristics of the ring. For example, a ring has right global dimension 0 if and only if it is a [[semisimple ring]], and a ring has weak global dimension 0 if and only if it is a [[von Neumann regular ring]].
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| === Graded modules and algebras ===
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| Let ''M'' be a [[graded module]] over a [[graded algebra]], which is generated over a field by its elements of positive degree. Then ''M'' has a free resolution in which the free modules ''E''<sub>''i''</sub> may be graded in such a way that the ''d''<sub>''i''</sub> and ε are [[Graded vector space#Linear maps|graded linear maps]]. Among these graded free resolutions, the '''minimal free resolutions''' are those for which the number of basis elements of each ''E''<sub>''i''</sub> is minimal. The number of basis elements of each ''E''<sub>''i''</sub> and their degrees are the same for all the minimal free resolutions of a graded module.
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| If ''I'' is a [[homogeneous ideal]] in a [[polynomial ring]] over a field, the [[Castelnuovo-Mumford regularity]] of the [[projective algebraic set]] defined by ''I'' is the minimal integer ''r'' such that the degrees of the basis elements of the ''E''<sub>''i''</sub> in a minimal free resolution of ''I'' are all lower than ''r-i''.
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| ===Examples===
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| A classic example of a free resolution is given by the [[Koszul complex]] of a [[regular sequence]] in a [[local ring]] or of a homogeneous regular sequence in a [[graded algebra]] finitely generated over a field.
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| Let ''X'' be an [[aspherical space]], i.e., its [[universal cover]] ''E'' is [[contractible]]. Then every [[singular homology|singular]] (or [[simplicial]]) chain complex of ''E'' is a free resolution of the module '''Z''' not only over the ring '''Z''' but also over the [[group ring]] '''Z''' [''π''<sub>1</sub>(''X'')].
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| ==Resolutions in abelian categories==
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| The definition of resolutions of an object ''M'' in an [[abelian category]] ''A'' is the same as above, but the ''E<sub>i</sub>'' and ''C<sup>i</sup>'' are objects in ''A'', and all maps involved are [[morphism]]s in ''A''.
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| The analogous notion of projective and injective modules are [[projective object|projective]] and [[injective object]]s, and, accordingly, projective and injective resolutions. However, such resolutions need not exist in a general abelian category ''A''. If every object of ''A'' has a projective (resp. injective) resolution, then ''A'' is said to have [[enough projectives]] (resp. [[enough injectives]]). Even if they do exist, such resolutions are often difficult to work with. For example, as pointed out above, every ''R''-module has an injective resolution, but this resolution is not [[functor]]ial, i.e., given a homomorphism ''M'' → ''M' '', together with injective resolutions
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| :<math>0 \rightarrow M \rightarrow I_*, \ \ 0 \rightarrow M' \rightarrow I'_*,</math>
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| there is in general no functorial way of obtaining a map between <math>I_*</math> and <math>I'_*</math>.
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| ==Acyclic resolution ==
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| In many cases one is not really interested in the objects appearing in a resolution, but in the behavior of the resolution with respect to a given [[functor]].
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| Therefore, in many situations, the notion of '''acyclic resolutions''' is used: given a [[left exact functor]] ''F'': ''A'' → ''B'' between two abelian categories, a resolution
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| :<math>0 \rightarrow M \rightarrow E_0 \rightarrow E_1 \rightarrow E_2 \rightarrow \dots</math>
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| of an object ''M'' of ''A'' is called ''F''-acyclic, if the [[derived functor]]s ''R''<sub>''i''</sub>''F''(''E''<sub>''n''</sub>) vanish for all ''i''>0 and ''n''≥0. Dually, a left resolution is acyclic with respect to a right exact functor if its derived functors vanish on the objects of the resolution.
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| For example, given a ''R'' module ''M'', the [[tensor product]] <math>\otimes_R M</math> is a right exact functor '''Mod'''(''R'') → '''Mod'''(''R''). Every flat resolution is acyclic with respect to this functor. A ''flat resolution'' is acyclic for the tensor product by every ''M''. Similarly, resolutions that are acyclic for all the functors '''Hom'''( ⋅ , ''M'') are the projective resolutions and those that are acyclic for the functors '''Hom'''(''M'', ⋅ ) are the injective resolutions.
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| Any injective (projective) resolution is ''F''-acyclic for any left exact (right exact, respectively) functor.
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| The importance of acyclic resolutions lies in the fact that the derived functors ''R''<sub>''i''</sub>''F'' (of a left exact functor, and likewise ''L''<sub>''i''</sub>''F'' of a right exact functor) can be obtained from as the homology of ''F''-acyclic resolutions: given an acyclic resolution <math>E_*</math> of an object ''M'', we have
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| :<math>R_i F(M) = H_i F(E_*),</math>
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| where right hand side is the ''i''-th homology object of the complex <math>F(E_*).</math>
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| This situation applies in many situations. For example, for the [[constant sheaf]] ''R'' on a [[differentiable manifold]] ''M'' can be resolved by the sheaves <math>\mathcal C^*(M)</math> of smooth [[differential form]]s:
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| <math>0 \rightarrow R \subset \mathcal C^0(M) \stackrel d \rightarrow \mathcal C^1(M) \stackrel d \rightarrow \dots \mathcal C^{dim M}(M) \rightarrow 0.</math>
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| The sheaves <math>\mathcal C^*(M)</math> are [[fine sheaf|fine sheaves]], which are known to be acyclic with respect to the [[global section]] functor <math>\Gamma: \mathcal F \mapsto \mathcal F(M)</math>. Therefore, the [[sheaf cohomology]], which is the derived functor of the global section functor Γ is computed as
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| <math>\mathrm H^i(M, \mathbf R) = \mathrm H^i( \mathcal C^*(M)).</math>
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| Similarly [[Godement resolution]]s are acyclic with respect to the global sections functor.
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| ==See also==
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| * [[Resolution (disambiguation)]]
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| * [[Hilbert–Burch theorem]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{Citation | author= Iain T. Adamson | title=Elementary rings and modules | series=University Mathematical Texts | publisher=Oliver and Boyd | year=1972 | isbn=0-05-002192-3 }}
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| *{{Citation | last1=Eisenbud | first1=David | author1-link=David Eisenbud | title=Commutative algebra. With a view toward algebraic geometry | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Graduate Texts in Mathematics]] | isbn=3-540-94268-8 | mr=1322960 | year=1995 | volume=150 | zbl=0819.13001 }}
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| *{{citation
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| | last=Jacobson
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| | first=Nathan
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| | author-link=Nathan Jacobson
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| | title=Basic algebra II
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| | year=2009
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| | edition=Second
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| | publisher=Dover Publications
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| | isbn=978-0-486-47187-7
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| | origyear=1985
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| }}
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| * {{Lang Algebra|edition=3}}
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| * {{Weibel IHA}}
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| [[Category:Homological algebra]]
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| [[Category:Module theory]]
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