Resolution (algebra) - Revision history

Bm319 at 18:21, 30 January 2024

2024-01-30T18:21:48Z

Нейроромант: /* Acyclic resolution */

2023-04-27T08:35:11Z

Acyclic resolution

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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:		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:

	: <math>0 \rightarrow R \subset \mathcal C^0(M) \stackrel d \rightarrow \mathcal C^1(M) \stackrel d \rightarrow \cdots \mathcal C^{\dim M}(M) \rightarrow 0.</math>		: <math>0 \rightarrow R \subset \mathcal C^0(M) \stackrel d \rightarrow \mathcal C^1(M) \stackrel d \rightarrow \cdots \stackrel d \rightarrow \mathcal C^{\dim M}(M) \rightarrow 0.</math>

	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		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

1234qwer1234qwer4: /* Resolutions in abelian categories */ inconsistent indentation

2022-05-29T10:55:07Z

Resolutions in abelian categories: inconsistent indentation

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	=== Abelian categories without projective resolutions in general ===		=== Abelian categories without projective resolutions in general ===
	One class of examples of Abelian categories without projective resolutions are the categories <math>\text{Coh}(X)</math> of [[Coherent sheaf\|coherent sheaves]] on a [[Scheme (mathematics)\|scheme]] <math>X</math>. For example, if <math>X = \mathbb{P}^n_S</math> is projective space, any coherent sheaf <math>\mathcal{M}</math> on <math>X</math> has a presentation given by an exact sequence~~<blockquote>~~<math>\bigoplus_{i,j=0} \mathcal{O}_X(s_{i,j}) \to \bigoplus_{i=0} \mathcal{O}_X(s_i) \to \mathcal{M} \to 0</math>~~</blockquote>~~The first two terms are not in general projective since <math>H^n(\mathbb{P}^n_S,\mathcal{O}_X(s)) \neq 0</math> for <math>s > 0</math>. But, both terms are locally free, and locally flat. Both classes of sheaves can be used in place for certain computations, replacing projective resolutions for computing some derived functors.		One class of examples of Abelian categories without projective resolutions are the categories <math>\text{Coh}(X)</math> of [[Coherent sheaf\|coherent sheaves]] on a [[Scheme (mathematics)\|scheme]] <math>X</math>. For example, if <math>X = \mathbb{P}^n_S</math> is projective space, any coherent sheaf <math>\mathcal{M}</math> on <math>X</math> has a presentation given by an exact sequence
			:<math>\bigoplus_{i,j=0} \mathcal{O}_X(s_{i,j}) \to \bigoplus_{i=0} \mathcal{O}_X(s_i) \to \mathcal{M} \to 0.</math>
			The first two terms are not in general projective since <math>H^n(\mathbb{P}^n_S,\mathcal{O}_X(s)) \neq 0</math> for <math>s > 0</math>. But, both terms are locally free, and locally flat. Both classes of sheaves can be used in place for certain computations, replacing projective resolutions for computing some derived functors.

	==Acyclic resolution ==		==Acyclic resolution ==

TakuyaMurata: /* See also */ lk

2022-02-24T05:47:06Z

Kaptain-k-theory: /* Resolutions in abelian categories */

2021-06-17T21:47:33Z

Resolutions in abelian categories

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	:<math>0 \rightarrow M \rightarrow I_, \ \ 0 \rightarrow M' \rightarrow I'_,</math>		:<math>0 \rightarrow M \rightarrow I_, \ \ 0 \rightarrow M' \rightarrow I'_,</math>
	there is in general no functorial way of obtaining a map between <math>I_</math> and <math>I'_</math>.		there is in general no functorial way of obtaining a map between <math>I_</math> and <math>I'_</math>.

			=== Abelian categories without projective resolutions in general ===
			One class of examples of Abelian categories without projective resolutions are the categories <math>\text{Coh}(X)</math> of [[Coherent sheaf\|coherent sheaves]] on a [[Scheme (mathematics)\|scheme]] <math>X</math>. For example, if <math>X = \mathbb{P}^n_S</math> is projective space, any coherent sheaf <math>\mathcal{M}</math> on <math>X</math> has a presentation given by an exact sequence<blockquote><math>\bigoplus_{i,j=0} \mathcal{O}_X(s_{i,j}) \to \bigoplus_{i=0} \mathcal{O}_X(s_i) \to \mathcal{M} \to 0</math></blockquote>The first two terms are not in general projective since <math>H^n(\mathbb{P}^n_S,\mathcal{O}_X(s)) \neq 0</math> for <math>s > 0</math>. But, both terms are locally free, and locally flat. Both classes of sheaves can be used in place for certain computations, replacing projective resolutions for computing some derived functors.

	==Acyclic resolution ==		==Acyclic resolution ==

Monkbot: Task 18 (cosmetic): eval 3 templates: hyphenate params (1×);

2021-01-13T13:51:23Z

Task 18 (cosmetic): eval 3 templates: hyphenate params (1×);

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	\| publisher=Dover Publications		\| publisher=Dover Publications
	\| isbn=978-0-486-47187-7		\| isbn=978-0-486-47187-7
	\| ~~origyear~~=1985		\| orig-year=1985
	}}		}}
	* {{Lang Algebra\|edition=3}}		* {{Lang Algebra\|edition=3}}

D.Lazard: Undid revision 888960145 by 2601:445:437F:FE66:87A:A686:FB9C:2CBB (talk) non useful

2019-03-22T14:42:01Z

Undid revision 888960145 by 2601:445:437F:FE66:87A:A686:FB9C:2CBB (talk) non useful

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	===Definitions===		===Definitions===
	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		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

	:<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{\varepsilon}{\longrightarrow}M\longrightarrow0.</math>		:<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{\varepsilon}{\longrightarrow}M\longrightarrow0.</math>

	The homomorphisms ''d<sub>i</sub>'' are called boundary maps. The map ε is called an '''augmentation map'''. For succinctness, the resolution above can be written as		The homomorphisms ''d<sub>i</sub>'' are called boundary maps. The map ε is called an '''augmentation map'''. For succinctness, the resolution above can be written as

	:<math>E_\bullet\overset{\varepsilon}{\longrightarrow}M\longrightarrow0.</math>		:<math>E_\bullet\overset{\varepsilon}{\longrightarrow}M\longrightarrow0.</math>

	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		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

	:<math>0\longrightarrow M\overset{\varepsilon}{\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>		:<math>0\longrightarrow M\overset{\varepsilon}{\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>

	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		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

	:<math>0\longrightarrow M\overset{\varepsilon}{\longrightarrow}C^\bullet.</math>		:<math>0\longrightarrow M\overset{\varepsilon}{\longrightarrow}C^\bullet.</math>

2601:445:437F:FE66:87A:A686:FB9C:2CBB: /* Acyclic resolution */

2019-03-22T14:31:43Z

Acyclic resolution

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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		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
	:<math>0 \rightarrow M \rightarrow E_0 \rightarrow E_1 \rightarrow E_2 \rightarrow \cdots</math>		:<math>0 \rightarrow M \rightarrow E_0 \rightarrow E_1 \rightarrow E_2 \rightarrow \cdots</math>
	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.		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.

	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.		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.

2601:445:437F:FE66:87A:A686:FB9C:2CBB: /* Definitions */

2019-03-22T14:30:21Z

Definitions

← Previous revision		Revision as of 14:30, 22 March 2019
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	===Definitions===		===Definitions===
	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		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

	:<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{\varepsilon}{\longrightarrow}M\longrightarrow0.</math>		:<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{\varepsilon}{\longrightarrow}M\longrightarrow0.</math>

	The homomorphisms ''d<sub>i</sub>'' are called boundary maps. The map ε is called an '''augmentation map'''. For succinctness, the resolution above can be written as		The homomorphisms ''d<sub>i</sub>'' are called boundary maps. The map ε is called an '''augmentation map'''. For succinctness, the resolution above can be written as

	:<math>E_\bullet\overset{\varepsilon}{\longrightarrow}M\longrightarrow0.</math>		:<math>E_\bullet\overset{\varepsilon}{\longrightarrow}M\longrightarrow0.</math>

	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		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

	:<math>0\longrightarrow M\overset{\varepsilon}{\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>		:<math>0\longrightarrow M\overset{\varepsilon}{\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>

	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		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

	:<math>0\longrightarrow M\overset{\varepsilon}{\longrightarrow}C^\bullet.</math>		:<math>0\longrightarrow M\overset{\varepsilon}{\longrightarrow}C^\bullet.</math>

2601:445:437F:FE66:1932:AB5D:38EF:41E2: /* Acyclic resolution */

2018-05-17T22:41:39Z

Acyclic resolution

← Previous revision		Revision as of 22:41, 17 May 2018
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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:		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:

	<math>0 \rightarrow R \subset \mathcal C^0(M) \stackrel d \rightarrow \mathcal C^1(M) \stackrel d \rightarrow \cdots \mathcal C^{\dim M}(M) \rightarrow 0.</math>		: <math>0 \rightarrow R \subset \mathcal C^0(M) \stackrel d \rightarrow \mathcal C^1(M) \stackrel d \rightarrow \cdots \mathcal C^{\dim M}(M) \rightarrow 0.</math>

	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		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
	<math>\mathrm H^i(M, \mathbf R) = \mathrm H^i( \mathcal C^*(M)).</math>		<math>\mathrm H^i(M, \mathbf R) = \mathrm H^i( \mathcal C^*(M)).</math>

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	* [[Hilbert's syzygy theorem]]		* [[Hilbert's syzygy theorem]]
	* [[Free presentation]]		* [[Free presentation]]
			* [[Matrix factorizations (algebra)]]

	==Notes==		==Notes==

← Previous revision		Revision as of 18:21, 30 January 2024
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	In [[mathematics]], and more specifically in [[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 of an [[abelian category]]), ~~which~~ is used to define [[invariant (mathematics)\|invariants]] characterizing the structure of a specific module or object of this category. When, as usually, arrows are oriented to the right, the sequence is supposed to be infinite to the left for (left) resolutions, and to the right for right resolutions. However, a '''finite resolution''' is one where only finitely many of the objects in the sequence are [[Zero object\|non-zero]]; it is usually represented by a finite exact sequence in which the leftmost object (for resolutions) or the rightmost object (for coresolutions) is the [[zero-object]].<ref>{{nlab\|id=projective+resolution\|title=projective resolution}}, {{nlab\|id=resolution}}</ref>		In [[mathematics]], and more specifically in [[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 of an [[abelian category]]) that is used to define [[invariant (mathematics)\|invariants]] characterizing the structure of a specific module or object of this category. When, as usually, arrows are oriented to the right, the sequence is supposed to be infinite to the left for (left) resolutions, and to the right for right resolutions. However, a '''finite resolution''' is one where only finitely many of the objects in the sequence are [[Zero object\|non-zero]]; it is usually represented by a finite exact sequence in which the leftmost object (for resolutions) or the rightmost object (for coresolutions) is the [[zero-object]].<ref>{{nlab\|id=projective+resolution\|title=projective resolution}}, {{nlab\|id=resolution}}</ref>

	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''. In particular, every module has '''free resolutions''', '''projective resolutions''' and '''flat resolutions''', which are left resolutions consisting, respectively of [[free module]]s, [[projective module]]s or [[flat module]]s. Similarly every module has '''injective resolutions''', which are right resolutions consisting of [[injective module]]s.		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''. In particular, every module has '''free resolutions''', '''projective resolutions''' and '''flat resolutions''', which are left resolutions consisting, respectively of [[free module]]s, [[projective module]]s or [[flat module]]s. Similarly every module has '''injective resolutions''', which are right resolutions consisting of [[injective module]]s.
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	===Definitions===		===Definitions===
	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		Given a module ''M'' over a [[ring (mathematics)\|ring]] ''R'', a '''left resolution''' (or simply '''resolution''') of ''M'' is an [[exact sequence]] (possibly infinite) of ''R''-modules
	:<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{\varepsilon}{\longrightarrow}M\longrightarrow0.</math>		:<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{\varepsilon}{\longrightarrow}M\longrightarrow0.</math>
	The homomorphisms ''d<sub>i</sub>'' are called boundary maps. The map ε is called an '''augmentation map'''. For succinctness, the resolution above can be written as		The homomorphisms ''d<sub>i</sub>'' are called boundary maps. The map ''ε'' is called an '''augmentation map'''. For succinctness, the resolution above can be written as
	:<math>E_\bullet\overset{\varepsilon}{\longrightarrow}M\longrightarrow0.</math>		:<math>E_\bullet\overset{\varepsilon}{\longrightarrow}M\longrightarrow0.</math>

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	Resolutions are used to define [[homological dimension (disambiguation)\|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.		Resolutions are used to define [[homological dimension (disambiguation)\|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.

	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]].		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]].

	=== Graded modules and algebras ===		=== Graded modules and algebras ===
	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.		Let ''M'' be a [[graded module]] over a [[graded algebra]], which is generated over a [[field (mathematics)\|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.

	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''.		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''.

	===Examples===		===Examples===
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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.		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.

	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.		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.

	Any injective (projective) resolution is ''F''-acyclic for any left exact (right exact, respectively) functor.		Any injective (projective) resolution is ''F''-acyclic for any left exact (right exact, respectively) functor.