Chain complex - Revision history

Vipul: /* Dualizing a chain complex */

2010-12-21T01:40:39Z

Dualizing a chain complex

← Older revision		Revision as of 01:40, 21 December 2010
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	===Dualizing a chain complex===		===Dualizing a chain complex===

	Given a chain complex of <math>R</math>-modules (for some ring <math>R</math>), we can associate, to each <math>R</math>-module, the module of homomorphisms from it to <math>R</math> (this is the dual module). The dual modules will form another chain complex with the directions of the arrows reversed. This is called a '''~~chain cocomplex~~'''. Of course, it can be made into a chain complex by simply negating all the indices.		Given a chain complex of <math>R</math>-modules (for some ring <math>R</math>), we can associate, to each <math>R</math>-module, the module of homomorphisms from it to <math>R</math> (this is the dual module). The dual modules will form another chain complex with the directions of the arrows reversed. This is called a '''cochain complex'''. Of course, it can be made into a chain complex by simply negating all the indices.

	The <math>(-n)^{th}</math> boundary group of the dual complex is termed the <math>n^{th}</math> coboundary group of the original chain complex, and the <math>(-n)^{th}</math> cycle group of the dual complex is termed the <math>n^{th}</math> cocycle group. The quotient of these is termed the <math>n^{th}</math> cohomology group.		The <math>(-n)^{th}</math> boundary group of the dual complex is termed the <math>n^{th}</math> coboundary group of the original chain complex, and the <math>(-n)^{th}</math> cycle group of the dual complex is termed the <math>n^{th}</math> cocycle group. The quotient of these is termed the <math>n^{th}</math> cohomology group.

Vipul: /* Dualizing a chain complex */

2010-12-21T01:40:19Z

Dualizing a chain complex

← Older revision		Revision as of 01:40, 21 December 2010
Line 35:		Line 35:
	===Dualizing a chain complex===		===Dualizing a chain complex===

	Given a chain complex of <math>R</math>-modules (for some ring <math>R</math>), we can associate, to each <math>R</math>-module, the module of homomorphisms from it to <math>R</math> (this is the dual module). The dual ~~modues~~ will form another chain complex with the directions of the arrows reversed. This is called a '''chain cocomplex'''. Of course, it can be made into a chain complex by simply negating all the indices.		Given a chain complex of <math>R</math>-modules (for some ring <math>R</math>), we can associate, to each <math>R</math>-module, the module of homomorphisms from it to <math>R</math> (this is the dual module). The dual modules will form another chain complex with the directions of the arrows reversed. This is called a '''chain cocomplex'''. Of course, it can be made into a chain complex by simply negating all the indices.

	The <math>(-n)^{th}</math> boundary group of the dual complex is termed the <math>n^{th}</math> coboundary group of the original chain complex, and the <math>(-n)^{th}</math> cycle group of the dual complex is termed the <math>n^{th}</math> cocycle group. The quotient of these is termed the <math>n^{th}</math> cohomology group.		The <math>(-n)^{th}</math> boundary group of the dual complex is termed the <math>n^{th}</math> coboundary group of the original chain complex, and the <math>(-n)^{th}</math> cycle group of the dual complex is termed the <math>n^{th}</math> cocycle group. The quotient of these is termed the <math>n^{th}</math> cohomology group.

Vipul: /* Constructs */

2010-12-21T01:39:13Z

Constructs

← Older revision		Revision as of 01:39, 21 December 2010
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	===Cycle groups for a chain complex===		===Cycle groups for a chain complex===

	For a chain complex, the kernel of the map <math>d_n</math> is a subgroup of <math>C_n</math>, and this subgroup is termed the <math>n^{th}</math> '''cycle group'''. Its elements are termed '''cycles'''. These groups are often denoted by <math>~~C_n~~</math>.		For a chain complex, the kernel of the map <math>d_n</math> is a subgroup of <math>C_n</math>, and this subgroup is termed the <math>n^{th}</math> '''cycle group'''. Its elements are termed '''cycles'''. These groups are often denoted by <math>Z_n</math>.

	===Boundary groups for a chain complex===		===Boundary groups for a chain complex===

Vipul: /* Constructs */

2010-12-21T01:38:33Z

Constructs

← Older revision		Revision as of 01:38, 21 December 2010
Line 23:		Line 23:
	===Cycle groups for a chain complex===		===Cycle groups for a chain complex===

	For a chain complex, the kernel of the map <math>d_n</math> is a subgroup of <math>~~H_n~~</math>, and this subgroup is termed the <math>n^{th}</math> '''cycle group'''. Its elements are termed '''cycles'''. These groups are often denoted by <math>C_n</math>.		For a chain complex, the kernel of the map <math>d_n</math> is a subgroup of <math>C_n</math>, and this subgroup is termed the <math>n^{th}</math> '''cycle group'''. Its elements are termed '''cycles'''. These groups are often denoted by <math>C_n</math>.

	===Boundary groups for a chain complex===		===Boundary groups for a chain complex===

	For a chain complex, the image of the map <math>d_{n+1}</math> is a subgroup of <math>~~H_n~~</math>, and this subgroup is termed the <math>n^{th}</math> '''boundary group'''. Its elements are termed '''boundaries'''. These groups are often denoted by <math>B_n</math>.		For a chain complex, the image of the map <math>d_{n+1}</math> is a subgroup of <math>C_n</math>, and this subgroup is termed the <math>n^{th}</math> '''boundary group'''. Its elements are termed '''boundaries'''. These groups are often denoted by <math>B_n</math>.

	===Homology groups for a chain complex===		===Homology groups for a chain complex===

Vipul: 1 revision

2008-05-11T19:40:24Z

1 revision

← Older revision	Revision as of 19:40, 11 May 2008
(No difference)

Vipul at 09:00, 22 May 2007

2007-05-22T09:00:54Z

New page

==Definition==

A '''chain complex''' over an [[Abelian category]] (for instance, the [[module category]] over a ring) is defined as follows.

Note that since any Abelian category can be viewed as a subcategory of the category of Abelian groups, we shall view all the objects as Abelian groups with some additional structure.

===Data===

* A sequence of objects in the Abelian category, aindexed by integers. In other words, for each integer <math>n</math>, an object <math>C_n</math>. These are termed the '''chain groups'''.
* A map from each object to its predecessor. In other words, a map <math>d_n:C_n \to C_{n-1}</math>. These are termed the '''boundary maps'''.

===Conditions===

* The boundary maps are all homomorphisms.
* The composite of boundary maps <math>d_{n-1} \circ d_n</math> is the zero map. In other words, the kernel of <math>d_{n-1}</math> contains the image of <math>d_n</math>.

===Miscellanea===

Note that we use define a chain complex only for positive integer values, or negative integer values. In this case, we can set the remaining groups to be trivial and all the maps to and from them to be trivial.

==Constructs==

===Cycle groups for a chain complex===

For a chain complex, the kernel of the map <math>d_n</math> is a subgroup of <math>H_n</math>, and this subgroup is termed the <math>n^{th}</math> '''cycle group'''. Its elements are termed '''cycles'''. These groups are often denoted by <math>C_n</math>.

===Boundary groups for a chain complex===

For a chain complex, the image of the map <math>d_{n+1}</math> is a subgroup of <math>H_n</math>, and this subgroup is termed the <math>n^{th}</math> '''boundary group'''. Its elements are termed '''boundaries'''. These groups are often denoted by <math>B_n</math>.

===Homology groups for a chain complex===

For a chain complex, the quotient of the <math>n^{th}</math> cycle group by the <math>n^{th}</math> boundary group is termed the <math>n^{th}</math> '''homology group'''. Its elements, viewed as cosets of the boundary group in the cycle group, are termed '''homology classes'''. The homology groups are often denoted as <math>H_n</math>.

===Dualizing a chain complex===

Given a chain complex of <math>R</math>-modules (for some ring <math>R</math>), we can associate, to each <math>R</math>-module, the module of homomorphisms from it to <math>R</math> (this is the dual module). The dual modues will form another chain complex with the directions of the arrows reversed. This is called a '''chain cocomplex'''. Of course, it can be made into a chain complex by simply negating all the indices.

The <math>(-n)^{th}</math> boundary group of the dual complex is termed the <math>n^{th}</math> coboundary group of the original chain complex, and the <math>(-n)^{th}</math> cycle group of the dual complex is termed the <math>n^{th}</math> cocycle group. The quotient of these is termed the <math>n^{th}</math> cohomology group.