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Difference between revisions of "Homology of a complex"

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The starting point for various homological constructions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h0478101.png" /> be an Abelian category and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h0478102.png" /> be a chain complex in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h0478103.png" />, i.e. a family of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h0478104.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h0478105.png" /> and morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h0478106.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h0478107.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h0478108.png" />. The quotient object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h0478109.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h04781010.png" />-th homology of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h04781011.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h04781012.png" />. The family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h04781013.png" /> is also denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h04781014.png" />. The concept of the homology of a complex serves as the base for a number of important constructions in homological algebra, commutative algebra, algebraic geometry, and topology. Thus, in topology, each topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h04781015.png" /> defines a chain complex in the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h04781016.png" /> of Abelian groups: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h04781017.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h04781018.png" /> is the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h04781019.png" />-dimensional singular chains of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h04781020.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h04781021.png" /> is the boundary homomorphism. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h04781022.png" />-th homology of this complex is said to be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h04781023.png" />-th singular homology group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h04781024.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047810/h04781025.png" />. The concept of the cohomology of a cochain complex is defined in a dual manner.
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The starting point for various homological constructions. Let $A$ be an Abelian category. A '''chain complex''' in $A$ is a family $K_\bullet = (K_n,d_n)$ of objects $(K_n)_{n \in \mathbf{Z}}$ in $A$ and morphisms $d_n : K_n \rightarrow K_{n-1}$ such that $d_{n-1} \circ d_n = 0$ for all $n$. The quotient object $\ker d_n / \text{im} d_{n+1}$ is called the $n$-th homology of the complex $K_\bullet$ and is denoted by $H_n(K_\bullet)$. The family $(H_n(K_\bullet))_{n \in \mathbf{Z}}$ is also denoted by $H_\bullet(K_\bullet)$. The concept of the homology of a complex serves as the base for a number of important constructions in homological algebra, commutative algebra, algebraic geometry, and topology. Thus, in topology, each topological space $X$ defines a chain complex in the category $\textsf{Ab}$ of Abelian groups: $(C_n(X),\partial_n)$. Here $C_n(X)$ is the group of $n$-dimensional [[singular chain]]s of $X$, while $\partial_n$ is the boundary homomorphism. The $n$-th homology of this complex is said to be the $n$-th [[singular homology]] group of $X$ and is denoted by $H_n(X)$.  
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The concept of the cohomology of a cochain complex is defined in a dual manner.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. MacLane, "Homology" , Springer (1963) {{MR|}} {{ZBL|0818.18001}} {{ZBL|0328.18009}} </TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top"> S. MacLane, "Homology" , Springer (1963) {{MR|}} {{ZBL|0818.18001}} {{ZBL|0328.18009}} </TD></TR>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) {{MR|0210112}} {{MR|1325242}} {{ZBL|0145.43303}} </TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top"> E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) {{MR|0210112}} {{MR|1325242}} {{ZBL|0145.43303}} </TD></TR>
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</table>
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{{TEX|done}}

Revision as of 21:45, 22 October 2016

The starting point for various homological constructions. Let $A$ be an Abelian category. A chain complex in $A$ is a family $K_\bullet = (K_n,d_n)$ of objects $(K_n)_{n \in \mathbf{Z}}$ in $A$ and morphisms $d_n : K_n \rightarrow K_{n-1}$ such that $d_{n-1} \circ d_n = 0$ for all $n$. The quotient object $\ker d_n / \text{im} d_{n+1}$ is called the $n$-th homology of the complex $K_\bullet$ and is denoted by $H_n(K_\bullet)$. The family $(H_n(K_\bullet))_{n \in \mathbf{Z}}$ is also denoted by $H_\bullet(K_\bullet)$. The concept of the homology of a complex serves as the base for a number of important constructions in homological algebra, commutative algebra, algebraic geometry, and topology. Thus, in topology, each topological space $X$ defines a chain complex in the category $\textsf{Ab}$ of Abelian groups: $(C_n(X),\partial_n)$. Here $C_n(X)$ is the group of $n$-dimensional singular chains of $X$, while $\partial_n$ is the boundary homomorphism. The $n$-th homology of this complex is said to be the $n$-th singular homology group of $X$ and is denoted by $H_n(X)$.

The concept of the cohomology of a cochain complex is defined in a dual manner.

References

[1] S. MacLane, "Homology" , Springer (1963) Zbl 0818.18001 Zbl 0328.18009


Comments

References

[a1] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) MR0210112 MR1325242 Zbl 0145.43303
How to Cite This Entry:
Homology of a complex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homology_of_a_complex&oldid=39499
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article