| Version 12 |
Version 11 |
| \PMlinkescapeword{exact} |
\PMlinkescapeword{exact} |
| \PMlinkescapeword{objects} |
\PMlinkescapeword{objects} |
|
|
| If $(\mathbf{A},d)$ is a chain complex |
If $(\mathbf{A},d)$ is a chain complex |
| $$\begin{CD} |
$$\begin{CD} |
| \cdots@<{d_{n-1}}<<A_{n-1} @<{d_{n}}<< A_n@<{d_{n+1}}<< A_{n+1} @<{d_{n+2}}<<\cdots |
\cdots@<{d_{n-1}}<<A_{n-1} @<{d_{n}}<< A_n@<{d_{n+1}}<< A_{n+1} @<{d_{n+2}}<<\cdots |
| \end{CD}$$ |
\end{CD}$$ |
| then the $n$-th \emph{homology group} (or \emph{homology module}) |
then the $n$-th \emph{homology group} (or \emph{homology module}) |
| $H_n(\mathbf{A},d)$ of $(\mathbf{A},d)$ |
$H_n(\mathbf{A},d)$ of $(\mathbf{A},d)$ |
| is the quotient module |
is the quotient module |
| \[ |
\[ |
| H_n(\mathbf{A},d)=\frac{\ker d_n}{\im d_{n+1}}. |
H_n(\mathbf{A},d)=\frac{\ker d_n}{\im d_{n+1}}. |
| \] |
\] |
|
|
| The chain complex is an \PMlinkname{exact sequence}{ExactSequence} if and only if |
The chain complex is an \PMlinkname{exact sequence}{ExactSequence} if and only if |
| all of the homology groups are trivial. |
all of the homology groups are trivial. |
| The homology groups can therefore be thought of |
The homology groups can therefore be thought of |
| as measuring the extent to which the chain complex fails to be exact. |
as measuring the extent to which the chain complex fails to be exact. |
|
|
| Homology groups of other objects are defined as the homology groups of an associated chain complex. (In particular, see the entry on the \PMlinkname{homology of topological spaces}{HomologyTopologicalSpace}.) |
Homology groups of other objects (such as topological spaces) |
|
are defined as the homology groups of an associated chain complex. |
