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Difference between revisions of "Free resolution"

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A special case of a projective resolution. Every module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041630/f0416301.png" /> over an associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041630/f0416302.png" /> is the quotient module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041630/f0416303.png" /> of a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041630/f0416304.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041630/f0416305.png" /> by a submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041630/f0416306.png" />. The submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041630/f0416307.png" /> has a similar representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041630/f0416308.png" />, etc. As a result one obtains an exact sequence of free modules
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A special case of a projective resolution. Every module $M$ over an associative ring $R$ is the quotient module $F_0/N_0$ of a free $R$-module $F_0$ by a submodule $N_0$. The submodule $N_0$ has a similar representation $F_1/N_1$, etc. As a result one obtains an exact sequence of free modules
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041630/f0416309.png" /></td> </tr></table>
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$$F_0\leftarrow F_1\leftarrow\ldots\leftarrow F_n\leftarrow\ldots,$$
  
called the free resolution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041630/f04163010.png" />. The canonical homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041630/f04163011.png" /> is called a supplementing homomorphism (or augmentation).
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called the free resolution of $M$. The canonical homomorphism $F_0\to M$ is called a supplementing homomorphism (or augmentation).
  
  

Revision as of 07:49, 21 June 2014

A special case of a projective resolution. Every module $M$ over an associative ring $R$ is the quotient module $F_0/N_0$ of a free $R$-module $F_0$ by a submodule $N_0$. The submodule $N_0$ has a similar representation $F_1/N_1$, etc. As a result one obtains an exact sequence of free modules

$$F_0\leftarrow F_1\leftarrow\ldots\leftarrow F_n\leftarrow\ldots,$$

called the free resolution of $M$. The canonical homomorphism $F_0\to M$ is called a supplementing homomorphism (or augmentation).


Comments

See also Free module.

How to Cite This Entry:
Free resolution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_resolution&oldid=12844
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article