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Difference between revisions of "Split sequence"

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''split exact sequence, split short exact sequence''
 
''split exact sequence, split short exact sequence''
  
An [[Exact sequence|exact sequence]]
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An [[exact sequence]]
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\begin{equation}
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0 \rightarrow A \stackrel{f}{\rightarrow} B \stackrel{g}{\rightarrow} C \rightarrow 0 \label{eq:1}
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\end{equation}
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in an [[Abelian category]] which is isomorphic to the direct sum sequence,
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$$
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0 \rightarrow A \rightarrow A \oplus C \rightarrow C \rightarrow 0
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$$
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by an isomorphism $B \rightarrow A \oplus C$ which induces the identity on $A$ and on $C$. Sufficient conditions for an exact sequence \eqref{eq:1} to be split are the existence of a right inverse $f'$ for $f$, or of a left inverse $g'$ for $g$. The class of split exact sequences is the zero of the group $\mathrm{Ext}_R^1(A,C)$ (see [[Baer multiplication]]). In a category of vector spaces (that is, of modules over a fixed field) every exact sequence splits.
  
<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/s/s086/s086840/s0868401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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For [[relative homological algebra]], the typical situation is to consider exact sequences in one category which split in another.
 
 
in an Abelian category which is isomorphic to the direct sum sequence,
 
 
 
<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/s/s086/s086840/s0868402.png" /></td> </tr></table>
 
 
 
by an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086840/s0868403.png" /> which induces the identity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086840/s0868404.png" /> and on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086840/s0868405.png" />. Sufficient conditions for an exact sequence (*) to be split are the existence of a right inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086840/s0868406.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086840/s0868407.png" />, or of a left inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086840/s0868408.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086840/s0868409.png" />. The class of split exact sequences is the zero of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086840/s08684010.png" /> (see [[Baer multiplication|Baer multiplication]]). In a category of vector spaces (that is, of modules over a fixed field) every exact sequence splits.
 
 
 
For [[Relative homological algebra|relative homological algebra]], the typical situation is to consider exact sequences in one category which split in another.
 
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)  pp. 16, 260</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)  pp. 16, 260 {{ZBL|0133.26502}}</TD></TR>
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</table>
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Latest revision as of 13:41, 2 September 2017

split exact sequence, split short exact sequence

An exact sequence \begin{equation} 0 \rightarrow A \stackrel{f}{\rightarrow} B \stackrel{g}{\rightarrow} C \rightarrow 0 \label{eq:1} \end{equation} in an Abelian category which is isomorphic to the direct sum sequence, $$ 0 \rightarrow A \rightarrow A \oplus C \rightarrow C \rightarrow 0 $$ by an isomorphism $B \rightarrow A \oplus C$ which induces the identity on $A$ and on $C$. Sufficient conditions for an exact sequence \eqref{eq:1} to be split are the existence of a right inverse $f'$ for $f$, or of a left inverse $g'$ for $g$. The class of split exact sequences is the zero of the group $\mathrm{Ext}_R^1(A,C)$ (see Baer multiplication). In a category of vector spaces (that is, of modules over a fixed field) every exact sequence splits.

For relative homological algebra, the typical situation is to consider exact sequences in one category which split in another.


Comments

References

[a1] S. MacLane, "Homology" , Springer (1963) pp. 16, 260 Zbl 0133.26502
How to Cite This Entry:
Split sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Split_sequence&oldid=41781
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article