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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r0824301.png" /> be a commutative ring with unit, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r0824302.png" /> be an augmented <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r0824303.png" />-algebra, i.e. there is given a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r0824304.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r0824305.png" />. A divided power structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r0824306.png" /> (or, more precisely, on the augmentation ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r0824307.png" />) is a sequence of mappings
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<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/r/r082/r082430/r0824308.png" /></td> </tr></table>
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{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
Let  $  R $
 +
be a commutative ring with unit, and let  $  A $
 +
be an augmented  $  R $-
 +
algebra, i.e. there is given a homomorphism of  $  R $-
 +
algebras  $  \epsilon : A \rightarrow R $.
 +
A divided power structure on  $  R $(
 +
or, more precisely, on the augmentation ideal  $  I( A)= \mathop{\rm Ker} ( \epsilon ) $)
 +
is a sequence of mappings
 +
 
 +
$$
 +
\gamma _ {r} : I( A)  \rightarrow  I( A),\  r = 1, 2 \dots
 +
$$
  
 
such that
 
such that
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r0824309.png" />;
+
1) $  \gamma _ {1} ( x) = x $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243010.png" />;
+
2) $  \gamma _ {r} ( x) \gamma _ {s} ( x) = ( {} _ { s }  ^ {r+ s } ) \gamma _ {r+} s ( x) $;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243011.png" />;
+
3) $  \gamma _ {t} ( x+ y)= \sum _ {r=} 0 ^ {t} \gamma _ {r} ( x) \gamma _ {t-} r ( y) $;
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243012.png" />;
+
4) $  \gamma _ {s} ( \gamma _ {r} ( x))= \epsilon _ {s,r} \gamma _ {rs} ( x) $;
  
5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243013.png" />;
+
5) $  \gamma _ {r} ( xy) = r! \gamma _ {r} ( x) \gamma _ {r} ( y) $;
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243014.png" /> in 3) and
+
where $  \gamma _ {0} ( x) = 1 $
 +
in 3) and
  
<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/r/r082/r082430/r08243015.png" /></td> </tr></table>
+
$$
 +
\epsilon _ {s,r}  = \left ( \begin{array}{c}
 +
r \\
 +
r- 1
 +
\end{array}
 +
\right )
 +
\left ( \begin{array}{c}
 +
2r \\
 +
r- 1
 +
\end{array}
 +
\right ) \dots
 +
\left ( \begin{array}{c}
 +
( s- 1) r \\
 +
r- 1
 +
\end{array}
 +
\right ) .
 +
$$
  
In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243016.png" /> is a graded commutative algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243017.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243018.png" />, these requirements are augmented as follows (and changed slightly):
+
In case $  A $
 +
is a graded commutative algebra over $  R $
 +
with $  A _ {0} = R $,  
 +
these requirements are augmented as follows (and changed slightly):
  
6) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243019.png" />,
+
6) $  \gamma _ {r} ( A _ {k} ) \subset  A _ {rk} $,
  
 
with 5) replaced by
 
with 5) replaced by
Line 27: Line 68:
 
5')
 
5')
  
<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/r/r082/r082430/r08243020.png" /></td> </tr></table>
+
$$
 +
 
 +
\begin{array}{ll}
 +
\gamma _ {r} ( xy)  = r! \gamma _ {r} ( x) \gamma _ {r} ( y)  & \textrm{ for }  r\geq  2  \textrm{ and }  x, y  \textrm{ of  even
 +
degree  } ; \\
 +
\gamma _ {r} ( xy)  = 0 & \textrm{ for }  r\geq  2  \textrm{ and }  x, y  \textrm{ of  odd
 +
degree  } . \\
 +
\end{array}
 +
 
 +
$$
  
Given an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243021.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243022.png" />, an algebra with divided powers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243023.png" /> is constructed as follows. It is generated (as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243024.png" />-algebra) by symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243027.png" /> and between these symbols the following relations are imposed:
+
Given an $  R $-
 +
module $  M $,  
 +
an algebra with divided powers $  \Gamma ( M) $
 +
is constructed as follows. It is generated (as an $  R $-
 +
algebra) by symbols $  m  ^ {(} r) $,
 +
$  m \in M $,  
 +
$  r= 1, 2 \dots $
 +
and between these symbols the following relations are imposed:
  
<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/r/r082/r082430/r08243028.png" /></td> </tr></table>
+
$$
 +
( m _ {1} + m _ {2} )  ^ {(} t)  = \sum _ { r= } 0 ^ { t }  m _ {1}  ^ {(} r) m _ {2}  ^ {(} t- r) ,
 +
$$
  
<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/r/r082/r082430/r08243029.png" /></td> </tr></table>
+
$$
 +
( \alpha m )  ^ {(} t)  = \alpha  ^ {t} m  ^ {(} t) ,\  \alpha \in R,
 +
$$
  
<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/r/r082/r082430/r08243030.png" /></td> </tr></table>
+
$$
 +
m  ^ {(} r) m  ^ {(} s)  = \left ( \begin{array}{c}
 +
r+ s \\
 +
r
 +
\end{array}
 +
\right ) m  ^ {(} r+ s) .
 +
$$
  
This <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243031.png" /> satisfies 1)–5). The augmentation sends <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243032.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243033.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243034.png" />). If one assigns to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243035.png" /> the degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243036.png" />, a graded commutative algebra is obtained with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243038.png" /> which satisfies 1)–4), 5'), 6).
+
This $  \Gamma ( M) $
 +
satisfies 1)–5). The augmentation sends $  m  ^ {(} r) $
 +
to 0 $(
 +
r> 0 $).  
 +
If one assigns to $  m  ^ {(} r) $
 +
the degree $  2r $,  
 +
a graded commutative algebra is obtained with $  \Gamma ( M) _ {0} = R $,  
 +
$  \Gamma ( M) _ {1} = M $
 +
which satisfies 1)–4), 5'), 6).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243039.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243040.png" />-algebra, divided powers can always be defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243041.png" />. The relations 1)–5) can be understood as a way of writing down the interrelations between such  "divided powers"  (such as the one resulting from the binomial theorem) without having to use division by integers.
+
If $  A $
 +
is a $  \mathbf Q $-
 +
algebra, divided powers can always be defined as $  a \mapsto ( r!)  ^ {-} 1 a  ^ {r} $.  
 +
The relations 1)–5) can be understood as a way of writing down the interrelations between such  "divided powers"  (such as the one resulting from the binomial theorem) without having to use division by integers.
  
A divided power sequence in a co-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243042.png" /> is a sequence of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243043.png" /> satisfying
+
A divided power sequence in a co-algebra $  ( C, \mu ) $
 +
is a sequence of elements $  y _ {0} = 1 , y _ {1} , y _ {2} \dots $
 +
satisfying
  
<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/r/r082/r082430/r08243044.png" /></td> </tr></table>
+
$$
 +
\mu ( y _ {n} )  = \sum _ {i+ j= n } y _ {i} \oplus y _ {j} .
 +
$$
  
 
Divided power sequences are used in the theories of Hopf algebras and formal groups (cf. [[Formal group|Formal group]]; [[Hopf algebra|Hopf algebra]]), [[#References|[a1]]]–[[#References|[a3]]]. Rings with divided powers occur in algebraic topology (where they provide a natural setting for power cohomology operations), [[#References|[a4]]], [[#References|[a5]]], and the theory of formal groups [[#References|[a3]]], [[#References|[a2]]].
 
Divided power sequences are used in the theories of Hopf algebras and formal groups (cf. [[Formal group|Formal group]]; [[Hopf algebra|Hopf algebra]]), [[#References|[a1]]]–[[#References|[a3]]]. Rings with divided powers occur in algebraic topology (where they provide a natural setting for power cohomology operations), [[#References|[a4]]], [[#References|[a5]]], and the theory of formal groups [[#References|[a3]]], [[#References|[a2]]].

Revision as of 14:55, 7 June 2020


Let $ R $ be a commutative ring with unit, and let $ A $ be an augmented $ R $- algebra, i.e. there is given a homomorphism of $ R $- algebras $ \epsilon : A \rightarrow R $. A divided power structure on $ R $( or, more precisely, on the augmentation ideal $ I( A)= \mathop{\rm Ker} ( \epsilon ) $) is a sequence of mappings

$$ \gamma _ {r} : I( A) \rightarrow I( A),\ r = 1, 2 \dots $$

such that

1) $ \gamma _ {1} ( x) = x $;

2) $ \gamma _ {r} ( x) \gamma _ {s} ( x) = ( {} _ { s } ^ {r+ s } ) \gamma _ {r+} s ( x) $;

3) $ \gamma _ {t} ( x+ y)= \sum _ {r=} 0 ^ {t} \gamma _ {r} ( x) \gamma _ {t-} r ( y) $;

4) $ \gamma _ {s} ( \gamma _ {r} ( x))= \epsilon _ {s,r} \gamma _ {rs} ( x) $;

5) $ \gamma _ {r} ( xy) = r! \gamma _ {r} ( x) \gamma _ {r} ( y) $;

where $ \gamma _ {0} ( x) = 1 $ in 3) and

$$ \epsilon _ {s,r} = \left ( \begin{array}{c} r \\ r- 1 \end{array} \right ) \left ( \begin{array}{c} 2r \\ r- 1 \end{array} \right ) \dots \left ( \begin{array}{c} ( s- 1) r \\ r- 1 \end{array} \right ) . $$

In case $ A $ is a graded commutative algebra over $ R $ with $ A _ {0} = R $, these requirements are augmented as follows (and changed slightly):

6) $ \gamma _ {r} ( A _ {k} ) \subset A _ {rk} $,

with 5) replaced by

5')

$$ \begin{array}{ll} \gamma _ {r} ( xy) = r! \gamma _ {r} ( x) \gamma _ {r} ( y) & \textrm{ for } r\geq 2 \textrm{ and } x, y \textrm{ of even degree } ; \\ \gamma _ {r} ( xy) = 0 & \textrm{ for } r\geq 2 \textrm{ and } x, y \textrm{ of odd degree } . \\ \end{array} $$

Given an $ R $- module $ M $, an algebra with divided powers $ \Gamma ( M) $ is constructed as follows. It is generated (as an $ R $- algebra) by symbols $ m ^ {(} r) $, $ m \in M $, $ r= 1, 2 \dots $ and between these symbols the following relations are imposed:

$$ ( m _ {1} + m _ {2} ) ^ {(} t) = \sum _ { r= } 0 ^ { t } m _ {1} ^ {(} r) m _ {2} ^ {(} t- r) , $$

$$ ( \alpha m ) ^ {(} t) = \alpha ^ {t} m ^ {(} t) ,\ \alpha \in R, $$

$$ m ^ {(} r) m ^ {(} s) = \left ( \begin{array}{c} r+ s \\ r \end{array} \right ) m ^ {(} r+ s) . $$

This $ \Gamma ( M) $ satisfies 1)–5). The augmentation sends $ m ^ {(} r) $ to $ 0 $( $ r> 0 $). If one assigns to $ m ^ {(} r) $ the degree $ 2r $, a graded commutative algebra is obtained with $ \Gamma ( M) _ {0} = R $, $ \Gamma ( M) _ {1} = M $ which satisfies 1)–4), 5'), 6).

If $ A $ is a $ \mathbf Q $- algebra, divided powers can always be defined as $ a \mapsto ( r!) ^ {-} 1 a ^ {r} $. The relations 1)–5) can be understood as a way of writing down the interrelations between such "divided powers" (such as the one resulting from the binomial theorem) without having to use division by integers.

A divided power sequence in a co-algebra $ ( C, \mu ) $ is a sequence of elements $ y _ {0} = 1 , y _ {1} , y _ {2} \dots $ satisfying

$$ \mu ( y _ {n} ) = \sum _ {i+ j= n } y _ {i} \oplus y _ {j} . $$

Divided power sequences are used in the theories of Hopf algebras and formal groups (cf. Formal group; Hopf algebra), [a1][a3]. Rings with divided powers occur in algebraic topology (where they provide a natural setting for power cohomology operations), [a4], [a5], and the theory of formal groups [a3], [a2].

References

[a1] N. Roby, "Les algèbres à puissances divisées" Bull. Soc. Math. France , 89 (1965) pp. 75–91
[a2] M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978)
[a3] P. Cartier, "Exemples d'hyperalgèbres" , Sem. S. Lie 1955/56 , 3 , Secr. Math. Univ. Paris (1957)
[a4] E. Thomas, "The generalized Pontryagin cohomology operations and rings with divided powers" , Amer. Math. Soc. (1957)
[a5] S. Eilenberg, S. MacLane, "On the groups , II" Ann. of Math. , 60 (1954) pp. 49–189
How to Cite This Entry:
Ring with divided powers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ring_with_divided_powers&oldid=49566