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Difference between revisions of "Ring with divided powers"

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1)  $  \gamma _ {1} ( x) = x $;
 
1)  $  \gamma _ {1} ( x) = x $;
  
2)  $  \gamma _ {r} ( x) \gamma _ {s} ( x) = ( {} _ { s }  ^ {r+ s } ) \gamma _ {r+} s ( x) $;
+
2)  $  \gamma _ {r} ( x) \gamma _ {s} ( x) = ( {} _ { s }  ^ {r+ s } ) \gamma _ {r+s} ( x) $;
  
3)  $  \gamma _ {t} ( x+ y)= \sum _ {r=} ^ {t} \gamma _ {r} ( x) \gamma _ {t-} r ( y) $;
+
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) $;
 
4)  $  \gamma _ {s} ( \gamma _ {r} ( x))= \epsilon _ {s,r} \gamma _ {rs} ( x) $;
Line 83: Line 83:
 
an algebra with divided powers  $  \Gamma ( M) $
 
an algebra with divided powers  $  \Gamma ( M) $
 
is constructed as follows. It is generated (as an  $  R $-
 
is constructed as follows. It is generated (as an  $  R $-
algebra) by symbols  $  m  ^ {(} r) $,  
+
algebra) by symbols  $  m  ^ {(r)} $,  
 
$  m \in M $,  
 
$  m \in M $,  
 
$  r= 1, 2 \dots $
 
$  r= 1, 2 \dots $
Line 89: Line 89:
  
 
$$  
 
$$  
( m _ {1} + m _ {2} )  ^ {(} t)  =  \sum _ { r= } 0 ^ { t }  m _ {1}  ^ {(} r) m _ {2}  ^ {(} t- r) ,
+
( 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,
+
( \alpha m )  ^ {(t)} =  \alpha  ^ {t} m  ^ {(t)} ,\  \alpha \in R,
 
$$
 
$$
  
 
$$  
 
$$  
m  ^ {(} r) m  ^ {(} s)  =  \left ( \begin{array}{c}
+
m  ^ {(r)} m  ^ {(s)} =  \left ( \begin{array}{c}
 
r+ s \\
 
r+ s \\
 
  r  
 
  r  
 
\end{array}
 
\end{array}
  \right ) m  ^ {(} r+ s) .
+
  \right ) m  ^ {(r+ s)}.
 
$$
 
$$
  
 
This  $  \Gamma ( M) $
 
This  $  \Gamma ( M) $
satisfies 1)–5). The augmentation sends  $  m  ^ {(} r) $
+
satisfies 1)–5). The augmentation sends  $  m  ^ {(r)}$
 
to  $  0 $(
 
to  $  0 $(
 
$  r> 0 $).  
 
$  r> 0 $).  
If one assigns to  $  m  ^ {(} r) $
+
If one assigns to  $  m  ^ {(r)} $
 
the degree  $  2r $,  
 
the degree  $  2r $,  
 
a graded commutative algebra is obtained with  $  \Gamma ( M) _ {0} = R $,  
 
a graded commutative algebra is obtained with  $  \Gamma ( M) _ {0} = R $,  
Line 116: Line 116:
 
If  $  A $
 
If  $  A $
 
is a  $  \mathbf Q $-
 
is a  $  \mathbf Q $-
algebra, divided powers can always be defined as  $  a \mapsto ( r!)  ^ {-} 1 a  ^ {r} $.  
+
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.
 
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.
  

Revision as of 08:27, 20 January 2024


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=55228