View source for Ring with divided powers

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

<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>

such that

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r0824309.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243010.png" />;

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243011.png" />;

4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243012.png" />;

5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243013.png" />;

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243014.png" /> in 3) and

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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):

6) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243019.png" />,

with 5) replaced by

5')

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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:

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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/r08243030.png" /></td> </tr></table>

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).

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.

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

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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]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Roby,   "Les algèbres à puissances divisées"  ''Bull. Soc. Math. France'' , '''89'''  (1965)  pp. 75–91</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Hazewinkel,   "Formal groups and applications" , Acad. Press  (1978)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Cartier,   "Exemples d'hyperalgèbres" , ''Sem. S. Lie 1955/56'' , '''3''' , Secr. Math. Univ. Paris  (1957)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Thomas,   "The generalized Pontryagin cohomology operations and rings with divided powers" , Amer. Math. Soc.  (1957)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Eilenberg,   S. MacLane,   "On the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082430/r08243045.png" />, II"  ''Ann. of Math.'' , '''60'''  (1954)  pp. 49–189</TD></TR></table>