Divided power structure

Inmathematics,specificallycommutative algebra,adivided power structureis a way of introducing items with similar properties as expressions of the form$x^{n}/n!$have, also when it is not possible to actually divide by$n!$.

Definition

LetAbe acommutative ringwith anidealI.Adivided power structure(orPD-structure,after the Frenchpuissances divisées) onIis a collection of maps$\gamma _{n}:I\to A$forn= 0, 1, 2,... such that:

1. $\gamma _{0}(x)=1$and$\gamma _{1}(x)=x$for$x\in I$,while$\gamma _{n}(x)\in I$forn> 0.
2. $\gamma _{n}(x+y)=\sum _{i=0}^{n}\gamma _{n-i}(x)\gamma _{i}(y)$for$x,y\in I$.
3. $\gamma _{n}(\lambda x)=\lambda ^{n}\gamma _{n}(x)$for$\lambda \in A,x\in I$.
4. $\gamma _{m}(x)\gamma _{n}(x)=((m,n))\gamma _{m+n}(x)$for$x\in I$,where$((m,n))={\frac {(m+n)!}{m!n!}}$is an integer.
5. $\gamma _{n}(\gamma _{m}(x))=C_{n,m}\gamma _{mn}(x)$for$x\in I$and$m>0$,where$C_{n,m}={\frac {(mn)!}{(m!)^{n}n!}}$is an integer.

For convenience of notation,$\gamma _{n}(x)$is often written as$x^{[n]}$when it is clear what divided power structure is meant.

The termdivided power idealrefers to an ideal with a given divided power structure, anddivided power ringrefers to a ring with a given ideal with divided power structure.

Homomorphisms of divided power algebras are ring homomorphisms that respects the divided power structure on its source and target.

Examples

• The free divided power algebra over$\mathbb {Z}$on one generator:

$\mathbb {Z} \langle {x}\rangle:=\mathbb {Z} \left[x,{\tfrac {x^{2}}{2}},\ldots,{\tfrac {x^{n}}{n!}},\ldots \right]\subset \mathbb {Q} [x].$

• IfAis an algebra over$\mathbb {Q},$then every idealIhas a unique divided power structure where$\gamma _{n}(x)={\tfrac {1}{n!}}\cdot x^{n}.$Indeed, this is the example which motivates the definition in the first place.

• IfMis anA-module, let$S^{\bullet }M$denote thesymmetric algebraofMoverA.Then its dual$(S^{\bullet }M)^{\vee }={\text{Hom}}_{A}(S^{\bullet }M,A)$has a canonical structure of divided power ring. In fact, it is canonically isomorphic to a naturalcompletionof$\Gamma _{A}({\check {M}})$(see below) ifMhas finite rank.

Constructions

IfAis any ring, there exists a divided power ring

$A\langle x_{1},x_{2},\ldots,x_{n}\rangle$

consisting ofdivided power polynomialsin the variables

$x_{1},x_{2},\ldots,x_{n},$

that is sums ofdivided power monomialsof the form

$cx_{1}^{[i_{1}]}x_{2}^{[i_{2}]}\cdots x_{n}^{[i_{n}]}$

with$c\in A$.Here the divided power ideal is the set of divided power polynomials with constant coefficient 0.

More generally, ifMis anA-module, there is auniversalA-algebra, called

$\Gamma _{A}(M),$

with PD ideal

$\Gamma _{+}(M)$

and anA-linear map

$M\to \Gamma _{+}(M).$

(The case of divided power polynomials is the special case in whichMis afree moduleoverAof finite rank.)

IfIis any ideal of a ringA,there is auniversal constructionwhich extendsAwith divided powers of elements ofIto get adivided power envelopeofIinA.

Applications

The divided power envelope is a fundamental tool in the theory ofPD differential operatorsandcrystalline cohomology,where it is used to overcome technical difficulties which arise in positivecharacteristic.

The divided power functor is used in the construction of co-Schur functors.

See also

• Crystalline cohomology

References

• Berthelot, Pierre;Ogus, Arthur(1978).Notes on Crystalline Cohomology.Annals of Mathematics Studies.Princeton University Press.Zbl0383.14010.
• Hazewinkel, Michiel(1978).Formal Groups and Applications.Pure and applied mathematics, a series of monographs and textbooks. Vol. 78.Elsevier.p. 507.ISBN0123351502.Zbl0454.14020.
• p-adic derived de Rham cohomology- contains excellent material on PD-polynomial rings andPD-envelopes
• What's the name for the analogue of divided power algebras for x^i/i- contains useful equivalence to divided power algebras as dual algebras