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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, specifically
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...
, a divided power structure is 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

Let ''A'' be a
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
with an ideal ''I''. A divided power structure (or PD-structure, after the French ''puissances divisées'') on ''I'' is a collection of maps \gamma_n : I \to A for ''n'' = 0, 1, 2, ... such that: #\gamma_0(x) = 1 and \gamma_1(x) = x for x \in I, while \gamma_n(x) \in I for ''n'' > 0. #\gamma_n(x + y) = \sum_^n \gamma_(x) \gamma_i(y) for x, y \in I. #\gamma_n(\lambda x) = \lambda^n \gamma_n(x) for \lambda \in A, x \in I. #\gamma_m(x) \gamma_n(x) = ((m, n)) \gamma_(x) for x \in I, where ((m, n)) = \frac is an integer. #\gamma_n(\gamma_m(x)) = C_ \gamma_(x) for x \in I and m > 0, where C_ = \frac is an integer. For convenience of notation, \gamma_n(x) is often written as x^ when it is clear what divided power structure is meant. The term ''divided power ideal'' refers to an ideal with a given divided power structure, and ''divided power ring'' refers 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 \Z on one generator: ::\Z\langle\rangle:=\Z\left ,\tfrac,\ldots,\tfrac,\ldots \rightsubset \Q * If ''A'' is an algebra over \Q, then every ideal ''I'' has a unique divided power structure where \gamma_n(x) = \tfrac \cdot x^n.The uniqueness follows from the easily verified fact that in general, x^n = n! \gamma_n(x). Indeed, this is the example which motivates the definition in the first place. * If ''M'' is an ''A''-module, let S^\bullet M denote the
symmetric algebra In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...
of ''M'' over ''A''. Then its dual (S^\bullet M)^\vee = \text_A(S^\bullet M, A) has a canonical structure of divided power ring. In fact, it is canonically isomorphic to a natural completion of \Gamma_A(\check) (see below) if ''M'' has finite rank.


Constructions

If ''A'' is any ring, there exists a divided power ring :A \langle x_1, x_2, \ldots, x_n \rangle consisting of ''divided power polynomials'' in the variables :x_1, x_2, \ldots, x_n, that is sums of ''divided power monomials'' of the form :c x_1^ x_2^ \cdots x_n^ with c \in A. Here the divided power ideal is the set of divided power polynomials with constant coefficient 0. More generally, if ''M'' is an ''A''-module, there is a universal ''A''-algebra, called :\Gamma_A(M), with PD ideal :\Gamma_+(M) and an ''A''-linear map :M \to \Gamma_+(M). (The case of divided power polynomials is the special case in which ''M'' is a
free module In mathematics, a free module is a module that has a ''basis'', that is, a generating set that is linearly independent. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commu ...
over ''A'' of finite rank.) If ''I'' is any ideal of a ring ''A'', there is a
universal construction In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
which extends ''A'' with divided powers of elements of ''I'' to get a divided power envelope of ''I'' in ''A''.


Applications

The divided power envelope is a fundamental tool in the theory of PD differential operators and crystalline cohomology, where it is used to overcome technical difficulties which arise in positive characteristic. The divided power
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
is used in the construction of co-Schur functors.


See also

* Crystalline cohomology


References

* * {{ cite book , title=Formal Groups and Applications , volume=78 , series=Pure and applied mathematics, a series of monographs and textbooks , first= Michiel , last= Hazewinkel , author-link = Michiel Hazewinkel , publisher=
Elsevier Elsevier ( ) is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as ''The Lancet'', ''Cell (journal), Cell'', the ScienceDirect collection of electronic journals, ...
, year=1978 , isbn=0123351502 , zbl=0454.14020 , page=507 * p-adic derived de Rham cohomology - contains excellent material on PD-polynomial rings an
PD-envelopesWhat's the name for the analogue of divided power algebras for x^i/i
- contains useful equivalence to divided power algebras as dual algebras Commutative algebra Polynomials