Divided power structure


In mathematics, specifically commutative algebra, a divided power structure is a way of introducing items with similar properties as expressions of the form have, also when it is not possible to actually divide by.

Definition

Let A be a commutative ring with an ideal I. A divided power structure on I is a collection of maps for n = 0, 1, 2,... such that:
  1. and for, while for n > 0.
  2. for.
  3. for.
  4. for, where is an integer.
  5. for and, where is an integer.
For convenience of notation, is often written as 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 respect the divided power structure on its source and target.

Examples

  • The free divided power algebra over on one generator:
  • If A is an algebra over then every ideal I has a unique divided power structure where Indeed, this is the example which motivates the definition in the first place.
  • If M is an A-module, let denote the symmetric algebra of M over A. Then its dual has a canonical structure of divided power ring. In fact, it is canonically isomorphic to a natural completion of if M has finite rank.

    Constructions

If A is any ring, there exists a divided power ring
consisting of divided power polynomials in the variables
that is sums of divided power monomials of the form
with. 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
with PD ideal
and an A-linear map
If I is any ideal of a ring A, there is a universal construction 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 is used in the construction of co-Schur functors.