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Wikipedia / Divided power structure

In mathematics, specifically commutative algebra, a divided power structure. Just like there are polynomial algebras on n generators, there the foundational object in this setting is the divided power algebra on n generators.

Definition of a free divided power algebra on n generators

. Then

to be

.

Another characterization

.

Definition of a divided power structure

The preceding section has the following generalization. Let A be a commutative ring with an ideal I. A divided power structure (or PD-structure, after the French puissances divisées) on I for n = 0, 1, 2, ... such that:

  1. for n > 0.

  2. .

  3. .

  4. is an integer.

  5. is an integer.

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, then every ideal I Indeed, this is the example which motivates the definition in the first place.

    If M is an A denote the symmetric algebra of M over A has a canonical structure of divided power ring. In fact, it is canonically isomorphic to a natural completion (see below) if M has finite rank.

Constructions

If A is any ring then we have the free divided power algebra on n generators

defined in the first section above. Concretely it consists of so called divided power polynomials in the variables

  1. ,

that is sums of divided power monomials of the form

. The structure of a divided power algebra(of the third section) is given by letting I be the ideal of divided power polynomials with no constant coefficient.

More generally, if M is an A-module, there is a free divided A-algebra on M, called

with PD ideal

and an A-linear map

(The case of divided power polynomials is the special case in which M is a free module over A of finite rank.)

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 concrete divided power algebras of the first two sections are ubiquitous in algebraic topology since a free divided power algebra on n generators is just the dual of a polynomial algebra. 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.

See also

References