In mathematics, specifically commutative algebra, a divided power structure is a way of making expressions of the form
Contents
Definition
Let A be a commutative ring with an ideal I. A divided power structure (or PD-structure, after the French puissances divisées) on I is a collection of maps
-
γ 0 ( x ) = 1 andγ 1 ( x ) = x forx ∈ I , whileγ n ( x ) ∈ I for n > 0. -
γ n ( x + y ) = ∑ i = 0 n γ n − i ( x ) γ i ( y ) forx , y ∈ I . -
γ n ( λ x ) = λ n γ n ( x ) forλ ∈ A , x ∈ I . -
γ m ( x ) γ n ( x ) = ( ( m , n ) ) γ m + n ( x ) forx ∈ I , where( ( m , n ) ) = ( m + n ) ! m ! n ! -
γ n ( γ m ( x ) ) = C n , m γ m n ( x ) forx ∈ I , whereC n , m = ( m n ) ! ( m ! ) n n !
For convenience of notation,
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
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
More generally, if M is an A-module, there is a universal A-algebra, 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 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.