In mathematics, specifically commutative algebra, a divided power structure is a way of introducing items with similar properties as expressions of the form
xn/n!
n!
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
\gamman:I\toA
\gamma0(x)=1
\gamma1(x)=x
x\inI
\gamman(x)\inI
\gamman(x+y)=
n | |
\sum | |
i=0 |
\gamman-i(x)\gammai(y)
x,y\inI
\gamman(λx)=λn\gamman(x)
λ\inA,x\inI
\gammam(x)\gamman(x)=((m,n))\gammam+n(x)
x\inI
((m,n))=
(m+n)! | |
m!n! |
\gamman(\gammam(x))=Cn,\gammamn(x)
x\inI
m>0
Cn,=
(mn)! | |
(m!)nn! |
For convenience of notation,
\gamman(x)
x[n]
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.
\Z
\Z\langle{x}\rangle:=\Z\left[x,\tfrac{x2}{2},\ldots,\tfrac{xn}{n!},\ldots\right]\subset\Q[x].
\Q,
\gamman(x)=\tfrac{1}{n!} ⋅ xn.
S\bulletM
(S\bulletM)\vee=
\bullet | |
Hom | |
A(S |
M,A)
\GammaA(\check{M})
If A is any ring, there exists a divided power ring
A\langlex1,x2,\ldots,xn\rangle
consisting of divided power polynomials in the variables
x1,x2,\ldots,xn,
that is sums of divided power monomials of the form
c
[i1] | |
x | |
1 |
[i2] | |
x | |
2 |
…
[in] | |
x | |
n |
with
c\inA
More generally, if M is an A-module, there is a universal A-algebra, called
\GammaA(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 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.
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.