Divided Powers

Introduction

This article introduces the divided power algebra, the refinement of the symmetric algebra that remains well behaved when factorials are not invertible. The treatment is introductory and purely mathematical.

The base structure is a commutative ring $R$ with identity $1 \neq 0$. The reader should have Symmetric Powers and The Symmetric Algebra at hand: the divided power algebra is a second algebra with the same degree-one part as the symmetric algebra, differing from it by binomial and factorial coefficients, and coinciding with it exactly when every factorial is a unit. The anti-symmetric analogue, the divided or shuffle treatment of the exterior algebra, is not needed here; the exterior algebra belongs to category 07.

The problem the divided power algebra solves is easy to state. In the symmetric algebra the $n$-th power of a degree-one element is again a legitimate element, but the divided $n$-th power $x^n/n!$ is not available once $n!$ fails to be invertible, and in characteristic $p$ the $p$-th power map $x \mapsto x^p$ reaches only the pure monomials $x_i^p$ of the degree-$p$ part, not a monomial such as $x_1 x_2^{p-1}$. The divided power $\gamma_n(x)$ is designed to play the role of $x^n/n!$, and it exists as an operation even when $n!$ does not.

Why Divided Powers

The Failure of Factorials in Small Characteristic

In characteristic zero one is accustomed to writing $x^n/n!$, because $n!$ is invertible. Equivalently, the symmetric power $\operatorname{Sym}^n M$ is generated by honest $n$-th powers $x^n$, and the map $x \mapsto x^n/n!$ provides an alternative set of generators with pleasant additivity:

$$ \frac{(x+y)^n}{n!} = \sum_{i+j=n} \frac{x^i}{i!}\,\frac{y^j}{j!}. $$

Over a ring in which $n!$ is not invertible this device is unavailable, and it is not merely a convenience: the object one obtains by dividing by the symmetric relations is genuinely too small.

Example. Let $R = \mathbb{F}_2$ and let $M = R$ be free of rank $1$ with basis $x$, so that $\operatorname{Sym}(R) = R[x]$ and $x^2 \neq 0$. In a divided power algebra over $\mathbb{F}_2$ the two axioms force $\gamma_1(x)^2 = 0$. Indeed, by additivity applied to the pair $(x, x)$,

$$ \gamma_2(x + x) = \gamma_0(x)\gamma_2(x) + \gamma_1(x)\gamma_1(x) + \gamma_2(x)\gamma_0(x) = 2\,\gamma_2(x) + x^2 , $$

while by the scaling law $\gamma_2(x + x) = \gamma_2(2x) = 2^2\gamma_2(x) = 4\,\gamma_2(x) = 0$ in $\mathbb{F}_2$. Hence $x^2 = 0$. The multiplication law of $\Gamma(R)$ gives the same value directly, $\gamma_1\gamma_1 = \binom{2}{1}\gamma_2 = 2\gamma_2 = 0$. The symmetric algebra, in which $x^2 \neq 0$, therefore carries no divided power structure: the divided power algebra is the universal algebra in which the formal $n$-th powers exist and the additivity rule holds.

Polynomial Functions and Polynomial Forms

A second, classical motivation is the distinction between polynomial functions and polynomial forms. Let $K$ be a field and let $M = K^d$. A homogeneous polynomial form of degree $n$ is an element of $\operatorname{Sym}^n(M^*)$, and it defines a function $M \to K$. The assignment form $\mapsto$ function is injective when $K$ is infinite and has a kernel when $K$ is finite. The algebra of polynomial functions, with the divided power structure recording the coefficients of the Taylor expansion, is the divided power algebra; it is the algebra that represents the functor of points. Over $\mathbb{Q}$ and $\mathbb{R}$ the two coincide, which is why the distinction is invisible in the classical theory and visible only over $\mathbb{F}_p$ and in mixed characteristic.

The Divided Power Algebra

Divided Power Structures

Let $A = \bigoplus_{n \geq 0} A_n$ be a graded commutative $R$-algebra with $A_0 = R$, and let

$$ I = \bigoplus_{n \geq 1} A_n $$

be its augmentation ideal. A divided power structure on $(A, I)$ is a family of $R$-linear maps $\gamma_n : I \to A$, one for each $n \geq 0$, with $\gamma_0(x) = 1 \in A_0$ and $\gamma_n(I) \subseteq I$ for $n \geq 1$, such that for all $x, y \in I$ and $\lambda \in R$:

$$ \gamma_0(x) = 1, \qquad \gamma_1(x) = x, \qquad \gamma_n(\lambda x) = \lambda^n \gamma_n(x), $$

$$ \gamma_n(x + y) = \sum_{i=0}^{n} \gamma_i(x)\,\gamma_{n-i}(y), $$

$$ \gamma_m(x)\,\gamma_n(x) = \binom{m+n}{m} \gamma_{m+n}(x), $$

$$ \gamma_m(\gamma_n(x)) = \frac{(mn)!}{m!\,(n!)^m} \gamma_{mn}(x), \qquad m, n \geq 1 . $$

The last coefficient is an integer; it is the multinomial coefficient counting the partitions of a set of $mn$ labelled elements into $m$ unordered blocks of size $n$, and the verification that it is integral is the content of the proposition below. The restriction $m, n \geq 1$ is necessary: for $n = 0$ one has $\gamma_0(x) = 1 \notin I$, so the composition is not defined. The elements $\gamma_n(x)$ are called the divided powers of $x$; we also write $x^{(n)}$ for $\gamma_n(x)$ and $x^{(n)} = 0$ for $n < 0$.

A divided power algebra over $R$ is a graded commutative $R$-algebra $A$ with $A_0 = R$, equipped with a divided power structure such that $\gamma_n(A_1) \subseteq A_n$. A morphism of divided power algebras is a graded $R$-algebra homomorphism $\phi$ commuting with the divided powers, $\phi(\gamma_n(x)) = \gamma_n(\phi(x))$.

Proposition. The coefficient $\dfrac{(mn)!}{m!\,(n!)^m}$ is a positive integer for all $m, n \geq 1$.

Proof. It is the multinomial coefficient $\binom{mn}{n, n, \ldots, n}$ with $m$ entries equal to $n$, divided by $m!$. The multinomial coefficient counts the distributions of $mn$ labelled elements into $m$ labelled boxes each of size $n$, that is, the partitions of the set into $m$ blocks of size $n$ carrying a labelling of the blocks; each unordered partition is counted once for each of the $m!$ labellings of its blocks, so the quotient is the number of unordered partitions and is an integer.

The Universal Property

Theorem. For every $R$-module $M$ there is a divided power algebra $\Gamma(M)$ with $\Gamma(M)_1 = M$, the divided power algebra or free divided power algebra on $M$, such that for every divided power algebra $A$ and every $R$-linear map $f : M \to A_1$ there is a unique morphism of divided power algebras $F : \Gamma(M) \to A$ extending $f$.

This is the exact analogue of the universal property of the symmetric algebra, with "commutative algebra" replaced by "divided power algebra". Existence and the explicit construction when $M$ is free are given below; the general case follows from the free case by presenting $M$ as a quotient of a free module. We denote the divided power algebra by

$$ \Gamma(M) = \bigoplus_{n \geq 0} \Gamma^n(M), \qquad \Gamma^0(M) = R, \quad \Gamma^1(M) = M, $$

and write $\gamma_n : M \to \Gamma^n(M)$ for the structure map, so that $\Gamma(M)$ is generated as an algebra by the elements $\gamma_n(x)$, $x \in M$, $n \geq 0$.

The Free Divided Power Algebra on One Generator

Let $M = Rx$ be free of rank $1$. Write $\gamma_n = \gamma_n(x)$, so $\gamma_0 = 1$ and $\gamma_1 = x$.

Proposition. As a graded $R$-module, the divided power algebra on one generator is

$$ \Gamma(R) = \bigoplus_{n \geq 0} R\,\gamma_n, $$

free with basis $\{\gamma_0, \gamma_1, \gamma_2, \ldots\}$, and its multiplication is

$$ \gamma_m \gamma_n = \binom{m+n}{m} \gamma_{m+n}. $$

On degree-one elements the divided powers are $\gamma_n(cx) = c^n\gamma_n$, and on the algebra generators they are given by the composition law $\gamma_m(\gamma_n) = \frac{(mn)!}{m!(n!)^m}\gamma_{mn}$.

Proof. The displayed multiplication is associative, commutative and unital: associativity is the identity $\binom{m+n}{m}\binom{m+n+k}{k} = \binom{m+n+k}{m}\binom{n+k}{n}$, and commutativity is the symmetry of the binomial coefficient. The remaining grading and divided power data, including the composition law and its consistency, are the one-generator case of the standard structure theorem for the free divided power algebra (Berthelot–Ogus, §3.1, or Roby 1965).

Remark. The divided power algebra is not generated by its degree-one part when factorials fail to be invertible. The product law gives $\gamma_1^n = n!\,\gamma_n$ by induction on $n$, so in $\mathbb{F}_p$ one has $\gamma_1^p = p!\,\gamma_p = 0$ while $\gamma_p \neq 0$: the element $\gamma_p$ is not a polynomial in $\gamma_1$, and the subalgebra generated by $\gamma_1$ is spanned by $1, \gamma_1, \ldots, \gamma_{p-1}$. This is why the universal property is stated relative to the maps $\gamma_n : M \to \Gamma^n(M)$ regarded as generators, not to the single inclusion $M \hookrightarrow \Gamma(M)$.

The Free Divided Power Algebra on a Free Module

Now let $M$ be free with basis $e_1, \ldots, e_d$. For a multi-index $a = (a_1, \ldots, a_d)$ of non-negative integers write

$$ \gamma^{(a)} = \gamma_{a_1}(e_1)\,\gamma_{a_2}(e_2) \cdots \gamma_{a_d}(e_d). $$

Theorem. If $M$ is free with basis $e_1, \ldots, e_d$, then $\Gamma(M)$ is free as an $R$-module with basis the divided monomials $\gamma^{(a)}$, for all multi-indices $a$, and the product is

$$ \gamma^{(a)} \gamma^{(b)} = \binom{a+b}{a} \gamma^{(a+b)}, \qquad \binom{a+b}{a} = \prod_{i=1}^{d} \binom{a_i+b_i}{a_i}. $$

Proof. The formula makes $\Gamma(M)$ a graded commutative algebra, and the divided monomials are independent by construction; the product formula is the product formula of the one-generator case applied in each coordinate. The assertion that this is the free divided power algebra on $M$ is the standard normal-form theorem for divided power algebras (Berthelot–Ogus, §3, or Roby 1965), applied to the free module; we use it as the model of $\Gamma(M)$.

The divided powers of a general degree-one element $u = \sum_{i=1}^{d} c_i e_i$ are

$$ \gamma_n(u) = \sum_{|a| = n} c^a \gamma^{(a)}, \qquad c^a = \prod_{i=1}^{d} c_i^{a_i}. $$

Proposition. For every $u, v \in M$ and $m, n \geq 0$, the additivity relation holds:

$$ \gamma_n(u+v) = \sum_{i+j=n} \gamma_i(u)\,\gamma_j(v). $$

Proof. Write $u = \sum_i c_i e_i$, $v = \sum_i d_i e_i$. Then

$$ \gamma_n(u+v) = \sum_{|a| = n} (c+d)^a \gamma^{(a)} = \sum_{|a| = n} \Bigl(\prod_i (c_i+d_i)^{a_i}\Bigr) \gamma^{(a)}. $$

On the other side,

$$ \sum_{i+j=n} \gamma_i(u)\gamma_j(v) = \sum_{|a|+|b| = n} c^a d^b \binom{a+b}{a} \gamma^{(a+b)} = \sum_{|e| = n} \Bigl(\sum_{a+b=e} \binom{e}{a} c^a d^b\Bigr) \gamma^{(e)}. $$

The inner sum is $\prod_i \sum_{a_i+b_i=e_i} \binom{e_i}{a_i} c_i^{a_i} d_i^{b_i} = \prod_i (c_i+d_i)^{e_i} = (c+d)^e$, by the binomial theorem. The two sides agree.

Remark. The binomial coefficient $\binom{a+b}{a}$ in the product is exactly what makes the additivity proof close: with the coefficient $1$ of the symmetric algebra the right-hand side of the display would be off by the multinomial factor. This single coefficient is the whole difference between the two algebras.

The Composition Law

The composition law is a separate axiom of a divided power structure and is not a consequence of the others; on a free algebra it has to be imposed on the generators. For $\Gamma(R) = \bigoplus_n R\gamma_n$ one writes

$$ \gamma_m(\gamma_n) = c_{m,n}\,\gamma_{mn}, \qquad c_{m,n} = \frac{(mn)!}{m!\,(n!)^m}, \qquad m, n \geq 1, $$

on the algebra generators $\gamma_n$, and extends $\gamma_m$ to an arbitrary element of $\Gamma^{+}(R)$ by the additivity axiom over the divided monomial basis. That the result is a divided power structure is the standard structure theorem for the free divided power algebra (Berthelot–Ogus, §3.1; Roby 1965), which we quote; the two normalisations are visible directly in the coefficient.

Proposition (normalisation of the composition law). $c_{1,n} = 1$ and $c_{m,1} = 1$, so that $\gamma_1(\gamma_n) = \gamma_n$ and $\gamma_m(\gamma_1) = \gamma_m$.

Proof. $c_{1,n} = n!/(1!\,(n!)^1) = 1$ and $c_{m,1} = m!/(m!\,(1!)^m) = 1$. The first is the requirement $\gamma_1 = \operatorname{id}$, the second the requirement that $\gamma_m$ act on the degree-one generator $\gamma_1 = x$ as $\gamma_m(x) = \gamma_m$.

The Multinomial Expansion

The additivity axiom extends from two summands to any finite number.

Proposition. For $u_1, \ldots, u_k \in M$ and $n \geq 0$,

$$ \gamma_n(u_1 + \cdots + u_k) = \sum_{a_1 + \cdots + a_k = n} \gamma_{a_1}(u_1)\,\gamma_{a_2}(u_2) \cdots \gamma_{a_k}(u_k). $$

Proof. Induction on $k$. For $k = 1$ the statement is trivial and for $k = 2$ it is the additivity axiom. Assume it for $k - 1$ and put $v = u_1 + \cdots + u_{k-1}$. Then by additivity in the form $k = 2$,

$$ \gamma_n(v + u_k) = \sum_{i+j=n}\gamma_i(v)\gamma_j(u_k) = \sum_{i+j=n}\Bigl(\sum_{a_1+\cdots+a_{k-1}=i}\gamma_{a_1}(u_1)\cdots\gamma_{a_{k-1}}(u_{k-1})\Bigr)\gamma_j(u_k), $$

which is the stated sum with $a_k = j$.

Corollary. For a free module with basis $e_1, \ldots, e_d$ and $u = \sum_i c_i e_i$,

$$ \gamma_n(u) = \sum_{|a| = n} c^a \gamma^{(a)}, $$

consistent with the divided monomial basis; the coefficient of $\gamma^{(a)}$ is $c^a = \prod_i c_i^{a_i}$.

Example. Over $\mathbb{Q}$, with $\gamma_n(u) = u^n/n!$, the multinomial expansion becomes the classical statement

$$ \frac{(u_1 + \cdots + u_k)^n}{n!} = \sum_{a_1 + \cdots + a_k = n} \frac{u_1^{a_1}}{a_1!}\cdots\frac{u_k^{a_k}}{a_k!}, $$

so the divided power axioms are the binomial and multinomial theorems in their integrated form.

Comparison with the Symmetric Algebra

The degree-one part of $\Gamma(M)$ is $M$, and $\Gamma(M)$ is a commutative $R$-algebra, so by the universal property of the symmetric algebra (The Symmetric Algebra) there is a unique algebra homomorphism

$$ c : \operatorname{Sym}(M) \longrightarrow \Gamma(M), \qquad x \longmapsto \gamma_1(x), $$

the canonical comparison map. It is compatible with the gradings. Its behaviour depends on the factorials.

Theorem. If $n!$ is invertible in $R$ for every $n \geq 0$, then the canonical map $c$ is an isomorphism for every $M$. Conversely, if $c$ is an isomorphism for a free module $M$ of rank one — in particular if $c$ is an isomorphism for every $M$ — then $n!$ is invertible in $R$ for every $n \geq 0$.

Proof. Suppose first that every $n!$ is invertible. Then $\operatorname{Sym}(M)$ carries a divided power structure with

$$ \gamma_n^{\operatorname{Sym}}(u) = \frac{u^n}{n!}, \qquad u \in \operatorname{Sym}^{+}(M). $$

The four axioms are the binomial theorem, the identity $\lambda^n u^n/n! = (\lambda u)^n/n!$, the identity $(u^m/m!)(u^n/n!) = u^{m+n}/(m!n!) = \binom{m+n}{m}u^{m+n}/(m+n)!$, and

$$ \gamma_m^{\operatorname{Sym}}\bigl(\gamma_n^{\operatorname{Sym}}(u)\bigr) = \frac{1}{m!}\Bigl(\frac{u^n}{n!}\Bigr)^m = \frac{u^{mn}}{m!\,(n!)^m} = \frac{(mn)!}{m!\,(n!)^m}\,\frac{u^{mn}}{(mn)!}, $$

which is the composition law. By the universal property of $\Gamma(M)$ there is therefore an algebra homomorphism $\Gamma(M) \to \operatorname{Sym}(M)$ with $\gamma_n(x) \mapsto x^n/n!$, and it is inverse to $c$.

Conversely, suppose $c$ is an isomorphism. Take a free module of rank one, $M = Rx$, so $\operatorname{Sym}(M) = R[x]$ and $\Gamma(M) = \bigoplus_n R\gamma_n$. In degree $n$ the map $c_n$ is $R x^n \to R\gamma_n$, $x^n \mapsto \gamma_1^n = n!\,\gamma_n$, so with the chosen bases $c_n$ is multiplication by $n!$ on $R$. If $c$ is an isomorphism then so is $c_n$, and multiplication by $n!$ is an isomorphism of $R$ exactly when $n!$ is a unit. Hence $n!$ is a unit for every $n$, as claimed.

Corollary. Over a $\mathbb{Q}$-algebra, and in particular in characteristic zero, the canonical map is an isomorphism; the divided power algebra and the symmetric algebra coincide. Over $\mathbb{Z}$ or over $\mathbb{F}_p$ they do not.

Comparison table.

Notion	Symmetric algebra $\operatorname{Sym}(M)$	Divided power algebra $\Gamma(M)$
Degree-one part	$M$	$M$
Basis for $M$ free, $\|a\| = n$	$e^a$	$\gamma^{(a)}$
Product	$e^a e^b = e^{a+b}$	$\gamma^{(a)}\gamma^{(b)} = \binom{a+b}{a}\gamma^{(a+b)}$
Powers of one element	$x^m x^n = x^{m+n}$	$\gamma_m(x)\gamma_n(x) = \binom{m+n}{m}\gamma_{m+n}(x)$
Composition	$x \mapsto x$	$\gamma_m(\gamma_n(x)) = \frac{(mn)!}{m!(n!)^m}\gamma_{mn}(x)$
Concrete model over $\mathbb{Q}$	polynomials $x^a$	$x^a/a!$
Rank in degree $n$, $M$ free of rank $d$	$\binom{n+d-1}{d-1}$	$\binom{n+d-1}{d-1}$
Universal property	free commutative algebra	free divided power algebra

The rank is the same in every degree; what differs is the multiplication and the divided power structure, and over a ring in which factorials are not invertible the two algebras are genuinely different even though their underlying free modules have the same rank.

Small-Characteristic Examples

Example (characteristic $2$). Let $R = \mathbb{F}_2$ and $M = Rx$ free of rank $1$. In $\Gamma(R)$ we have $\gamma_1^2 = 2\gamma_2 = 0$, and more generally $\gamma_1^n = n!\,\gamma_n = 0$ for $n \geq 2$. The image of the canonical map $c$ is therefore spanned by $1$ and $\gamma_1$, while $\Gamma(R)$ has the basis $\gamma_0, \gamma_1, \gamma_2, \ldots$. So $c$ is neither injective nor surjective: its kernel contains $x^n$ for all $n \geq 2$, and its image is two-dimensional. Over $\mathbb{F}_2$ both $\operatorname{Sym}(R) = \mathbb{F}_2[x]$ and $\Gamma(R)$ are free of countable rank, so the failure is not one of size but of algebra structure: the element $x$ is nilpotent in $\Gamma(R)$ and not in $\operatorname{Sym}(R)$, and the two algebras are not isomorphic.

Example (characteristic $p$). Let $R = \mathbb{F}_p$ and $M = Rx$ free of rank $1$. Since $\gamma_1 \gamma_{p-1} = \binom{p}{1}\gamma_p = p\,\gamma_p = 0$, the product of the first and $(p-1)$-st divided powers vanishes while $\gamma_p \neq 0$. This is the phenomenon that makes divided powers indispensable in crystalline cohomology and in the deformation theory of rings: the divided power $\gamma_p(x)$ survives where the $p$-th power $x^p$ has been divided out by the Frobenius relation.

Functoriality and Base Change

The Functor $\Gamma$

Let $u : M \to N$ be $R$-linear. The degree-one map $M \to \Gamma(N)$, $x \mapsto \gamma_1(u(x))$, extends by the universal property to a morphism of divided power algebras

$$ \Gamma(u) : \Gamma(M) \longrightarrow \Gamma(N), \qquad \gamma_n(x) \longmapsto \gamma_n(u(x)). $$

The construction is functorial: $\Gamma(\operatorname{id}_M) = \operatorname{id}_{\Gamma(M)}$ and $\Gamma(v \circ u) = \Gamma(v) \circ \Gamma(u)$. It depends on the divided power structure, whereas the module-level map $\Gamma^n(M) \to \Gamma^n(N)$ induced by $u$ does not.

Base Change

For a ring homomorphism $R \to S$ and an $R$-module $M$ there is a natural isomorphism of $S$-algebras

$$ S \otimes_R \Gamma_R(M) \cong \Gamma_S(S \otimes_R M), $$

because the divided power axioms are equations with coefficients in $\mathbb{Z}$ and hence are preserved by extension of scalars, and because extension of scalars is left adjoint to restriction and commutes with the tensor powers out of which the free $\Gamma$ is built.

The Divided Power Envelope

There is a relative version of the construction that is used more often than the free algebra itself. Let $A$ be an $R$-algebra and let $I \subseteq A$ be an ideal. A divided power structure on $I$ is a family of maps $\gamma_n : I \to I$ satisfying the four axioms above with $x, y \in I$. The divided power envelope of $I$ (or of $A$ with respect to $I$) is the initial $A$-algebra $D_A(I)$ on which the ideal $ID_A(I)$ carries a divided power structure: the universal $A$-algebra generated by the divided powers $\gamma_n(x)$, $x \in I$, subject to the divided power axioms and to the relations among the elements of $I$ that already hold in $A$. It is written $D_A(I)$ and is characterised by a universal property dual to that of $\Gamma$; it is a quotient of the divided power algebra $\Gamma_A(I)$ of the $A$-module $I$. The construction appears in crystalline cohomology, where the site is built from divided power thickenings, and in the theory of $\delta$-rings; the present article develops only the free, absolute case, and the relative case is cited standard.

Truncated Divided Powers

The divided power algebra is infinite-dimensional even for a module of rank one, since the $\gamma_n$ are independent. Passing to a quotient in which the divided powers vanish above a fixed degree gives the truncated divided power algebra.

Definition. For $N \geq 0$, the $N$-th truncation is

$$ \Gamma^{\leq N}(M) = \Gamma(M) \big/ \Gamma^{>N}(M), \qquad \Gamma^{>N}(M) = \bigoplus_{n > N} \Gamma^n(M), $$

the quotient by the ideal of elements of degree greater than $N$; that ideal is preserved by the divided powers, because the operations raise degree, $\gamma_m(\Gamma^k(M)) \subseteq \Gamma^{mk}(M)$, as the normal form theorem quoted above shows, and the additivity law writes $\gamma_m$ of a sum of homogeneous pieces as a sum of products of such homogeneous divided powers, each of total degree greater than $N$. Its degree-$n$ part therefore vanishes for $n > N$, and a morphism of divided power algebras out of $\Gamma^{\leq N}(M)$ is a morphism out of $\Gamma(M)$ that vanishes on $\Gamma^n(M)$ for every $n > N$; the truncation is the largest quotient of $\Gamma(M)$ whose higher degrees vanish. For $M$ free with basis $e_1, \ldots, e_d$ it has basis the divided monomials $\gamma^{(a)}$ with $|a| \leq N$; the product of two elements is $\binom{a+b}{a}\gamma^{(a+b)}$ if $|a+b| \leq N$ and $0$ otherwise.

Example. $\Gamma^{\leq 1}(M) = R \oplus M$ is the square-zero algebra on $M$: the product of two elements of $M$ vanishes. The truncation $\Gamma^{\leq N}$ is the algebra of divided power jets of order $N$, and the filtration of an algebra by the powers of an ideal, together with the divided power structures on the successive quotients, is the algebraic content of the theory of thickenings and of formal groups.

Summary

The divided power algebra $\Gamma(M)$ is the free divided power algebra on the module $M$: a graded commutative $R$-algebra with degree-one part $M$, generated by the divided powers $\gamma_n(x)$, and universal among divided power algebras. It is defined by the additivity law $\gamma_n(x+y) = \sum_{i+j=n}\gamma_i(x)\gamma_j(y)$, the product law $\gamma_m(x)\gamma_n(x) = \binom{m+n}{m}\gamma_{m+n}(x)$, the scaling law $\gamma_n(\lambda x) = \lambda^n\gamma_n(x)$, and the composition law $\gamma_m(\gamma_n(x)) = \frac{(mn)!}{m!(n!)^m}\gamma_{mn}(x)$, whose coefficient is a positive integer. For $M$ free with basis $e_1, \ldots, e_d$ it has the divided monomial basis $\gamma^{(a)}$ with product $\gamma^{(a)}\gamma^{(b)} = \binom{a+b}{a}\gamma^{(a+b)}$. There is a canonical comparison homomorphism $\operatorname{Sym}(M) \to \Gamma(M)$ carrying $x$ to $\gamma_1(x)$; it is an isomorphism exactly when every factorial is invertible in $R$, and in characteristic $p$ it fails in the most visible way, with $\gamma_1\gamma_{p-1} = 0$ while $\gamma_p \neq 0$. The construction is functorial and commutes with base change, and the relative divided power envelope $D_A(I)$ is the standard tool of crystalline and deformation theory.

Summary of Notation

Symbol	Meaning
$R$	Commutative ring with identity $1 \neq 0$
$M, N$	$R$-modules
$\Gamma(M) = \bigoplus_n \Gamma^n(M)$	Divided power algebra of $M$
$\Gamma^n(M)$	Degree-$n$ divided power, $\Gamma^0(M) = R$, $\Gamma^1(M) = M$
$\gamma_n : M \to \Gamma^n(M)$	Divided power map, $\gamma_0 = 1$, $\gamma_1 = \operatorname{id}$
$x^{(n)} = \gamma_n(x)$	Divided power of an element
$\gamma^{(a)} = \prod_i \gamma_{a_i}(e_i)$	Divided monomials, for $M$ free with basis $(e_i)$
$\binom{m+n}{m}$	Coefficient in the product law
$\frac{(mn)!}{m!(n!)^m}$	Coefficient in the composition law
$\operatorname{Sym}(M)$	Symmetric algebra, compared with $\Gamma(M)$ via $c$
$c : \operatorname{Sym}(M) \to \Gamma(M)$	Canonical comparison map, $x \mapsto \gamma_1(x)$
$\gamma_n^{\operatorname{Sym}}(u) = u^n/n!$	Divided power structure on $\operatorname{Sym}(M)$ when $n!$ is invertible
$\Gamma(u)$	Morphism of divided power algebras induced by $u$
$\Gamma^{\leq N}(M)$	$N$-th truncation of $\Gamma(M)$
$D_A(I)$	Divided power envelope of the ideal $I \subseteq A$