Universal Divided Power Algebra

• Universal divided power algebra is an algebraic structure that formalizes divided powers of module elements with canonical combinatorial relations and a graded algebra structure.
• The construction, introduced by Roby, underpins key developments in algebraic geometry, deformation theory, and cohomological frameworks like crystalline cohomology and p-adic Hodge theory.
• Its universal property with respect to polynomial laws and divided power axioms provides a systematic approach for computations in both classical and derived algebraic settings.

The universal divided power algebra formalizes the concept of "divided powers" of module elements over a commutative ring, yielding a canonical algebraic structure central to algebraic geometry, deformation theory, and $p$-adic cohomological frameworks. The construction, originally due to Roby (1965), provides the minimal commutative algebra in which the divided powers of a module or ideal satisfy precise arithmetic and combinatorial relations. Its graded pieces encode the functorial theory of polynomial laws, and it realizes a universal property with respect to divided-power structures. The universal divided power algebra, denoted $\Gamma_R(M)$ for an $R$-module $M$, admits extensive generalizations to derived and filtered contexts and plays a key role in the study of crystalline cohomology, $p$-adic Hodge theory, and the theory of Kähler differentials.

1. Explicit Construction and Defining Relations

Let $R$ be a commutative ring and $M$ an $R$-module. The universal divided power algebra $\Gamma_R(M)$ is defined as the quotient

$$\Gamma_R(M) := R[X_{n,x} \mid n\in\mathbb{N}, x\in M] / I,$$

where $I$ is the ideal generated by the following relations for all $x, y\in M$, $r\in R$, and $m, n \in \mathbb{N}$:

• $X_{0,x} - 1$,
• $X_{n, x + y} - \sum_{i + j = n} X_{i, x} X_{j, y}$,
• $X_{n, r x} - r^n X_{n, x}$,
• $X_{m, x} X_{n, x} - \binom{m+n}{m} X_{m+n, x}$.

Write $x^{[n]} :=$ the class of $X_{n,x}$ in the quotient. Thus, the $R$-algebra $\Gamma_R(M)$ is generated by the symbols $x^{[n]}$ subject to the divided power relations (Chambert-Loir et al., 5 Dec 2025, Kmail et al., 9 Feb 2025).

2. Graded Algebra Structure and Divided Power Axioms

Each relation is homogeneous of total degree $n$, conferring a canonical grading: $\Gamma_R(M) = \bigoplus_{d \geq 0} \Gamma^d_R(M),$ where $x^{[n]}$ has degree $n$. Notably, $\Gamma^0_R(M) \cong R$ and $\Gamma^1_R(M) \cong M$ via $1_R \mapsto 1 = 1^{[0]},\ x \mapsto x^{[1]}$.

The basic divided power operations satisfy for all $x$ in the augmentation ideal $\Gamma^+_R(M)$, $r \in R$, and $m, n \geq 0$: $\gamma_0(x) = 1, \quad \gamma_1(x) = x,$

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

The map $x^{[1]} \mapsto x^{[n]}$ realizes the divided powers on the augmentation ideal. When $R$ is a $\mathbb{Q}$-algebra, one recovers $x^{[n]} = x^n / n!$ (Chambert-Loir et al., 5 Dec 2025, Kmail et al., 9 Feb 2025).

3. Universal Property and Polynomial Laws

A divided power $R$-algebra is a commutative $R$-algebra $A$, an ideal $J$, and maps $\gamma_n : J \to A$ satisfying the Cartan axioms. The universal property asserts: for any $R$-linear map $\varphi : M \to J$, there exists a unique $R$-algebra homomorphism

$\tilde{\varphi}:\; \Gamma_R(M) \longrightarrow A$

such that $\tilde{\varphi}(x^{[n]}) = \gamma_n(\varphi(x))$ for all $x\in M$, $n \geq 0$.

Roby's perspective identifies the graded pieces $\Gamma^n_R(M)$ as the $R$-module of degree-$n$ polynomial laws $M \Rightarrow R$. More precisely, the assignment $x \mapsto x^{[n]}$ is universal among degree-$n$ homogeneous polynomial laws, yielding a concrete description: $\Gamma^n_R(M) \simeq \left\{ \text{degree-%%%%53%%%% polynomial laws } M \Rightarrow R \right\}$ as $R$-modules (Chambert-Loir et al., 5 Dec 2025, Kmail et al., 9 Feb 2025).

4. Augmentation Ideal and Combinatorial Formulas

The augmentation ideal $\Gamma^+_R(M) = \bigoplus_{n > 0} \Gamma^n_R(M)$ admits a canonical divided power structure. For $M$ free with basis $\{b_i\}$ and $x = \sum x_i b_i$: $$\gamma_n(x) = \sum_{i_1 + \dots + i_r = n} \frac{n!}{i_1! \cdots i_r!} \prod_j (x_j b_j)^{[i_j]}.$$ This formula realizes the multinomial combinatorics required by the Cartan axioms. For arbitrary $M$, divided powers descend from a free presentation via a quotient, with careful attention to well-definedness and divisibility (Chambert-Loir et al., 5 Dec 2025).

5. Universal Enveloping Algebra and Kähler Differentials

For any divided power algebra $A$, its universal enveloping algebra is expressed as

$U(A) = A^+ \otimes_R U(0)$

where $A^+ = A \oplus R$ and $$U(0) = R \langle e_p \mid p \text{ prime} \rangle / (e_p^p)$$.

A divided power (DP) derivation $D:A\to M$ satisfies: $D(ab) = a D(b) + b D(a)$

$$D(\gamma^n(a)) = \gamma^n(D(a)) + \sum_{i+j=n,\, i,j>0} \gamma^i(a)\gamma^j(D(a))$$

There exists a universal DP-$A$-module $\Omega^1_{A/R}$ and DP-derivation $d : A \to \Omega^1_{A/R}$ fulfilling the universal property.

In the free case $A = \Gamma_R(V)$, one finds $\Omega^1_{A/R} \cong U(A) \otimes_R V$, and explicit formulas describe the DP-differentials, e.g., for $V = R x$: $$d\,\gamma^n(x) = \sum_{k=1}^n (-1)^{k-1} \binom{n}{k} \gamma^{n-k}(x) \otimes x$$ with further specializations in characteristic $p$ (Kmail et al., 9 Feb 2025).

6. Derived and Filtered Contexts, Cohomological Applications

In derived algebraic geometry, the free derived divided power monad generalizes to spectra and $\infty$-categories. On a base $R$, the derived divided power algebra is

$$\Gamma_R^+(M) = \bigoplus_{n \geq 1} (M^{\otimes n})^{\Sigma_n}$$

The filtered derived divided power algebra $\Gamma_R^{\geq *}(A)$ carries the Hodge filtration, with the associated graded pieces identified as the exterior powers of the cotangent complex $L_{A/R}$. The functor $\Gamma_R^*(A) = L^{\geq *}_{A/R}$ is characterized by a universal property with respect to filtered DP-thickenings.

This formalism recovers, in the smooth or regular quotient case, the classical divided power envelope used in the construction of crystalline cohomology and the period rings of $p$-adic Hodge theory (Magidson, 8 May 2024).

7. Formalization and Computational Aspects

The universal divided power algebra and associated polynomial laws have been formalized in Lean/Mathlib, encompassing definitions, graded algebra structures, base-change isomorphisms, and multinomial coefficients. Key difficulties addressed include management of universe levels (to avoid contradictions such as Russell's paradox), extensions to semirings, and explicit encoding of tensor product associators and scalar towers required for polynomial laws. This formalization supports computer-assisted reasoning about DP-algebras and polynomial laws, essential for rigorous developments in modern algebraic geometry (Chambert-Loir et al., 5 Dec 2025).

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References:

• [Formalizing Polynomial Laws and the Universal Divided Power Algebra, (Chambert-Loir et al., 5 Dec 2025)]
• [Divided powers and Kähler differentials, (Kmail et al., 9 Feb 2025)]
• [Divided Powers and Derived De Rham Cohomology, (Magidson, 8 May 2024)]