Differential Graded Algebra with Divided Powers

• Differential graded algebras with divided powers are graded-commutative DGAs equipped with compatible divided power operations, enhancing resolution and cohomology analyses.
• They enable explicit factorization theorems and semi-free constructions that underpin derived functor theory, minimal resolutions, and homotopy Lie algebra computations.
• Their applications span algebraic topology, commutative algebra, and derived algebraic geometry, offering practical methods for studying de Rham cohomology and crystalline complexes.

A differential graded algebra (DGA) with divided powers is an algebraic structure combining a graded-commutative DGA with an additional system of "divided power" operations on its even-degree elements or ideals, compatible with the grading, differential, and multiplication. This enhances the algebraic control over resolutions, cohomology theories, and deformation phenomena, with critical applications in commutative algebra, algebraic topology, and derived algebraic geometry. The past decade has seen significant progress in the construction, characterization, and application of such objects, with a focus on their structural theorems, resolutions, universality properties, and categories.

1. Core Definitions and Fundamental Properties

Let $A = \bigoplus_{i\in\mathbb{Z}} A^i$ be a graded-commutative associative algebra over a commutative ring, with differential $d_A: A^i \to A^{i+1}$ satisfying $d_A \circ d_A = 0$ and a (graded) Leibniz rule: $$d_A(a \cdot b) = d_A(a) \cdot b + (-1)^{\deg a} a \cdot d_A(b)$$. Odd elements are strictly square-zero, $a^2 = 0$ if $\deg(a)$ is odd.

A divided power structure on a DG ideal $I \subset A$ is a family of maps $\gamma_n: I \to A$, $n\ge 0$, defined on homogeneous elements and of appropriate cohomological degree, subject to the following axioms:

• $\gamma_0(x) = 1$, $\gamma_1(x) = x$;
• $\gamma_n(a x) = a^n \gamma_n(x)$ for $a\in A^0$;
• $$\gamma_m(x)\gamma_n(x) = \binom{m+n}{m}\gamma_{m+n}(x)$$;
• $$\gamma_n(x + y) = \sum_{i=0}^n \gamma_i(x) \gamma_{n-i}(y)$$;
• $d_A(\gamma_n(x)) = d_A(x)\gamma_{n-1}(x)$ for all $n \ge 1$;
• If $\mathrm{char}(A^0) = 0$, then $\gamma_n(x) = x^n/n!$.

These axioms generalize the classical divided power algebra formalism to the DG context and (when defined on the entire algebra) provide a pd–DGA in the sense of André–Roby. Strict graded-commutativity, specifically $a^2 = 0$ for odd $a$, is essential to the coherence of the differential and divided power operations (Yekutieli, 2023, Caradot et al., 27 Nov 2025).

2. Structural Theorems and Factorizations

One of the central advances is Yekutieli's factorization theorem: any quasi-isomorphism $f: A \to B$ of commutative DG rings, with $B$ admitting a PD-structure on its augmentation ideal, factors as $$A \xrightarrow{e} \widetilde{B} \xrightarrow{\tilde{f}} B$$ where $e$ is split-injective quasi-iso and $\tilde{f}$ is a surjective quasi-iso, with $\widetilde{B}$ semi-free as a DG-$A$-module (Yekutieli, 2023). The construction involves a controlled cell attachment procedure: free DG cells for odd cohomological degree generators and PD–cells for even degrees. This provides explicit semi-free CDG constructions with PD-structures, facilitating homotopical and derived invariants.

An immediate consequence is the existence of functorial cofibration–fibration factorizations in the model category of commutative DG rings, and hence the possibility of derived functors (tensor, Hom, cotangent complex) and the formation of the derived category $\mathrm{Ho}(\mathrm{CDGRng})$ reflecting quasi-isomorphism classes.

3. Canonical Constructions and Free pd–DG Algebras

The construction of free pd–DG algebras is achieved via the symmetric tensor (shuffle) algebra on a complex: for a cochain complex $X^*$, the functor $$TS(X) = \bigoplus_{n\ge0} (X^{\otimes n})^{\Sigma_n}$$ with the shuffle product and differential defines a strictly graded-commutative DGA with natural divided power operations $\gamma_n(x) = x^{(n)}$—symmetrized tensor powers normalized by $n!$. This structure is universal: $TS$ is left adjoint to the forgetful functor from pd–DG algebras to complexes (Caradot et al., 27 Nov 2025).

This underpins standard constructions of minimal (Koszul–Tate-type) resolutions, including in non-Noetherian settings, by successively adjoining generators (even or odd) to kill cohomology in higher degrees (Caradot et al., 27 Nov 2025). All structural and functorial axioms, including compatibility with the differential and external functoriality under base change, follow from the combinatorics of the shuffle product and binomial/Catalan coefficients.

4. Distinguished Examples and Resolutions

Resolutions of length four that are self-dual and acyclic (over a commutative Noetherian ring $P$) always carry the structure of a DGA with divided powers (a DGT-algebra), and their multiplication realizes Poincaré duality (Kustin, 2019):

• The multiplication uses perfect pairings arising from self-duality.
• The divided power structure is declared on the even-degree summands, with axioms verified directly.
• Neither locality, Gorenstein, nor minimality assumptions are needed.

Similarly, for monomial ideals in polynomial rings, the Taylor resolution and its generalizations admit DG $\Gamma$-algebra structures (i.e., DG-algebra with divided powers), with uniqueness and functoriality results for the attached (homotopical) invariants (Ferraro et al., 8 Jul 2025).

5. Derived De Rham, Crystalline Cohomology, and Theoretical Functors

In derived algebraic geometry, derived divided power DGAs (PD–DGAs) underpin the modern formalism of derived de Rham and derived crystalline cohomology (Magidson, 8 May 2024):

• A derived PD–DGA is equivalently given via Smith ideals, filtered algebras, or DG–algebras with divided powers, with rigorous translation between these models.
• The derived de Rham complex $L^\bullet_{A/R}$ is characterized as the initial object among PD–DGAs under $A$ and satisfies a universal property.
• The construction recovers classical De Rham cohomology in the smooth case, matching the classical Hodge filtration and connecting to the crystalline site via PD–envelopes. Computations are informed by bar–Cech totalizations and the Rees/decalage equivalence.

The existence of these universal objects ensures the compatibility of de Rham and higher PD-operations in derived settings, crucial for both algebraic geometry and deformation theories.

6. Applications: Homotopy Lie Algebras and Resolutions

DGAs with divided powers provide canonical presentations for the homotopy Lie algebra $\pi^*(R, R/\mathfrak{m})$ of a local ring, via coderivation/derivation constructions on the pd–DG resolution (Caradot et al., 27 Nov 2025, Ferraro et al., 8 Jul 2025). For complete intersections the resulting pd–DG algebra truncates, yielding explicit quadratic generators and relations for the Ext algebra and homotopy Lie algebra (Milnor–Moore, Koszul duality paradigm).

In combinatorial commutative algebra, the divided power structures on generalized Taylor complexes enable functorial identification of homotopy Lie algebras and establish a uniform framework for resolutions of (squarefree) monomial ideals, showing that the DG $\Gamma$–structure descends to minimal resolutions and to key subcomplexes such as the Scarf complex (Ferraro et al., 8 Jul 2025).

7. Model Structures, Uniqueness, and Canonical Lifting

The model category structure on commutative DG rings with divided powers (cofibrations as retracts of semi-free extensions, weak equivalences as quasi-isos, fibrations as surjections in nonpositive degrees) is tightly controlled by the existence and properties of free and functorial pd–DG resolutions (Yekutieli, 2023, Caradot et al., 27 Nov 2025). Key results include:

• Any pd–DG-algebra resolution is unique up to pd–DG-isomorphism once bases are fixed in cohomology (Caradot et al., 27 Nov 2025).
• Lifting and minimality results: any quasi-surjection carrying the pd–ideal surjectively admits a lift from the free pd–DG algebra, and any two pd–DG algebras resolving the same object are canonically isomorphic up to the divided power structure.

These properties ensure the canonicity and universality necessary for derived functor theory, cotangent complexes, and functorial invariants in both algebraic and topological contexts.