Differential Graded Algebras with Divided Powers

• Differential graded algebras with divided powers are graded commutative dg algebras endowed with divided-power operations on even-degree elements that satisfy precise axioms.
• They utilize ∞-categorical and symmetric-tensor constructions to create robust frameworks for derived de Rham cohomology, prismatic cohomology, and cotangent complex computations.
• Explicit realizations in minimal and free resolutions, along with applications in homotopy Lie algebras and Poincaré duality, underscore their significance in modern derived algebraic geometry.

A differential graded algebra with divided powers (commonly abbreviated as PD-DGA, DGΓ-algebra, or "DGT-algebra" for Editor’s term) is a commutative differential graded algebra endowed with a compatible structure of divided-power operations on its even-degree elements. This notion generalizes both classical divided-power (pd) algebras and the more homotopical framework of dg-algebras, establishing the foundational algebraic environment for derived De Rham theory, prismatic cohomology, homotopy Lie algebras, and explicit constructions in commutative algebra and algebraic geometry. The development of the theory in recent work extends from ∞-categorical definitions and universal constructions in derived algebraic geometry (Magidson, 2024), through explicit combinatorial and symmetric-tensor resolutions (Caradot et al., 27 Nov 2025, Ferraro et al., 8 Jul 2025), to applications in Poincaré duality, minimal free resolutions, and cotangent complex computations (Kustin, 2019, Yekutieli, 2023).

1. Formal Definition and Core Axioms

A differential graded algebra with divided powers consists of a cochain complex $A^*=\bigoplus_{i\in\mathbb{Z}}A^i$ over a commutative base ring, equipped with an associative, unitary, graded-commutative multiplication, and a differential $d$ satisfying the graded Leibniz rule: $$d(m(a\otimes b)) = m(da \otimes b) + (-1)^{|a|} m(a \otimes db).$$ The algebra is graded-commutative: $a\cdot b = (-1)^{|a|\,|b|}b\cdot a$ and $a^2=0$ for $|a|$ odd.

A PD-DGA includes, for every even-degree $x$ in a distinguished dg-ideal $I$ (generated in positive degrees), a family of "divided power" operations $\Gamma^n(x)$ satisfying the following axioms (Caradot et al., 27 Nov 2025, Ferraro et al., 8 Jul 2025):

• $\Gamma^0(x)=1$, $\Gamma^1(x)=x$;
• $$\Gamma^n(x)\Gamma^m(x)=\binom{n+m}{n}\Gamma^{n+m}(x)$$;
• $\Gamma^n(a\,x)=a^n\Gamma^n(x)$ for $a$ even;
• $$\Gamma^n(x+y)=\sum_{i=0}^n\Gamma^i(x)\Gamma^{n-i}(y)$$;
• $$\Gamma^p(\Gamma^q(x))=\frac{(pq)!}{p!(q!)^p}\Gamma^{pq}(x)$$;
• $\Gamma^n(I^i)\subseteq I^{ni}$;
• If $|x|$ or $|y|$ odd, $\Gamma^n(xy)=0$ for $n\ge2$;
• $d(\Gamma^n(x))=\Gamma^{n-1}(x)d(x)$.

These axioms ensure compatibility between the graded structure, differential, and divided-power operations.

2. Derived ∞-Categorical and Monadic Constructions

In the context of derived algebraic geometry, the modern approach realizes non-unital derived divided-power algebras as algebras over a sifted-colimit-preserving monad $L^+_R$ on the symmetric-monoidal ∞-category of $R$-module spectra (Magidson, 2024). Explicitly,

$$L^+_R(M) \cong \bigoplus_{n\ge1} (M^{\otimes n})^{\Sigma_n},$$

where tensors are taken modulo symmetric group action, encoding homotopy-invariant divided powers.

Both unital and filtered versions are obtained via monadicity on the category of arrows and filtered complexes, allowing for universal filtered pd thickenings with functorial Hodge filtration and Rees-type equivalence to certain dg-algebra categories.

Free derived pd-algebras are realized as direct sums of symmetric invariants, and their universal property establishes initial objects among filtered pd $R$-algebras equipped with fixed degree-zero base.

3. PD-DGA Structures in Classical and Resolution Contexts

Beyond abstract definitions, divided-power structures are concretely realized in minimal resolutions and free resolutions of quotient modules, notably over monomial ideals and in complete intersection cases (Caradot et al., 27 Nov 2025, Ferraro et al., 8 Jul 2025). The symmetric-tensor construction (Koszul–Tate resolution) produces DGΓ–algebras: $P^* = \bigoplus_{n\ge0} TS^n(X^*)$ with shuffle multiplication and normalized shuffle powers for $\Gamma^n(x)$.

The generalized Taylor complex for resolutions of monomial ideals admits a unique DGΓ–structure defined on tensor products of DGΓ–algebras, adhering to the axiomatic framework for divided powers and correcting classic sign ambiguities in product formulas.

In Gorenstein grade-4 situations, as well as arbitrary self-dual acyclic complexes of length four (Kustin, 2019), there exist explicit constructions of DG-algebra structures with divided powers and perfect pairings, exhibiting Poincaré duality and extending previously local or minimal assumptions.

4. Universal Properties, Quasi-Isomorphism, and Transfer

The existence and uniqueness of PD-DGA structures on resolutions is secured by universal properties. Any free resolution of a module over a base ring admits a functorial extension to a strictly graded-commutative PD–dg algebra, with the property that any two such extensions are uniquely isomorphic (Caradot et al., 27 Nov 2025). For commutative DG ring quasi-isomorphisms where the target admits a PD–structure, Yekutieli proves a factorization theorem: every such $f: A \to B$ factors via a split injective and surjective quasi-isomorphism, with an intermediate algebra constructed as a semi-free extension by divided-power generators (Yekutieli, 2023). This process enables transfer of PD-structures across quasi-isomorphisms, facilitating the construction of bar-cobar resolutions, cotangent complexes, and derived functorial models for $\mathrm{Ext}$ and $\mathrm{Tor}$.

5. Interactions with Derived De Rham Cohomology and Filtrations

PD-DGAs constitute the natural environment for derived De Rham and crystalline cohomology. In this setup, the derived De Rham cohomology of a derived commutative algebra $A$ over $R$ is characterized by the universal property as the largest filtered divided-power thickening of $A$ (Magidson, 2024). The associated graded of the filtered pd complex recovers the ordinary derived De Rham algebra, which reduces to classical De Rham in the smooth case and aligns with prismatic and crystalline cohomology when appropriate divided-power bases are chosen.

The adic pd filtration recovers the Hodge filtration, and specific truncations produce square-zero extensions by the cotangent complex, integrating seamlessly with the derived algebraic geometric framework.

6. Homotopy Lie Algebras and Ext-Algebra Calculations

PD-DGAs facilitate the construction of homotopy Lie algebras via the complex of PD-derivations

$\Der^{*,pd}_R(P^*,P^*) = \left\{ D | D(\Gamma^n(x)) = \Gamma^{n-1}(x)D(x) \right\},$

with well-defined differentials and quadratic operations. The induced maps on cohomology embed into the Yoneda algebra $\mathrm{Ext}^*_R(R/\mathfrak{m}, R/\mathfrak{m})$, and in the complete intersection case, explicit presentations of Ext-algebras showcase the control of Massey-type products and quadratic relations by the divided powers (Caradot et al., 27 Nov 2025, Ferraro et al., 8 Jul 2025). Homotopy Lie algebra structures are thus tightly woven with the PD–derivation complex.

7. Examples, Limitations, and Broader Implications

Explicit low-degree examples illustrate DGΓ-structure on resolutions of principal or nonprincipal ideals, confirming the vanishing of high divided powers on odd-degree generators and the compatibility of corrected sign conventions. The existence of Poincaré duality in graded-commutative PD-DGAs of length four is established for non-local, non-Gorenstein, and non-minimal resolutions (Kustin, 2019). However, for lengths $\geq 5$ or non-self-dual resolutions, obstructions generally arise, and associative multiplicative structures may fail. In characteristic zero, the PD-structure is uniquely determined by $\gamma_k(a)=a^k/k!$, but non-existence phenomena are evident over $\mathbb{Z}$ due to relations such as $\gamma_2(x)^2=2\gamma_2(x)$ (Yekutieli, 2023).

The integration of DG-algebra, divided-power, and homotopical techniques in this setting underpins the current landscape of algebraic and geometric computations in the derived, prismatic, and cohomological context. The formalism enables derived functor constructions, affirms rigidity and transfer properties under quasi-isomorphism, and elucidates universal properties essential to the structure of de Rham and crystalline cohomology in both classical and derived settings.