DG-algebras and derived A-infinity algebras

A differential graded algebra can be viewed as an A-infinity algebra. By a theorem of Kadeishvili, a dga over a field admits a quasi-isomorphism from a minimal A-infinity algebra. We introduce the notion of a derived A-infinity algebra and show that any dga A over an arbitrary commutative ground ring k is equivalent to a minimal derived A-infinity algebra. Such a minimal derived A-infinity algebra model for A is a k-projective resolution of the homology algebra of A together with a family of maps satisfying appropriate relations. As in the case of A-infinity algebras, it is possible to recover the dga up to quasi-isomorphism from a minimal derived A-infinity algebra model. Hence the structure we are describing provides a complete description of the quasi-isomorphism type of the dga.

💡 Research Summary

The paper addresses a fundamental limitation of classical A‑infinity (A‑∞) algebra theory: while any differential graded algebra (dga) over a field admits a minimal A‑∞ model by Kadeishvili’s theorem, the same result does not automatically extend to dga’s over an arbitrary commutative ground ring k. To overcome this obstacle the authors introduce the concept of a derived A‑∞ algebra (also called a “derived A‑infinity algebra”) and prove that every dga A over any commutative ring k is quasi‑isomorphic to a minimal derived A‑∞ algebra.

A derived A‑∞ algebra is defined on a bigraded complex C^{,} equipped with two commuting differentials: a vertical differential d_vert and a horizontal differential d_hor. Their sum d_total = d_vert + d_hor makes C into a total chain complex. In addition to the usual binary product m_{0,2}, a family of higher operations m_{p,q}: C^{\otimes p} → C^{q} (p ≥ 1, q ≥ 0) is specified. These operations satisfy a collection of relations that generalise the Stasheff identities to the bi‑graded setting; the relations guarantee that the total differential is a derivation with respect to all higher multiplications. The “minimal” condition is that d_vert = 0, m_{0,1}=0, and the only non‑trivial unary operation is the horizontal differential. Consequently the first non‑trivial higher operation m_{0,2} reproduces the ordinary multiplication on the homology algebra H_*(A).

The main existence theorem proceeds as follows. One first chooses a k‑projective resolution P_* → H_(A) of the homology algebra. The resolution is regarded as a vertically concentrated complex (so d_vert = 0). Using homological perturbation techniques adapted to the bi‑graded context, the authors construct higher maps m_{p,q} on P_ that turn (P_, m_{p,q}) into a derived A‑∞ algebra. The construction is inductive: assuming the operations have been defined up to a certain total degree, the next operation is obtained by solving a cohomological equation whose obstruction lives in Ext^{>0}k(H(A),H_(A)). Because P_ is projective, these obstructions vanish, guaranteeing the existence of a full family of operations. The resulting derived A‑∞ algebra is minimal by construction.

Having built a minimal model, the paper shows how to recover the original dga up to quasi‑isomorphism. A transfer map f: (P_, m_{p,q}) → A is defined by extending the canonical projection P_ → H_(A) to a morphism of derived A‑∞ algebras. The higher components of f are obtained by a recursive homotopy formula that mirrors the classical A‑∞ transfer but respects the vertical/horizontal decomposition. One checks that f is a quasi‑isomorphism: it induces an isomorphism on homology and respects all higher multiplications up to homotopy. Conversely, a homotopy inverse g: A → (P_, m_{p,q}) can be constructed, establishing a zig‑zag of quasi‑isomorphisms between A and its minimal derived model.

The authors illustrate the theory with several concrete examples. For dga’s over the integers ℤ, they exhibit an explicit projective resolution of H_*(A) and compute the first few higher operations, showing how torsion phenomena appear naturally in the m_{p,q} terms. In the case of the cohomology ring of the projective line ℙ^1 over a polynomial ring k