Connected Cochain DG Algebra Overview

• Connected cochain DG algebras are N-graded associative algebras equipped with a +1 differential that squares to zero and have A^0 equal to the base field.
• They are pivotal in rational homotopy, noncommutative geometry, and algebraic topology by supporting derived duality and classification through explicit matrix-encoded differentials.
• Their properties, such as homological smoothness, Gorenstein, and Calabi–Yau conditions, underpin advanced invariants and equivalences in derived categories and representation theory.

A connected cochain differential graded (DG) algebra is a $\mathbb{N}$-graded associative $k$-algebra $A$ equipped with a degree $+1$ differential $d$ satisfying $A^0 = k$ (for a field $k$), $A^i = 0$ for $i < 0$, and $d^2 = 0$. The cohomology $H(A)$ is a graded algebra, itself connected when $A$ is connected. Connected cochain DG algebras arise in contexts such as rational homotopy, noncommutative geometry, algebraic topology, and representation theory. They serve as foundational objects for derived categories, duality theories, and invariants that generalize classical constructions to more intricate noncommutative or homotopical frameworks.

1. Structural Definition and Examples

The structural definition requires the underlying graded algebra $A^{\#}$ to be connected: $A^0 = k$ and $A^i = 0$ for $i < 0$. The differential $d$ increases degree by $+1$ and obeys the Leibniz rule: $d(ab) = d(a) b + (-1)^{|a|} a d(b)$. Notable subclasses include:

• DG polynomial algebras: $A^{\#} = k[x_1, \ldots, x_n]$ with $|x_i| = 1$ and $d$ determined by $n$ parameters, yielding formal, homologically smooth, and Gorenstein DGAs, often Calabi–Yau when $n$ is odd or the differential is nontrivial (Mao et al., 2017).
• DG free algebras: $A^{\#} = k\langle x_1, \ldots, x_n \rangle$ with $|x_i|=1$ and differential given by "crisscross ordered" $n$-tuples of $n \times n$ matrices; isomorphism classification is governed by matrix conjugation conditions, and in low dimensions all nontrivial DG free algebras are Koszul and Calabi–Yau (Mao et al., 2018).
• DG Sklyanin and quantum affine space algebras: Connected DGAs where $A^{\#}$ is a noncommutative regular algebra (e.g., Sklyanin), with differential data encoded by matrices and additional parameters, generalizing smooth projective geometry settings (Mao et al., 2020, Mao et al., 2020).
• Singular Hochschild and string topology complexes: Constructed from symmetric Frobenius or self-injective input algebras, serving as cochain or mixed chain–cochain DGAs realizing Tate–Hochschild cohomology and string topology operations (Rivera et al., 2017, Wang, 2018).

2. Duality and the Derived Category

A complete duality picture in the DG setting hinges on the construction of a dualizing DG module $D$, formalized as $$D = (K^* \otimes_A^L K)^{\vee} = \operatorname{Hom}_k(K^* \otimes_A^L K, k)$$, where $K$ is a chosen compact, K-projective DG module generating the relevant derived localizing subcategory. This module induces a contravariant equivalence: $$\operatorname{RHom}_A(-, D) : D_f(A) \longleftrightarrow D_f(A^{\mathrm{op}})$$ between the derived categories of DG left- and right-modules with finitely generated cohomology (Jorgensen, 2010). This construction generalizes Grothendieck duality and works robustly in noncommutative contexts.

Further, the existence of semi-free resolutions of the canonical module $A/A^{\geq 1}$ (often identified with $k$) with generators in degrees bounded above implies analogous bounded semi-free resolutions for any DG module with finitely generated cohomology, controlling Ext-regularity, Castelnuovo–Mumford regularity, and facilitating computations:

$$\dots \to \bigoplus_{s} A(-s)^{\oplus \beta_s} \to \dots \to A/A^{\geq 1}$$

Boundedness propagates from the canonical module to all finitely generated modules.

3. Homological Smoothness, Gorenstein, and Calabi–Yau Properties

Homological smoothness for a connected cochain DG algebra is equivalent to several categorical and homological properties:

• The cone length $\operatorname{cl}_A k < \infty$.
• The left global dimension $\mathrm{I.Gl.dim}\,A < \infty$.
• $D^c(A) = D_{fg}(A)$ (compact objects coincide with cohomologically finite ones).
• Singularity category vanishes: $D_{sg}(A) = 0$ (Mao, 20 Jul 2024, Mao et al., 2013).

If $A$ is additionally Gorenstein, for any $M \in D_{fg}(A)$ one derives the regularity formula: $$\mathrm{CMreg}\,M = \mathrm{depth}_A + \mathrm{Ext.reg}\,M < \infty$$ where $\mathrm{CMreg}$ is Castelnuovo–Mumford regularity, $\mathrm{depth}_A$ is the depth of $A$, and $\mathrm{Ext.reg}\,M$ is the Ext-regularity (Mao, 20 Jul 2024).

A categorical characterization links these properties to the structure of the Ext-algebra $E = H(\operatorname{RHom}_A(k, k))$:

• $A$ is homologically smooth and Gorenstein if and only if $E$ is a Frobenius graded algebra.
• $A$ is Calabi–Yau $\iff$ $E$ is a symmetric Frobenius graded algebra. This generalizes earlier results requiring a Koszul hypothesis (Mao, 20 Jul 2024).

4. Derived Invariants, Picard Groups, and Equivalence

Derived Picard groups $\operatorname{DPic}(A)$ classify self-equivalences of the derived category $D(A)$ generated by tilting DG $A^e$-modules. For homologically smooth Koszul DG algebras, $$\operatorname{DPic}(A) \cong \operatorname{DPic}(E)^{\mathrm{op}}$$ where $E$ is the Ext-algebra. Explicit computations yield groups such as $\mathbb{Z} \times GL_n(k)$ for DG polynomial algebras (Mao et al., 2018, Mao et al., 2018).

However, the derived category $D(A)$ is not determined by the cohomology ring $H(A)$ alone. The existence of non-derived equivalent DG algebras with isomorphic cohomology rings answers a question of Dugas in the negative; derived Picard groups distinguish such cases, underscoring the necessity for finer invariants or A$_\infty$-structures on $H(A)$ (Mao et al., 25 Aug 2025).

5. Explicit Differential Structures and Classification by Matrices

For DG free algebras and quantum affine DGAs, differentials are encoded by tuples of matrices subject to strict compatibility ("crisscross" orderings or quasi-permutation equivalence):

• For $A^{\#} = k\langle x_1, \ldots, x_n \rangle$, $d(x_i) = (x_1, \ldots, x_n) M^i$.
• For quantum affine $n$-space $A^{\#} = \mathcal{O}_{-1}(k^n)$, $d(x_i) = \sum_j m_{ij} x_j^2$, $M \in M_n(k)$. Isomorphism classes correspond to orbits under suitable group actions on $M_n(k)$ (change of basis by quasi-permutation matrices), and homological properties (Koszulity, Calabi–Yau) are established via computations on the chain-level resolutions and associated Ext-algebras (Mao et al., 2018, Mao et al., 2020).

6. Topological Models and Cobar Constructions

Connected cochain DG algebras arise naturally from algebraic topology via chain and cochain models:

• The cobar construction on the normalized chains of a space yields a DG algebra model for based loop spaces; e.g., for a path-connected $(X, b)$:

$$\Omega Q_\Delta(\operatorname{Sing}(X, b)) \simeq S_*(\Omega X; k)$$

• Such constructions admit quasi-isomorphisms of DG bialgebras, preserving both product and coproduct structures (e.g., Szczarba's comultiplicative twisting cochain) (Rivera, 2019, Franz, 2020).
• Derived functors (e.g., rigidification, cubical chains) underly categorical equivalences and allow algebraic computations in homotopy theory (Rivera et al., 2016).

7. Local Cohomology, Duality, and Automorphism Invariants

An advanced theory of local cohomology and local duality extends to Noetherian connected cochain DG algebras. The local cohomology functor $R\Im(-)$ detects Gorensteinness: $A$ is Gorenstein if and only if $R\Im(A)$ is isomorphic (up to shift and twist by an invertible DG bimodule) to the Matlis dual $A'$. The homological determinant $$H\operatorname{det}: \operatorname{Aut}_{dg}(A) \to k^\times$$ quantifies the effect of DG automorphisms on local cohomology; when finite group actions have trivial homological determinant, invariant subalgebras inherit Gorenstein properties (Mao et al., 2021).

8. Reflexivity and Koszul Duality

Reflexive DG categories abstract the duality between bounded and perfect derived categories, ensuring Morita equivalence between $A$ and its category of perfectly valued modules. For (co)chain DG algebras from topology, reflexivity is equivalent to derived completeness at the canonical augmentation; Koszul duality underpins this equivalence, directly relating topological and algebraic invariants such as Hochschild cohomology and derived autoequivalence groups (Booth et al., 12 Jun 2025).

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Connected cochain DG algebras thus form a highly structured framework wherein algebraic, categorical, and topological dualities conjoin, with matrix-encoded differentials enabling explicit classification, and derived invariants superseding mere cohomological data. Their paper yields robust generalizations of classical regularity, duality, and compactness, uniting commutative and noncommutative perspectives and opening pathways for deep applications in geometry, topology, and representation theory.