Hochschild cohomology of the second kind: Koszul duality and Morita invariance

1. Problem Statement and Motivation

The Core Problem:
Hochschild cohomology ($HH$) is a fundamental invariant for associative algebras and differential graded (dg) categories, intimately linked to deformation quantization and derived categories. However, standard Hochschild cohomology (of the "first kind") is defined via the totalization of the Hochschild complex using direct products. For dg algebras, this definition often fails to capture the full geometric information encoded in the algebra's derived category.

A "second kind" of Hochschild cohomology has been proposed (by Positselski and Polishchuk), utilizing direct sums for totalization. While this version is more sensitive to geometric structures (e.g., in matrix factorizations or local systems), the existing definitions suffer from two main issues:

1. Lack of Morita Invariance: Standard definitions are not always invariant under "Morita equivalences of the second kind" (equivalences of twisted derived categories).
2. Incompatibility with Koszul Duality: The relationship between the second-kind cohomology of an algebra and its Koszul dual (a coalgebra) was not fully established, particularly for non-conilpotent cases.

The Goal:
The authors aim to define a robust version of Hochschild cohomology of the second kind that:

• Is invariant under suitable Morita equivalences of the second kind.
• Is compatible with Koszul duality (relating algebras to their bar/cobar duals).
• Computes the ordinary Hochschild cohomology of geometrically significant dg categories (like derived categories of coherent sheaves or $\infty$-local systems).

2. Methodology and Framework

The paper operates within the framework of homological algebra of the second kind, where the notion of weak equivalence is refined beyond standard quasi-isomorphisms.

Key Definitions and Tools:

• Twisted Derived Category ($D^{\text{II}}_c(A)$): The authors work with the "twisted derived category" (formerly called the compactly generated derived category of the second kind). This category is generated by perfect modules of the second kind ($\text{Perf}^{\text{II}}(A)$), which are retracts of finitely generated twisted complexes.
• Model Structures: They utilize specific model structures on dg modules where weak equivalences are defined via coacyclic or contra-acyclic cones, rather than just quasi-isomorphisms.
• Koszul Duality: The paper extends Koszul duality to the bimodule setting. They establish Quillen equivalences between:
  • Bimodules over a dg algebra $A$ and bicomodules over its bar construction $BA$ (or extended bar construction $qBA$).
  • Modules over a pseudocompact algebra $C$ and modules over its cobar construction $\Omega C$.
• Curved Algebras: The theory is generalized to curved algebras (where $d^2 \neq 0$ but $d^2 = [h, -]$), which are essential for applications like matrix factorizations.

3. Key Contributions

A. Definition of Hochschild Cohomology of the Second Kind

The authors define the Hochschild cohomology complex of the second kind for a dg (or curved) algebra $A$ as:
$$HC^{\text{II}}_c(A) = \mathbb{R}\text{Hom}_{D^{\text{II}}_c(A \otimes A^{\text{op}})}(A, A)$$
Here, the derived hom is taken within the twisted derived category of $A$-bimodules. The cohomology is $HH^{\text{II}}_c(A) = H(HC^{\text{II}}_c(A))$.

B. Morita Invariance

The paper proves that this new invariant is robust under two types of equivalences:

1. MC-Equivalences: Maps inducing bijections on Maurer-Cartan moduli sets.
2. Bimodule II-Morita Equivalences: Equivalences induced by bimodules $P$ and $Q$ such that $P \otimes^L_B Q \simeq A$ and $Q \otimes^L_A P \simeq B$ in the twisted derived category.
   Result: If $A$ and $B$ are bimodule II-Morita equivalent, then $HC^{\text{II}}_c(A) \simeq HC^{\text{II}}_c(B)$.

C. Bimodule Koszul Duality

A major technical achievement is the construction of bimodule Koszul duality. The authors show that Koszul duality functors (Bar/Cobar) extend to equivalences of categories of bimodules.

• Theorem: There is a Quillen equivalence between the model category of $A$-bimodules (with the second-kind structure) and the model category of $qBA$-bicomodules (where $qBA$ is the extended bar construction).
• Consequence: This leads to a quasi-isomorphism of dg algebras:
  $$HC^{\text{II}}_c(A) \simeq HC(qBA)$$
  This connects the second-kind cohomology of an algebra directly to the ordinary Hochschild cohomology of its Koszul dual coalgebra.

D. Comparison with Existing Theories

The authors clarify the relationship between their definition and the "compactly supported" Hochschild cohomology ($HH^{\text{II}}_{PP}$) of Positselski and Polishchuk.

• They show that $HH^{\text{II}}_c(A)$ lies "between" the ordinary Hochschild cohomology ($HH(A)$) and $HH^{\text{II}}_{PP}(A)$.
• Specifically, $HH^{\text{II}}_c(A)$ is a pseudocompact completion of $HH^{\text{II}}_{PP}(A)$, while $HH(A)$ is a further completion. In favorable geometric cases, these definitions coincide.

4. Main Results and Applications

The paper demonstrates that for specific geometric algebras, $HH^{\text{II}}_c(A)$ recovers the ordinary Hochschild cohomology of the associated geometric dg category. This is achieved by proving that the natural functor $\text{Perf}^{\text{II}}(A) \otimes \text{Perf}^{\text{II}}(E) \to \text{Perf}^{\text{II}}(A \otimes E)$ is a Morita equivalence in these cases.

Specific Examples:

1. Complex Algebraic Manifolds:

   • Let $X$ be a smooth complex projective variety and $A(X)$ its Dolbeault algebra.
   • Result: $HH^{\text{II}}_c(A(X)) \cong HH(\text{Perf}(X))$, where $\text{Perf}(X)$ is the dg category of perfect complexes on $X$. This links the algebraic cohomology of the Dolbeault algebra to the geometry of coherent sheaves.

2. Topological Spaces:

   • Let $X$ be a topological space and $C^*(X)$ its singular cochain algebra (or $A^*(X)$ the de Rham algebra for manifolds).
   • Result: $HH^{\text{II}}_c(C^*(X)) \cong HH(\mathcal{L}C(X))$, where $\mathcal{L}C(X)$ is the dg category of $\infty$-local systems on $X$.

3. Matrix Factorizations:

   • Let $R$ be a regular commutative algebra and $w$ a potential, defining a curved algebra $R_w$.
   • Result: $HH^{\text{II}}_c(R_w) \cong HH(\text{MF}(R, w))$, where $\text{MF}(R, w)$ is the category of matrix factorizations. This confirms that the second-kind cohomology correctly captures the singularity structure of the hypersurface defined by $w$.

5. Significance and Impact

• Unification: The paper unifies disparate areas of homological algebra (twisted complexes, curved algebras, Koszul duality) to provide a coherent definition of second-kind cohomology.
• Geometric Insight: It resolves the mystery of why second-kind cohomology is "geometrically meaningful." It shows that for many algebras arising in geometry, the second-kind cohomology is not a new, obscure invariant, but rather the ordinary Hochschild cohomology of the underlying geometric dg category.
• Technical Robustness: By establishing Morita invariance and compatibility with Koszul duality, the authors provide a stable framework for computing these invariants in non-trivial settings (e.g., non-conilpotent cases and curved algebras) where previous methods failed.
• Curved Algebra Handling: The construction of an auxiliary dg algebra $A'$ for any curved algebra $A$ (such that their twisted derived categories are equivalent) is a novel technical tool that allows the transfer of results from the dg setting to the curved setting, bypassing the lack of a Yoneda embedding for curved algebras.

In summary, this work provides the definitive framework for Hochschild cohomology of the second kind, proving its invariance properties and demonstrating its power as a tool to compute the Hochschild cohomology of major geometric and topological categories.