Hochschild Cohomology Group

• Hochschild Cohomology Group is defined via a cochain complex that captures deformation and symmetry aspects of associative algebras.
• It features a graded-commutative cup product and a Gerstenhaber bracket, which formalize the algebra's multiplicative and derivational properties.
• Applications extend to noncommutative geometry, representation theory, and topological field theory through explicit combinatorial and categorical frameworks.

The Hochschild cohomology group, denoted $HH^n(A, M)$ for an associative algebra $A$ and an $A$-bimodule $M$, is a central invariant in the cohomology theory of associative algebras and more generally in noncommutative geometry, representation theory, and homological algebra. It encodes deformation, structural, and symmetry data via its algebraic, functorial, and higher categorical constructions.

1. Definition, Construction, and First Properties

Let $A$ be an associative $K$-algebra, $M$ an $A$-bimodule, and $n \geq 0$. The $n$-th Hochschild cohomology group is defined via the cochain complex

$$C^n(A, M) := \operatorname{Hom}_K(A^{\otimes n}, M),$$

with $C^0(A, M) = M$ and $C^{-k}(A, M) = 0$ for $k>0$, equipped with the differentials

$$(d^n f)(a_1, ..., a_{n+1}) = a_1 \cdot f(a_2, ..., a_{n+1}) + \sum_{i=1}^n (-1)^i f(a_1, ..., a_i a_{i+1}, ..., a_{n+1}) + (-1)^{n+1} f(a_1, ..., a_n) \cdot a_{n+1}.$$

The cohomology groups are then

$HH^n(A, M) := H^n(C^\bullet(A, M), d).$

In particular, for $M = A$ (the regular bimodule), $HH^0(A) = Z(A)$ is the center of $A$, and $HH^1(A)$ is the space of outer derivations modulo inner derivations. This construction also generalizes to the setting of ring objects in monoidal categories, where the same model persists and the central interpretations in degree $0$ and derivations in degree $1$ remain valid (Hellstrøm-Finnsen, 2016).

2. Algebraic Structures: Cup Product, Gerstenhaber Bracket, and Graded-Commutativity

The Hochschild cohomology $HH^*(A, A) = \bigoplus_n HH^n(A, A)$ is naturally endowed with a graded-commutative algebra structure via the cup product

$$(f \smile g)(a_1, ..., a_{m+n}) = f(a_1, ..., a_m) \cdot g(a_{m+1}, ..., a_{m+n}).$$

This product is graded-commutative: $[f] \smile [g] = (-1)^{mn} [g] \smile [f],$ and $HH^*(A, A)$ is a Gerstenhaber algebra. Explicitly, the Gerstenhaber bracket is the degree $-1$ graded Lie bracket

$[f, g] = f \circ g - (-1)^{(m-1)(n-1)} g \circ f,$

where circle denotes insertion of one cochain into another. This bracket measures the failure of strict associativity in the cup product and governs the deformation theory of $A$ (Wang, 2015).

In symmetric algebras, $\Delta$-type operators further promote $HH^*(A,A)$ or singular $HH^*_{sg}(A,A)$ to a Batalin–Vilkovisky (BV) algebra, with the bracket generated by the BV operator reflecting string topology phenomena (Wang, 2015).

3. Hochschild Cohomology for Structured Algebras and Categories

3.1. Hopf Algebras and Modular Group Action

For finite-dimensional factorizable ribbon Hopf algebras $A$, the modular group $SL(2, \mathbb{Z})$ acts projectively on $Z(A) = HH^0(A)$ and, via explicit functorial lifts of Drinfel’d and Radford maps, this action extends up to homotopy to the entire Hochschild cochain complex and the cohomology groups $HH^n(A)$ (Lentner et al., 2017). Generators of $SL(2,\mathbb{Z})$ map to projective automorphisms $[G_n], [I_n] \in PGL(HH^n(A))$ satisfying the modular relations up to prescribed scalars.

3.2. Group Algebras and Topological Field Theory

For $A = k[G]$ a group algebra, $HH^*(A)$ can be interpreted via the classifying space of the adjoint groupoid of $G$, encoding geometric and representation theoretic information (Mishchenko, 2018). The Hochschild cochains admit a natural Frobenius algebra structure extending the $(1,2)$-TQFT on $HH^0$ and a homotopy-commutative Gerstenhaber product, with the group cohomology complex embedded canonically as a DG subalgebra (Lodder, 2014). This perspective underlies both the topological field theory quantization of finite group data and the structure of modular functors in quantum algebra.

3.3. Algebras with Finiteness and Representation-Theoretic Constraints

In gentle, special biserial, and string algebras, Hochschild cohomology displays strong combinatorial control, often admitting closed formulas for dimensions and explicit vanishing or degeneracy of the multiplicative structure (Ladkani, 2012, Furuya, 2014, Redondo et al., 2013). In many such classes (e.g., triangular string algebras), the cup product on positive-degree elements is trivial, rendering $HH^{>0}$ a square-zero ideal.

3.4. Gluing, Extensions, and Derived Categories

The behavior of $HH^n$ under gluing of idempotents in monomial algebras or extensions by bimodules has been described precisely: dimensions of $HH^1$ differ by explicit combinatorial invariants associated to special paths and pairs, and higher degrees are controlled by path-length arguments (Liu et al., 2022). In cluster-tilted and related algebras, the first Hochschild cohomology admits a direct sum decomposition reflecting the trivial extension structure, splitting derivations into those inherited from the base algebra and those detecting the extension data (Assem et al., 2012).

4. Geometric and Categorical Generalizations

Hochschild cohomology generalizes to schemes and (triangulated, DG, or stable $\infty$-) categories, with central roles in homological algebra, deformation theory, and noncommutative geometry.

• On smooth schemes $X$, Hochschild–Kostant–Rosenberg (HKR) theory gives

$$HH^n(X) \cong \bigoplus_{p+q = n} H^p(X, \wedge^q T_X),$$

and this decomposition characterizes smoothness in large classes (e.g., projective hypersurfaces) (Liu et al., 2015).

• For $k$-linear enhanced categories, the Hochschild cohomology is computed as $\operatorname{Ext}^*$ of the identity functor, with group actions decomposing $HH^*$ into invariant and twisted summands reflecting the category's autoequivalences (Perry, 2018).

In singular settings or for algebras of infinite global dimension, "singular Hochschild cohomology" replaces $\operatorname{Ext}$ in $D^b(A)$ by its singular quotient $D_{sg}(A^e)$, thus extending the Gerstenhaber and (for symmetric algebras) BV structures to the negative and Tate-theoretic region (Wang, 2015).

5. Applications in Deformation Theory and Algebraic Geometry

The space $HH^2(A)$ controls infinitesimal deformations of the algebra $A$ (Gerstenhaber), classifies deformations of presheaf/multiplication structures (in the geometric context), and detects obstructions. Explicit descriptions associate summands of $HH^2$ to "local multiplicative", "restriction map", or "twist" deformations for schemes, or to extension classes, as in the cluster-tilted context. In several important classes, dimension formulas for $HH^2$ are related to the existence or absence of nontrivial deformations, reflecting rigidity or moduli phenomena (e.g., standard one-parametric self-injective algebras have $HH^2$ of dimension one, yielding a unique nontrivial deformation direction) (Al-Kadi, 2011).

6. Further Structures and Invariants

Tables and structural theorems provide formulas for $HH^n$ in various algebraic classes:

Algebra/Class	$\dim HH^1$	$\dim HH^2$	Cup Product Structure
Triangular string	Combinatorial (Theorem 4.3)	Combinatorial	All cup products vanish for $n,m>0$
Gentle	Derived from $\phi_A$ invariant	Derived from $\phi_A$	All higher products trivial
Self-injective biserial	Closed form in $n\ (\bmod\ 4)$	Closed form, linear growth	Multiplicative structure governed by periodicity in Koszul case

Hochschild cohomology is thus a highly structured, functorial, and categorically robust invariant, reflecting deformation theory, algebraic and topological symmetries, representation-theoretic data, and the deeper homotopy-theoretic properties of associative and ring-like objects across classical and higher contexts.

References:

(Lentner et al., 2017, Lodder, 2014, Wang, 2015, Mishchenko, 2018, Hellstrøm-Finnsen, 2016, Assem et al., 2012, Redondo et al., 2013, Liu et al., 2022, Al-Kadi, 2011, Liu et al., 2015, Furuya, 2014, Ladkani, 2012, Perry, 2018, Itaba et al., 2019)