Second Hochschild Cohomology Groups

• Second Hochschild Cohomology Groups are a central invariant that classify first-order (square-zero) deformations by encoding equivalence classes of extensions.
• They are computed via the Hochschild complex using projective bimodule resolutions and combinatorial methods, yielding explicit dimension formulas and invariance under derived equivalence.
• HH² bridges algebraic and geometric theories, as seen in its role in the Hochschild–Kostant–Rosenberg decomposition, which connects deformation theory to smooth projective varieties.

The second Hochschild cohomology group, denoted $HH^2$, is a central object in the deformation theory of associative and more general algebraic structures. It is defined for a wide range of settings: algebras, schemes, ring objects in monoidal categories, differential graded (dg) algebras, and certain structured categories. $HH^2$ governs first-order (infinitesimal) deformations and classifies equivalence classes of extensions by square-zero ideals or modules. Its detailed behavior varies widely depending on the ambient category and the specific object of study.

1. General Definition and Categorical Frameworks

In the classical case for an associative $k$-algebra $A$, $HH^2(A)$ is the degree-2 cohomology of the Hochschild complex: $$HH^2(A) = \operatorname{Ext}^2_{A \text{-bimod}}(A, A)$$ For a ring object $R$ in a monoidal $\mathbb{Ab}$-enriched category $\mathcal{C}$, the Hochschild cochain complex $C^\bullet(R)$ is defined by:

• $C^k(R) = 0$ for $k<0$
• $C^0(R) = \operatorname{Hom}_\mathcal{C}(I, R)$ (unit object $I$)
• $$C^k(R) = \operatorname{Hom}_\mathcal{C}(R^{\otimes k}, R)$$ for $k\ge 1$

The differential $d^k : C^k \to C^{k+1}$ is specified recursively (see Section 3 below). In all such contexts, the second cohomology group is: $HH^2(R) = \ker d^2 / \mathrm{im}\, d^1$ The elements of $HH^2$ are represented by equivalence classes of 2-cocycles modulo 2-coboundaries (Hellstrøm-Finnsen, 2016).

2. Cohomological Calculations in Algebraic Models

In explicit algebraic contexts, such as finite-dimensional algebras or path algebras with relations, $HH^2$ is realized through projective bimodule resolutions. For $A = KQ/I$, where $Q$ is a quiver and $I$ is admissible, one computes a minimal bimodule resolution

$$\cdots \to Q_3 \xrightarrow{A_3} Q_2 \xrightarrow{A_2} Q_1 \xrightarrow{A_1} Q_0 \to A \to 0$$

and then applies $\operatorname{Hom}_{A-A}(-, A)$ to obtain a cochain complex. Here, $HH^2(A)$ is the quotient of $\ker d_3$ by $\operatorname{im} d_2$ (Al-Kadi, 2011).

For gentle algebras, the dimension of $HH^2$ is given combinatorially in terms of the Avella–Alaminos–Geiss invariant $\varphi_A$ counting bands and permitted threads,

$$\dim_K HH^2(A) = \varphi_A(1,2) + \varphi_A(0,1) + \varphi_A(0,2)$$

(characteristic $\neq 2$) (Ladkani, 2012).

In monomial and radical-square-zero algebras, parallel-path or combinatorial complexes enable computations and dimension estimates, with modifications under algebraic operations such as "gluing arrows" always satisfying $\dim HH^2(B) \geq \dim HH^2(A)$ in such constructions (Liu et al., 2023).

3. Higher Categorical Settings and Derived Interpretations

The notion of $HH^2$ extends to ring objects in arbitrary $\mathbb{Ab}$-enriched monoidal categories. Here, the differential for $f \in C^2(R)$ takes the form

$$d^2(f) = \mu \circ (1 \otimes f) \circ \alpha^{0,1}_3 - f \circ (\alpha^{0,1}_2)^{-1} \circ \mu^0_3 \circ \alpha^{0,2}_3 + f \circ (\alpha^{1,1}_2)^{-1} \circ \mu^1_3 \circ \alpha^{1,2}_3 - \mu \circ (f \otimes 1)$$

and the 2-cocycle condition $d^2(f)=0$ encodes the commutativity of the associativity square involving $f$. The set of 2-cocycles modulo coboundaries gives $HH^2(R)$ (Hellstrøm-Finnsen, 2016).

For dg or curved algebras, the "Hochschild cohomology of the second kind" is defined as

$$HH^{II}(A) := \mathbb{R}\operatorname{Hom}_{D^{II}(A \otimes A^{\operatorname{op}})} (A, A)$$

using the compactly generated derived category of the second kind, $D^{II}$; this definition enjoys invariance under second-kind Morita equivalence and is compatible with Koszul duality (Guan et al., 2023).

Geometrically, $HH^2$ in many settings can be interpreted via the Hochschild–Kostant–Rosenberg (HKR) decomposition, decomposing $HH^2$ into parts parametrizing bivector fields (commutative deformations), ordinary scheme deformations, and gerbe classes (Liu et al., 2015). For smooth projective hypersurfaces $X$,

$$HH^2(X) \cong H^0(X, \wedge^2 T_X) \oplus H^1(X, T_X) \oplus H^2(X, \mathcal{O}_X)$$

Obstructions to this splitting precisely detect singularities.

4. Deformation Theory and Classification Role

$HH^2$ universally classifies first-order (square-zero) deformations:

• For monoidal category ring objects $R$, any $[f] \in HH^2(R)$ determines a unique (up to equivalence) square-zero extension $R \ltimes_f R$ with multiplication built from $\mu$ and $f$ (Hellstrøm-Finnsen, 2016).
• In associative algebras, $HH^2(A)$ classifies associative deformations of the multiplication. Given $\mu_0: A \otimes A \to A$, a first-order deformation has $$\mu(a,b) = \mu_0(a,b) + \epsilon \cdot \varphi(a,b)$$, $\varphi \in Z^2(A,A)$, $[\varphi] \in HH^2(A)$ (Al-Kadi, 2011).
• In geometric settings, the summands of $HH^2$ correspond to differentiated types of nontrivial formal or infinitesimal deformations.

This classification role holds in highly structured and derived settings, such as Brauer graph algebras, where cocycle types (semisimple, multiplicity, homology, bigon) correlate with explicit geometric operations on the associated surface models (Liu et al., 11 Jan 2026).

5. Explicit Calculations and Dimension Formulas

A range of explicit formulas for $HH^2$ in various contexts are summarized below:

Algebra/Structure	$\dim HH^2$	Reference
Standard one-parametric, not weakly symmetric, self-injective $A$	$=1$ or $=2$ depending on family	(Al-Kadi, 2011)
Gentle algebra $A$	$\varphi_A(1,2)+\varphi_A(0,1)+\varphi_A(0,2)$	(Ladkani, 2012)
Projective hypersurface $X$ (smooth)	$$\dim H^0(\wedge^2 T_X) + \dim H^1(T_X) + \dim H^2(\mathcal{O}_X)$$	(Liu et al., 2015)
Symmetric group algebra $kS_n$	$$\sum_{j=0}^{\lfloor (n-p)/p \rfloor} \phi_{2,j} p(n-p-pj)$$ ($p$ part.)	(Benson et al., 2023)
Reduced incidence algebra (formal/exponential/Eulerian series)	$0$	(Kanuni et al., 2016)
Formal Dirichlet series algebra	Infinite countable, $\Lambda^2(k^{\{\text{primes}\}})$	(Kanuni et al., 2016)
Brauer graph algebra $\Lambda$	$$1 + \sum_{v}(m(v)-1) + (|E|-|V|+1) + 2|{\rm bigons}|$$	(Liu et al., 11 Jan 2026)

The nature of $HH^2$—finite vs infinite, vanishing vs non-trivial—encodes important algebraic or geometric information. For example, vanishing $HH^2$ signals formal rigidity.

6. Functoriality, Invariance, and Derived Equivalence

Hochschild cohomology, including $HH^2$, is invariant under Morita equivalence (for algebras) and derived equivalence (for more general contexts), making it an essential derived invariant. In deeply structured settings, such as the "second kind" theory for dg or curved algebras, $HH^{II,2}$ is invariant under Morita equivalence in the category $D^{II}$ and compatible with Koszul duality (Guan et al., 2023).

The relationship between $HH^2$ and other algebraic invariants is also seen in exact sequences and injectivity of cohomology under algebraic operations (e.g., gluing arrows in quiver algebras strictly increases or preserves $HH^2$ dimension (Liu et al., 2023)).

7. Broader Contexts and Geometric Interpretation

In geometric representation theory and algebraic geometry, $HH^2$ governs the local structure of derived categories. Notably,

• For dg algebras modeling categories of $\infty$-local systems or matrix factorizations, the (second-kind) Hochschild cohomology computes the ordinary $HH^2$ of the relevant dg category (Guan et al., 2023).
• For Brauer graph algebras, each standard cocycle in $HH^2$ corresponds to specific types of geometric deformations of the surface model: orientability, local multiplicity, monodromy (via $H^1$), or smoothing/orbifold-izing boundaries (Liu et al., 11 Jan 2026).

The second Hochschild cohomology thus stands at the intersection of homological algebra, deformation theory, noncommutative geometry, and representation theory, encapsulating both algebraic and geometric deformation data in a unified cohomological invariant.