Hochschild Cohomology of Triangular Algebras

• Hochschild cohomology for triangular algebras is a framework combining homological algebra and representation theory to study deformation theory and algebraic invariants.
• Minimal projective resolutions, such as Bardzell’s construction in monomial and string cases, provide explicit bases for computing cochain complexes and reveal cup product vanishing phenomena.
• Non-monomial examples, including incidence algebras and geometric models, exhibit nontrivial cup products linking algebraic structures with topological and intersection properties.

Hochschild cohomology for triangular algebras is a central topic in the interface of homological algebra, representation theory, and noncommutative geometry. Triangular algebras encompass a broad class—including both path algebras of acyclic quivers with relations (triangular monomial, string, or special biserial) and upper triangular matrix algebras—and the structure of their Hochschild cohomology ring reveals combinatorial and homological features fundamental in deformation theory, derived equivalence, and the paper of algebraic invariants.

1. Triangular Algebras: Definitions and Basic Properties

A finite-dimensional algebra $A=\mathbb{k} Q/I$ is called triangular if the underlying quiver $Q$ is acyclic, i.e., $Q$ has no oriented cycles. For $\mathbb{k}$ a field and $I\subseteq \mathbb{k} Q$ a two-sided ideal generated by paths, $A$ is monomial if $I$ is generated by paths themselves, and string if $I$ satisfies special biserial conditions—each vertex is the source/target of at most two arrows, and each arrow extends in at most one way to a nonzero path on either side.

The class of upper triangular matrix algebras $$T=\begin{pmatrix}A & M\ 0 & B\end{pmatrix}$$, where $A$, $B$ are algebras and $M$ is a bimodule, is a prototypical example. Triangular algebras also include more general 2×2 or block matrix algebras and their quiver generalizations with “zero-composition” condition (all paths of length $\geq 2$ vanish) (Claude et al., 2017).

2. Minimal Projective Resolutions for Monomial and String Cases

For monomial algebras, the Bardzell resolution provides a minimal free $A$-$A$-bimodule resolution:

$$P_{n+1} = A \otimes_E \mathbb{k} \Gamma_n \otimes_E A$$

where $E = \mathbb{k} Q_0$, and $\Gamma_n$ is the set of $n$-ambiguities (combinatorial analogues of relations) (Artenstein et al., 2023). Differentials are defined via explicit combinatorics of sub-ambiguities, preserving minimality and acyclicity.

For string (e.g., special biserial) algebras, the resolution also uses path combinatorics—$A\otimes_E \mathbb{k} \mathrm{AP}_n \otimes_E A$, where $\mathrm{AP}_n$ indexes supports of $n$-concatenations of relations (Redondo et al., 2013). Differentials split into cases by parity of $n$, governed by the combinatorics of supports and immediate sub-supports, ensuring resolution exactness and minimality.

In both cases, applying $\mathrm{Hom}_{A\text{--}A}(-,A)$ replaces the resolution with a concrete cochain complex, often with a basis given by parallel path pairs (“parallel generators” or “supports”).

3. Hochschild Cohomology of Triangular Matrix and Quiver Algebras

For an upper triangular matrix algebra $$T=\begin{pmatrix}A & M\ 0 & B\end{pmatrix}$$, Hochschild cohomology fits into a long exact (Happel) sequence:

$$\cdots \to HH^n(B) \to HH^n(T) \to HH^n(A, M) \xrightarrow{\delta_n} HH^{n+1}(B) \to \cdots$$

Here, $HH^n(A, M) = \mathrm{Ext}^n_{A^e}(A, M)$ is Hochschild cohomology with bimodule coefficients. The connecting map $\delta_n$ is expressed in terms of the cup-product as a graded commutator with the identity cocycle of $M$:

$$\delta_n(f_A,f_B) = f_A \smile 1_M - (-1)^n 1_M \smile f_B$$

(Claude et al., 2017, Santiago-Vargas et al., 2023). Under suitable projectivity hypotheses (on $M$ as $A$-$B$-bimodule), the sequence collapses to short exact sequences, and the Hochschild cohomology in degree $n\geq 2$ becomes $HH^n(A)\oplus HH^n(B)$ (i.e., the direct sum of diagonal cohomologies) (Claude et al., 2017).

4. Structure of the Hochschild Cohomology Ring

A distinctive phenomenon for triangular monomial algebras is the vanishing of all cup-products in positive degree. Precisely, if $A=\mathbb{k} Q/I$ is triangular and monomial, then for any $x\in HH^n(A)$, $y\in HH^m(A)$ with $n,m>0$:

$x\smile y = 0 \in HH^{n+m}(A)$

(Artenstein et al., 2023). The proof relies on:

• Selection of cocycle representatives with a shared central subpath.
• Analysis that the nonvanishing of both $x\smile y$ and $y\smile x$ would force an oriented cycle in $Q$, contradicting acyclicity.
• Graded commutativity of the Hochschild cup product.

For triangular string algebras, complete computations of the Hochschild cohomology groups and explicit bases appear in (Redondo et al., 2013). There, the cup product is checked directly to vanish in positive degree:

$$HH^m(A) \smile HH^n(A) = 0 \quad\text{for all}\ m,n>0$$

The center is one-dimensional, $HH^0(A)\cong\mathbb{k}$, and explicit dimension formulas for $HH^1(A)$ and $HH^n(A)$ ($n\geq2$) are provided.

However, this vanishing phenomenon is not universal. In derived/categorical analogues, or for quadratic but non-monomial relations, nontrivial cup products arise (see below).

5. Cohomology and Cup Product in Non-Monomial and Geometric Examples

In the context of non-monomial triangular algebras, particularly those modeling algebraic geometry/topology (e.g., incidence algebras of cell decompositions of $T^2$ or $P^1\times P^1$), the cup product structure can be highly nontrivial (Dotsenko et al., 22 Nov 2025):

• For incidence algebras $\Lambda = I(P)$ corresponding to minimal cell decompositions of the torus, $HH^*\left(\Lambda\right)$ is isomorphic to the singular cohomology $H^*(T^2, k)$, and the Hochschild cup product agrees with the topological cup product.
• In these examples, $$HH^1(\Lambda)\otimes HH^1(\Lambda)\to HH^2(\Lambda)$$ may be nonzero, in direct contradiction with the “monomial cup-vanishing” intuition.
• Analogous behavior appears for certain “quadric surface” models, where the Hochschild cup product mirrors the geometric intersection products in $H^*(P^1\times P^1)$.

Beyond specific models, any extension of the vanishing theorem for cup products to non-monomial, acyclic algebras requires explicit construction of small projective resolutions and suitable diagonal maps—a task known only in select quadratic cases or for pointed Koszul algebras (Artenstein et al., 2023). Thus, the combinatorial vanishing property does not generalize beyond the monomial setting.

6. Influence of Resolutions, Diagonals, and Deformation Theory

The existence of a minimal resolution with an explicit diagonal is central to calculations of Hochschild cohomology ring structure. In the monomial case, Bardzell’s resolution and a well-structured diagonal allow for precise formulas for the cup product, and underpin the vanishing phenomenon (Artenstein et al., 2023).

For more general triangular algebras:

• The Hochschild-Mitchell framework manages the passage to categories and matrix algebras, providing long exact sequences linking cohomology of the building blocks and their interactions via bimodule extensions (Santiago-Vargas et al., 2023, Claude et al., 2017).
• Deformation theory: In geometric/quadratic models, the unobstructedness of certain $HH^2$ classes yields explicit deformations (quantum deformations of torus or quadric surfaces) whose Hochschild cohomology algebras interpolate between classical and degenerate (collapsed) cases (Dotsenko et al., 22 Nov 2025).

Table: Summary of Cup-Product Behavior

Algebra Class	Cup Product in $HH^{>0}$	Explicit Source
Triangular monomial	Vanishes ($x\smile y=0$)	(Artenstein et al., 2023)
Triangular string (special biserial)	Vanishes in $HH^{>0}$	(Redondo et al., 2013)
Non-monomial (incidence, quadratic)	May be nontrivial, topological	(Dotsenko et al., 22 Nov 2025)
Upper triangular matrix	Given by long exact sequence, may mix in low degrees	(Claude et al., 2017, Santiago-Vargas et al., 2023)

7. Open Problems and Outlook

The combinatorial vanishing of the cup product is provably robust in triangular monomial and string algebras, where minimal resolutions and diagonals are well-understood. For arbitrary acyclic algebras with non-monomial (e.g., binomial or quadratic) relations, construction of explicit diagonals, and thus precise computation of $HH^*(A)$ ring structure, remains open except in derived or geometric cases with additional structure (Artenstein et al., 2023, Dotsenko et al., 22 Nov 2025).

Extension of these vanishing results likely requires either combinatorial breakthroughs in small resolutions or use of abstract homotopy/homological methods not yet fully developed for arbitrary acyclic settings. The relationship between topological (e.g., simplicial) cohomology and Hochschild cohomology in the incidence/quadratic case exemplifies the limits of “monomial intuition.”

A plausible implication is that, for practitioners, explicit combinatorics of the quiver and relations are decisive for cohomological computations and understanding deformation theory, derived equivalence, and the cup product structure in triangular settings.