Cup Product on Hochschild Cohomology

• Cup product on Hochschild cohomology is a multiplicative structure defined via minimal resolutions and diagonal maps that encapsulate both algebraic relations and derived geometric aspects.
• Explicit formulas for quiver algebras, such as Λq, illustrate how the cup product unveils nontrivial commutative relations and provides evidence against finite generation conjectures.
• The study shows that variations in parameters like q yield either nilpotent behavior or infinite algebra generators, impacting the broader structure theory of Hochschild cohomology.

The cup product on Hochschild cohomology encodes the multiplicative structure on the cohomology ring, reflecting both the algebraic relations of the underlying algebra and its derived geometry. For finite-dimensional algebras, this product often interacts intricately with the choice of projective resolution, the comultiplicative structure at the chain level, and can have dramatic consequences for the structure theory of the cohomology ring. Recent research provides explicit formulas for the cup product in complex settings, notably for families of quiver algebras, and employs these tools to test and sometimes refute conjectures about finite generation and structure of Hochschild cohomology rings.

1. Definition and Construction of the Cup Product

Given a finite-dimensional associative algebra $A$ over a field $k$, the Hochschild cohomology groups $HH^n(A)$ are defined using a projective resolution $K_\bullet$ of $A$ as an $A$-bimodule: $HH^n(A) = H^n(\mathrm{Hom}_{A^e}(K_\bullet, A)).$ The cup product $\smile: HH^m(A) \times HH^n(A) \to HH^{m+n}(A)$ is induced by an explicit chain-level operation: $$\varphi \smile \psi := \mu \circ (\varphi \otimes \psi) \circ \Delta,$$ where $\mu: A \otimes A \to A$ is multiplication, $$\Delta: K_{m+n} \to \bigoplus_{i+j=m+n} K_i \otimes K_j$$ is a diagonal chain map lifting the bar-codiagonal, and $\varphi, \psi$ are cocycles in $\mathrm{Hom}_{A^e}(K_m, A)$, $\mathrm{Hom}_{A^e}(K_n, A)$, respectively (Oke, 2020).

In minimal resolutions $K_\bullet$ adapted to certain algebras (e.g., quiver algebras with relations), explicit formulas for $\Delta$ and thus for the cup product can be given. These are crucial for computations and for understanding the ring structure in concrete cases.

2. Explicit Formula: The Case of Quiver Algebras $\Lambda_q$

For the family of algebras $\Lambda_q = kQ / (a^2, b^2, ab - qba, ac)$, where $Q$ is a quiver with vertices $\{e_1, e_2\}$, loops $a,b$ at $e_1$, and an arrow $c: e_1 \to e_2$, the cup product on $HH^*(\Lambda_q)$ is given via a resolution with basis elements $f_i^n$ for the $K_n$.

The comultiplicative structure $\Delta$ and the cup product satisfy: $$(\phi\!\cup\!\psi)_k = \begin{cases} (-1)^{mn}\,\phi_0\,\psi_k, & k=0, \ (-1)^{mn}\sum_{j=\max(0,k-n)}^{\min(m,k)}(-q)^{j(n-k+j)}\phi_j\psi_{k-j}, & 0<k<m+n, \ (-1)^{mn}\,\phi_m\,\psi_n,&k=m+n, \ (-1)^{mn}\,\phi_m\,\psi_{n+1},&k=m+n+1. \end{cases}$$ for $\phi: K_m \to \Lambda_q$ and $\psi: K_n \to \Lambda_q$ [(Oke, 2020), Proposition 3.5]. The coefficients $(-q)^{j(n-k+j)}$ encode the nontrivial commutation relations among the paths in $Q$.

3. Structure of the Cohomology Ring and Counterexample to Finite Generation

The explicit cup product formula provides not only computational access but structural insight. For $q = \pm 1$, the algebra $HH^*(\Lambda_q)/N$, where $N$ is the ideal of all homogeneous nilpotent elements, can be identified with the subalgebra

$k[x^2, y^2] y^2 \subset k[x,y],$

with $y$ in degree 1, $xy$ in degree 2, and the isomorphism mapping $x^{2n-r}y^2$ to the unique cocycle $\phi_r^{2n} \in HH^{2n}(\Lambda_q)$.

Crucially, one constructs cocycles $\phi^{2n}_r$ in even degree which, for $q = \pm 1$, are nonnilpotent and not generated by elements in lower homological degree. This yields an infinite set of algebra generators in positive degree, so $HH^*(\Lambda_q)/N$ is not finitely generated as a $k$-algebra, explicitly disproving the Snashall–Solberg finite generation conjecture for this family [(Oke, 2020), Proposition 3.8, Theorem 3.10].

4. Nilpotence and Special Cases

The structure of the Hochschild cohomology ring via the cup product is sensitive to the parameter $q$:

• If $q \neq \pm 1$, every positive-degree cocycle is nilpotent, so $HH^*(\Lambda_q)/N \cong k$.
• For $q = 1$, the algebra $HH^*(\Lambda_q)/N$ admits a basis compatible with the identification above, and products like $\phi^2_2 \cup \phi^2_2 = \phi^4_4$ encode powers in the $y$ variable.

This dichotomy means the algebraic structure of the cup product is tightly linked to the underlying algebra parameters and relations. The nilpotent or non-nilpotent nature of the classes follows from explicit computations using the given formula.

5. Implications and Broader Significance

The explicit realization of the cup product on Hochschild cohomology in these quiver algebras serves as a model for analyzing the multiplicative structure in more general finite-dimensional algebras, especially those with nontrivial relations or symmetries. The identification of infinite algebra generators in positive degrees illustrates that the classical intuition regarding finite generation can fail, and such explicit counterexamples are crucial for sharpening conjectures and frameworks within homological algebra.

Moreover, the techniques—construction of explicit minimal resolutions, combinatorial analysis of cocycles, and careful tracking of the cup product—provide templates for computation in other settings, such as monomial algebras, incidence algebras, or graded algebras associated with geometric objects.

6. Comparative Framework and Extensions

The combinatorial cup product formula in (Oke, 2020) stands in contrast to the vanishing phenomena in triangular monomial algebras, where graded-commutativity forces all positive-degree products to vanish (Artenstein et al., 2023), and to the situation in certain geometric and deformation-theoretic settings where the cohomology ring can be described topologically or via deformation of incidence relations (Dotsenko et al., 22 Nov 2025). The employment of minimal projective resolutions equipped with a comultiplicative (diagonal) structure is a recurring tool in all these contexts, underpinning the calculation of the cup product and thus the algebra structure.

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References

• T. Oke, "Cup product on Hochschild cohomology of a family of quiver algebras" (Oke, 2020)
• M. Redondo et al., "The Hochschild cohomology ring of monomial algebras" (Artenstein et al., 2023)
• R. Rouquier and collaborators, "Algebraic versions of $\mathbb{T}^2$ and of $\mathbb{P}^1\times\mathbb{P}^1$ and Hochschild cohomology" (Dotsenko et al., 22 Nov 2025)