A counterexample to DG version of Han's conjecture

Abstract: In 2004, Han proposed the following conjecture: let $B$ be a finite-dimensional $k$-algebra. If $\mathrm{HH}{n}(B)\neq 0$ for only finitely many $n\in \mathbb{Z}$, then $B$ is smooth. This conjecture can be generalized to the DG setting: let $B$ be a finite-dimensional DG $k$-algebra. If $\mathrm{HH}{n}(B)\neq 0$ for only finitely many $n\in \mathbb{Z}$, then $B$ is smooth. In this note, we show that the DG generalization of Han's conjecture is false.

• The paper provides a counterexample showing that a finite-dimensional DG algebra with finite Hochschild homology does not guarantee smoothness.
• It employs a construction starting from a finite-cell DG algebra and uses acyclicity of DG ideals to derive the counterexample.
• The work highlights the divergence between properness and smoothness in the DG setting, challenging numeric invariants as sole indicators.

A Counterexample to the DG Version of Han's Conjecture

Introduction

The work "A counterexample to DG version of Han's conjecture" (2512.12460) addresses the validity of a conjecture posed by Han, which connects homological smoothness of finite-dimensional algebras to the vanishing pattern of their Hochschild homology (HH). Specifically, Han conjectured that for finite-dimensional algebras over an algebraically closed field, if their Hochschild homology is nonzero in only finitely many degrees, the algebra should be homologically smooth. This paper extends the scope of investigation to finite-dimensional differential graded (DG) algebras and establishes a counterexample, disproving the natural DG generalization of Han’s conjecture.

Formulation of the DG Han’s Conjecture

Formally, Han’s conjecture in the setting of DG algebras can be stated as: for a finite-dimensional DG $k$-algebra $B$ over an algebraically closed field $k$, if $\operatorname{HH}_n(B) \neq 0$ for only finitely many $n \in \mathbb{Z}$, then $B$ should be smooth over $k$. For DG algebras, smoothness is defined via the perfection of $B$ as a left DG module over its enveloping algebra $B^e = B \otimes_k B^{op}$. Unlike in the non-DG case, global dimension is not available, making smoothness the appropriate property of interest.

Main Result and Construction

The principal contribution of the paper is an explicit construction showing the failure of this conjecture in the DG setting. The authors provide a finite-dimensional DG $k$-algebra $B$ such that $\operatorname{HH}_{*}(B)$ is concentrated in finitely many degrees (in fact, in just two consecutive degrees), yet $B$ fails to be smooth.

The construction proceeds as follows:

1. Start with a suitable finite-cell DG algebra $A$, specifically, $A = k\langle x_1, x_2, x_3 \rangle$ with certain gradings ($|x_1| = |x_2| = 0$, $|x_3| = -1$) and differentials defined such that $d x_1 = d x_2 = 0$, and $d x_3 = x_1 x_2 - x_2 x_1 - 1$.
2. Study the two-sided DG ideal $I = (x_3, x_1 x_2 - x_2 x_1 - 1)$. Advanced orderings on monomials and acyclicity criteria are established to demonstrate that $I$ is an acyclic DG ideal via homological techniques.
3. Quotient by the acyclic ideal: $A / I$ is isomorphic, as a DG algebra, to the classical Weyl algebra $A_1$. The smoothness properties and Hochschild homology of $A_1$ play a central role in the argument.
4. Smooth categorical compactification: Leveraging results on noncommutative resolutions and categorical quotients, the existence of a finite-dimensional, proper, but non-smooth DG algebra $B$ is shown. The construction of $B$ arises from the categorical context as the kernel of a certain localization functor from a proper pretriangulated DG category with a full exceptional collection.

Numerical and Structural Results

The key numerical result is:

• $$HH_{i}(\mathscr{P}\mathrm{erf}(B)) \cong k^{\oplus m}$$ if $i=0$, $k$ if $i=1$, and $0$ otherwise, for some finite $m$.

Critically, $B$ is not smooth (by reduction to the infinitude of cohomology of $A_1$) even though its Hochschild homology is finite-dimensional and nonzero in only two neighboring degrees. This directly refutes the DG version of Han’s conjecture. The construction is robust for any field $k$ of characteristic zero.

Theoretical Implications

This counterexample demonstrates that the vanishing of Hochschild homology in almost all degrees is not sufficient to guarantee homological smoothness in the DG setting, even for finite-dimensional algebras. The method highlights the differences between properness and smoothness for DG algebras—two notions that are equivalent in the classical setting but diverge in the DG context.

The authors develop new tools for tracking acyclicity in DG ideals, utilizing orderings on monomials (degree order rather than right lex order) and a "unique order property" for pairs. This allows them to control differentials and decompose complex DG ideals to rigorously establish acyclicity.

Additionally, the theory is connected to noncommutative algebraic geometry via categorical compactifications, and the construction leverages smooth proper DG categories and categorical quotients, referencing deep results from Efimov and Orlov.

Practical and Further Implications

Practically, this result warns against using numeric or support-based invariants (such as the degrees of nonzero Hochschild homology) as sufficient proxies for smoothness in derived–or DG–contexts. Existing homological techniques that rely on such invariants require careful re-examination when extended to derived or noncommutative settings.

Theoretically, this opens paths for further investigation:

• The search for geometric (rather than purely algebraic) counterexamples remains open.
• Determining finer invariants or conditions that guarantee DG smoothness is an important open direction, perhaps involving deeper structural or categorical properties.
• The techniques may inspire analogous constructions in related settings (triangulated categories, noncommutative geometry, categorified settings), especially where compactness and smoothness diverge.
• The construction by Efimov via derived $\mathscr{D}$-module categories on projective spaces suggests new relationships between noncommutative resolutions and geometric representation theory.

Conclusion

The paper "A counterexample to DG version of Han's conjecture" (2512.12460) definitively refutes the direct DG generalization of Han’s conjecture by constructing explicit finite-dimensional DG algebras with finitely-supported Hochschild homology that are not smooth. The result rests on intricate homological algebra in the DG setting and utilizes modern categorical methods. It signals the necessity for more nuanced invariants for DG smoothness and has implications for derived algebraic geometry, representation theory, and homological algebra. Future research is likely to explore refined conditions for smoothness and to seek geometric analogues of the constructed counterexamples.