DG Generalization of Han's Conjecture

• The paper establishes a counterexample showing that a finite-dimensional DG algebra with finitely many nonzero Hochschild homology groups can be non-smooth.
• It employs categorical compactification and localization techniques to translate classical smoothness notions into the differential graded context.
• The analysis distinguishes DG smoothness from properness, highlighting that finite homology does not ensure a finite bimodule resolution.

The DG generalization of Han's conjecture concerns the interplay between Hochschild homology and the notion of smoothness for differential graded (DG) algebras over an algebraically closed field. Han's conjecture in the classical setting posits that a finite-dimensional algebra with Hochschild homology nonzero in only finitely many degrees must be (homologically) smooth. This statement, when naively extended to the setting of DG algebras, claims that if a finite-dimensional DG algebra also has this property, then it too must be smooth. This generalization, however, is shown to be false through a construction that produces a finite-dimensional DG algebra with finitely many nonzero Hochschild homology groups but which fails to be smooth (Liu et al., 13 Dec 2025).

1. Classical Han's Conjecture and Smoothness

Han's conjecture, formulated for finite-dimensional $k$-algebras $B$, states that if

$\dim_k \left(\HH_n(B)\right) < \infty \;\; \text{and}\;\; \HH_n(B) \neq 0 \text{ only for finitely many } n \in \mathbb{Z},$

then $B$ is homologically smooth, that is, $B$ has finite global dimension over a perfect field. Here, Hochschild homology is defined strictly in the case of associative algebras (with no additional grading or differential).

A $k$-algebra $B$ is called smooth if, viewed as a bimodule over its enveloping algebra $B^e = B \otimes_k B^{\mathrm{op}}$, $B$ is perfect in the derived category $\Der(B^e)$—equivalently, the diagonal bimodule $B$ admits a finite resolution by finitely generated projective $B^e$-modules.

2. DG Generalization and Precise Formulation

The direct analogue for DG algebras (cohomologically $\mathbb{Z}$-graded $k$-algebras with differential) recasts Hochschild homology as

$\HH_n(B) = H^{-n}\left(B \otimes^{\mathbf{L}}_{B^e} B\right),$

where the derived tensor product is taken over the DG enveloping algebra $B^e$. The DG generalization of Han's conjecture posits: for a finite-dimensional DG $k$-algebra $B$ with $\HH_n(B) \neq 0$ for only finitely many $n$, is $B$ necessarily smooth?

Smoothness for DG algebras means that $B$ is perfect as a $B^e$-module in the derived category. This condition is substantially more flexible in the DG context and, as shown by Liu and Shen, can fail to coincide with finiteness properties of Hochschild homology (Liu et al., 13 Dec 2025).

3. Construction of the Counterexample

The construction of the counterexample relies on a sequence of categorical and homological arguments:

• Weyl Algebra $A_1$: Begin with the first Weyl algebra $$A_1 = k \langle x_1, x_2 \rangle / (x_1 x_2 - x_2 x_1 - 1)$$. It is a smooth but not proper ordinary algebra, with one-dimensional Hochschild homology concentrated in degree $2$.
• Finite-cell DG Algebra $A$: Construct a finite-cell DG algebra $A = \bigl(k\langle x_1, x_2, x_3 \rangle, d \bigr)$, with $|x_1| = |x_2| = 0$, $|x_3| = -1$, and differentials $d(x_1) = d(x_2) = 0$, $d(x_3) = x_1 x_2 - x_2 x_1 - 1$. The DG ideal $I = (x_3, x_1 x_2 - x_2 x_1 - 1)$ is acyclic, so the quotient $A/I$ is quasi-isomorphic (as DG algebras) to $A_1$.
• Categorical Compactification: Leveraging a theorem of Efimov (and Orlov–Efimov–Lunts), construct a proper pretriangulated DG category $C$ with a full exceptional collection, such that $\Perf(A) \simeq C / \langle S \rangle$, where $\langle S \rangle \simeq \Perf(B)$ for some finite-dimensional DG algebra $B$.
• Localization Sequence: Form a short exact localization sequence of DG categories: $\Perf(B) \rightarrow C \rightarrow \Perf(A_1).$

This methodology produces a finite-dimensional DG algebra $B$ with explicit categorical and homological properties.

4. Hochschild Homology and Smoothness Failure

Applying Keller’s invariance and localization theorem for Hochschild homology yields a long exact sequence: $\cdots \rightarrow \HH_i(\Perf(B)) \rightarrow \HH_i(C) \rightarrow \HH_i(\Perf(A_1)) \rightarrow \HH_{i-1}(\Perf(B)) \rightarrow \cdots$

Given that $C$ has a length-$m$ exceptional collection ($\HH_i(C) \cong k^{\oplus m}$ only for $i=0$; zero otherwise), and the Weyl algebra calculation ($\HH_i(A_1) \cong k$ for $i=2$, zero otherwise), the exact sequence forces

$\HH_i(B) = \begin{cases} k^{\oplus m}, & i=0, \ k, & i=1, \ 0, & \text{otherwise}. \end{cases}$

Thus, $B$ has finite-dimensional Hochschild homology, nonzero only for $i=0,1$.

However, smoothness of $B$ would render $\Perf(B)$ admissible in the proper $C$, implying $\Perf(A)$ (and so $\Perf(A_1)$) is proper, which is impossible because $H^0(A_1) = A_1$ is infinite-dimensional. Thus, $B$ is not smooth, disproving the DG generalization.

5. Structural Lemmas and Propositions

The argument utilizes several categorical and homological facts:

Lemma/Proposition	Statement (abridged)	Reference Usage
Finite-cell DG algebra smoothness	Any finite-cell DG algebra $A$ is smooth over $k$	Ensures $A$ is smooth
Degree-order acyclicity	In $k\langle x_1,x_2,x_3\rangle$ with $d(x_3)=x_1x_2-x_2x_1-1$, the ideal $(x_3, x_1x_2-x_2x_1-1)$ is acyclic	Shows $A/I \simeq_{\textrm{DG}} A_1$
Keller’s Localization Theorem	For $\mathcal{A} \to \mathcal{B} \to \mathcal{C}$ exact, long exact sequence exists for Hochschild homology	Governs calculation of $\HH_*(B)$

These results enable the explicit construction of $B$ with the desired finiteness and smoothness-failure properties.

6. Distinction Between Smoothness and Properness in DG Context

In classical algebra, finiteness of Hochschild homology is typically coupled with finite global dimension. In contrast, in the DG context, smoothness—equating to the perfectness of the diagonal bimodule—is distinct from properness (having finite total cohomology). The counterexample leverages this distinction: the Weyl algebra is smooth but not proper, and its embedding via a finite-cell DG algebra and subsequent compactification allows the isolation of a DG algebra kernel $B$ retaining finite Hochschild homology but not smoothness. No analogue of proofs in the ordinary algebra case persists, revealing that “small” Hochschild homology does not ensure the existence of a finite bimodule resolution in the DG case (Liu et al., 13 Dec 2025).

7. Implications and Consequences

The explicit counterexample illustrates that in the DG setting, the relationship between Hochschild homological finiteness and smoothness is significantly more subtle and flexible compared to the classical case. This suggests that proving smoothness in the DG world requires criteria beyond the finiteness of Hochschild homology groups. A plausible implication is that future research on DG algebra smoothness must carefully account for the distinction between properness and perfectness in the context of derived categories and cannot rely on classical homological dimension arguments. The counterexample also demonstrates the utility of categorical compactification and localization techniques for constructing and analyzing pathologies in DG homological algebra (Liu et al., 13 Dec 2025).