Han's Conjecture on Algebraic Smoothness

• Han's Conjecture is a homological characterization asserting that a finite-dimensional algebra has finite global dimension if its Hochschild homology vanishes beyond a certain degree.
• The conjecture has driven research through explicit computations, categorical reductions, and extension techniques to verify algebraic smoothness in various classes of algebras.
• Recent advances have leveraged extension principles and recollement frameworks, while counterexamples in DG settings highlight the conjecture's limitations in non-classical contexts.

Han's Conjecture formulates a homological characterization of smoothness for finite-dimensional associative algebras over a field. It asserts that a finite-dimensional algebra has finite global dimension if and only if its Hochschild homology vanishes in all but finitely many degrees. The conjecture provides a bridge between homological algebra and structural properties, motivating substantial analysis of Hochschild homology, extension techniques, and categorical reductions. This summary details the conjecture’s development, central results, extension phenomena, technical tools for proofs, categorical perspectives, and recent advances and limitations.

1. Formulation and Background

Let $k$ be a field and $A$ a finite-dimensional $k$-algebra. The Hochschild homology groups are defined by

$$HH_n(A) = \mathrm{Tor}_n^{A^e}(A, A),\quad A^e = A \otimes_k A^{\mathrm{op}}$$

and the global dimension is

$$\mathrm{gldim}~A = \sup \{\mathrm{pd}_A M\mid M~\text{finitely generated}~A\text{-module}\}$$

Han’s Conjecture (2006) posits: $$HH_n(A) = 0~\text{for all}~n \gg 0\;\; \Longrightarrow\;\; \mathrm{gldim}\,A < \infty$$ Under mild hypotheses (e.g. $k$ perfect, or $A/J(A)$ separable), the converse—finite global dimension implies vanishing of Hochschild homology in all positive degrees—holds via Keller’s theorem (Cruz, 2023). Han’s conjecture extends the vanishing principle to imply smoothness.

Originally motivated by Happel’s question regarding Hochschild cohomology, Han’s homological reformulation was necessitated by explicit counterexamples in cohomology (Cruz, 2023), but positive results and computational evidence for homology led to the conjecture’s focus on $HH_*$.

2. Classes of Algebras Satisfying the Conjecture

Numerous algebraic classes have been shown to verify Han’s property, typically by explicit computation of $HH_*$, projective resolutions, or reduction to known smooth base cases:

• Group Algebras: $kG$ with $G$ finite, Han’s property holds by Maschke’s theorem and Burghelea–Swan (Cruz, 2023).
• Commutative Noetherian Algebras: Han’s property is equivalent to regularity/smoothness (Vigué-Poirrier, Loday–Quillen).
• Monomial, Exterior, Quantum Complete Intersection Algebras: Direct calculation and Cartan determinant techniques (Bergh–Erdmann, Bergh–Madsen, Solotar–Vivas, Solotar–Westreich) show infinite global dimension forces $HH_n$ nonzero for infinitely many $n$ (Cruz, 2023).
• Trivial Extensions and Local/Graded/Self-Injective Algebras: Hochschild homology computations confirm infinite support implies infinite global dimension.

Corollary results show the conjecture is closed under standard extension constructs, such as triangular matrix algebras, split bounded extensions, strongly stratifying chains, and recollements (Cibils et al., 2023, Cibils et al., 2017, Wang et al., 2024, Cibils et al., 2019).

3. Extension, Reduction, and Inductive Principles

Central advances have established that Han’s property is preserved under various extension paradigms, allowing reduction from complicated algebras to their subcomponents:

• Bounded and Split Bounded Extensions: If $B\subset A$ is a bounded extension (where $A/B$ is tensor nilpotent, has finite $B^e$-projective dimension, and vanishing higher $\mathrm{Tor}$) then $A$ and $B$ are singularly equivalent of Morita type with level, preserving $HH_n$ in large degrees (Qin et al., 2024, Cibils et al., 2019, Cibils et al., 2021).
• Strongly Stratifying Ideals and Morita Contexts: Han’s conjecture holds for $A$ if and only if it holds for each block in a strongly stratifying or co-stratifying chain, reducing verification to diagonal/local components (Cibils et al., 2023).
• Null-square Projective and Triangular Algebras: If blocks on the diagonal satisfy Han’s property, so does the full algebra even when off-diagonal products are nilpotent and bimodules are projective (Cibils et al., 2017).
• Recollement of Derived Categories: Han’s conjecture for a ring in the middle of a recollement holds if and only if it holds for rings on the sides, enabling reduction to derived 2-simple constituents (Wang et al., 2024).

Table: Extension Types and Han's Conjecture Preservation

Extension Type	Conditions for Preservation	Reference
Bounded / Split bounded	Tensor nilpotent, finite proj. dim	(Qin et al., 2024, Cibils et al., 2019, Cibils et al., 2021)
Strongly stratifying chains	Blocks (diagonals) verify Han	(Cibils et al., 2023)
Null-square projective	Projective bimodules, zero corner	(Cibils et al., 2017)
Derived recollement	Extension ladder exists	(Wang et al., 2024)

4. Jacobi–Zariski Sequence and Technical Tools

The proof strategies uniformly rely on the Jacobi–Zariski long exact sequence and related homological machinery (Cibils et al., 2019, Cibils et al., 2021, Cibils et al., 2023):

• Relative Hochschild Homology: Fits into a long exact (or nearly exact) sequence relating the homology of an extension and its base (or side) algebra.

$$\cdots \to HH_m(B, X) \to HH_m(A, X) \to HH_m(A|B, X) \to HH_{m-1}(B, X) \to \cdots$$

• Spectral Sequence Degeneration: Under boundedness, spectral sequences collapse in high degrees, forcing isomorphisms between $HH_n$ of $A$ and $B$ for $n\gg0$.
• Tensor Nilpotence and Projectivity Criteria: Explicit combinatorial criteria for quiver extensions ensure boundedness and thus equivalence of the conjecture between algebras (Cibils et al., 2021).
• Mapping Cone and Resolutions: In null-square or triangular extensions, mapping cone construction yields finite projective resolutions implying finite global dimension when vanishing holds.

5. Categorical, Derived, and DG Perspectives

Recent research has explicated the conjecture’s categorical and derived nuances:

• Singular Equivalences of Morita Type: Bounded extensions induce singularity equivalences, transferring all major homological conjectures (Han, Auslander–Reiten, Keller, finiteness) (Qin et al., 2024).
• Stable and Defect Category Equivalences: Under mild hypotheses, bounded extensions yield equivalence in Gorenstein projective stable and defect categories, linking homological and categorical properties (Qin et al., 2024).
• DG and Nonclassical Counterexamples: The generic DG generalization is shown to be false—there exist finite-dimensional DG algebras $B$ with Hochschild homology supported only in finitely many degrees, yet $B$ is not DG-smooth (Liu et al., 13 Dec 2025).

6. Recent Advances, Counterexamples, and Open Questions

Progress includes both the positive extension of Han’s property to intricate classes via recollement, stratification, and boundedness, and the demarcation of limits:

• Recollement Reduction Results: Han’s property is provably inherited by skew-gentle algebras, EGL category algebras, GLS Cartan triple algebras, via recollement and matrix reduction (Wang et al., 2024).
• DG Counterexamples: The DG analogue fails—non-smooth DG algebras with finitely supported Hochschild homology exist (Liu et al., 13 Dec 2025), suggesting stricter conditions are needed in DG settings.
• $\tau$-Hochschild Homology: Higher Auslander–Reiten translates, introduced as $\tau$-Hochschild (co)homology, cleanly separate the classical conjecture from deeper Tor patterns between simples, reducing Han’s conjecture to a statement about the persistence of Tor between certain pairs of vertices in bound quiver algebras (Cibils et al., 5 Sep 2025).

7. Perspectives and Directions

• Classification Problems: Identification and classification of derived 2-simple and non-smooth DG algebras remain central (Wang et al., 2024, Liu et al., 13 Dec 2025).
• Refined Invariants: Potential extension to cyclic homology, $K$-theory, and higher categorical invariants.
• Extension to Broader Classes: Extending closure properties to non-split, infinite-dimensional, or more general types of extensions.
• Geometric Analogues: Investigation of geometric varieties exhibiting similar vanishing phenomena in their categorical and Hochschild invariants.

The ongoing research clarifies that Han’s Conjecture serves as a touchstone for understanding the interplay between homological invariants, categorical structures, and algebraic smoothness. Its proven extensions, technical reductions, and pinpointed restrictions guide both theory and computational practice (Cruz, 2023, Cibils et al., 2019, Qin et al., 2024, Cibils et al., 2023, Wang et al., 2024, Cibils et al., 2017, Liu et al., 13 Dec 2025, Cibils et al., 2021, Cibils et al., 5 Sep 2025).