A counterexample to DG version of Han's conjecture

2025/12/13 by Yeqin Liu, Yu Shen, Liu, Yeqin +1
Computer Science · Mathematics · #14F08 #16E10 #16E40 #16E45 #Advanced Algebra and Logic #Advanced Operator Algebra Research #Algebraic Geometry (math.AG) #FOS: Mathematics #K-Theory and Homology (math.KT) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2512.12460

openalex publication_date 2025/12/13 · openalex created_date 2025/12/17 · openalex updated_date 2026/07/28

Abstract

In 2004, Han proposed the following conjecture: let B be a finite-dimensional k-algebra. If HH_n(B)≠ 0 for only finitely many n∈ ℤ, then B is smooth. This conjecture can be generalized to the DG setting: let B be a finite-dimensional DG k-algebra. If HH_n(B)≠ 0 for only finitely many n∈ ℤ, then B is smooth. In this note, we show that the DG generalization of Han's conjecture is false.

Citations

• A Survey on Han's Conjecture
• Perverse Bundles and Calogero-Moser Spaces
• On differential graded categories
• Hochschild (co)homology dimension
• Hochschild homology and cohomology of generalized Weyl algebras

Related

• A Survey on Han's Conjecture
• A recollement approach to Han's conjecture
• Protected corners and a trichotomy for Han's conjecture
• Formality of DG algebras (after Kaledin)
• Homotopical finiteness of smooth and proper dg-algebras