# Vanishing of Hochschild homology of a category

Let $$A$$ be a dg- or $$A_{\infty}$$-category (with $$\mathbb{Z}$$-graded Hom sets, over a field of characteristic $$0$$). Let $$HH_*(A)$$ be the Hochschild homology of $$A$$.

Suppose that $$HH_n(A)=0$$ for all $$n \in \mathbb{Z}$$. Does this imply that $$A$$ is the zero category?

If not, then what assumptions can I add to $$A$$ to make this true (e.g. smoothness, properness, ect)?

Remark: I have heard the heuristic that Hochschild homology can be viewed as "differential forms" on the "spectrum" of the non-commutative category $$A$$ (this heuristic is presumably motivated by the Hochschild-Kostant-Rosenberg theorem). Hence it's natural to expect that $$A$$ vanishes if it admits no non-zero differential forms.

Note also that if $$A$$ is a (possibly non-commutative) $$k$$-algebra, then one can check that $$HH_0(A) =0$$ iff $$A=0$$.

This precise question was phrased as the vanishing conjecture in Hochschild homology and semiorthogonal decompositions. But we now know that there exist so called (quasi)phantom categories, which give counterexamples. These are categories with vanishing Hochschild homology, and vanishing (resp. torsion) Grothendieck group. As they are admissible subcategories of derived categories of smooth projective varieties, they have all nice properties you could want for their dg enhancements. An overview of some constructions:

Gorchinskiy, Sergey; Orlov, Dmitri, Geometric phantom categories, Publ. Math., Inst. Hautes Étud. Sci. 117, 329-349 (2013). ZBL1285.14018.

Böhning, Christian; Graf von Bothmer, Hans-Christian; Katzarkov, Ludmil; Sosna, Pawel, Determinantal Barlow surfaces and phantom categories, J. Eur. Math. Soc. (JEMS) 17, No. 7, 1569-1592 (2015). ZBL1323.14014.

Galkin, Sergey; Katzarkov, Ludmil; Mellit, Anton; Shinder, Evgeny, Derived categories of Keum’s fake projective planes, Adv. Math. 278, 238-253 (2015). ZBL1327.14081.

Galkin, Sergey; Shinder, Evgeny, Exceptional collections of line bundles on the Beauville surface, Adv. Math. 244, 1033-1050 (2013). ZBL1408.14068.

Böhning, Christian; Graf von Bothmer, Hans-Christian; Sosna, Pawel, On the derived category of the classical Godeaux surface, Adv. Math. 243, 203-231 (2013). ZBL1299.14015.

Alexeev, Valery; Orlov, Dmitri, Derived categories of Burniat surfaces and exceptional collections, Math. Ann. 357, No. 2, 743-759 (2013). ZBL1282.14030.

What is interesting is that Hochschild cohomology can detect their non-vanishing, and all kinds of interesting behavior regarding deformation theory arises, see

Kuznetsov, Alexander, Height of exceptional collections and Hochschild cohomology of quasiphantom categories, J. Reine Angew. Math. 708, 213-243 (2015). ZBL1331.14024.

• Based on your comment regarding Hochschild COhomology detecting nonvanishing in the above examples, am I correct to infer that there are no known examples of non-zero categories with HH_*(A)=HH^*(A)=0? Are there any conjectures regarding whether such examples should exist? – user142700 Jan 31 at 18:40
• If a category is non-zero then the zeroth Hochschild cohomology will be nonzero, as it contains the identity morphism between the diagonals. – pbelmans Jan 31 at 20:07