Singular Hochschild cohomology via the singularity category

Abstract

We show that for a noetherian algebra $A$ whose bounded dg derived category is smooth, the singular Hochschild cohomology (=Tate--Hochschild cohomology) is isomorphic, as a graded algebra, to the Hochschild cohomology of the dg singularity category of $A$. The existence of such an isomorphism is suggested by recent work of Zhengfang Wang.

v4: 7 pages, statement of Lemma 2.4 corrected, v5: error at the beginning of the proof of Theorem 1.1 corrected, v6: commutative ground ring re-established, v7: Proof of Theorem 1.3 expanded and corrected, v8: references corrected, v9: Smoothness hypothesis added following a counterexample by Xiao-Wu Chen, v10: Hyp. on finite gldim of ground ring added following a remark by Xiao-Wu Chen