Relations Between Derived Hochschild Functors via Twisting

Abstract

Let be a regular ring, and let A, B be essentially finite type -algebras. For any functor F: D(ModA) × ⋅ × D(ModA) → D(ModB) between their derived categories, we define its twist F^!: D(ModA) × ⋅ × D(ModA) → D(ModB) with respect to dualizing complexes, generalizing Grothendieck's construction of f^!. We show that relations between functors are preserved between their twists, and deduce that various relations hold between derived Hochschild (co)-homology and the f^! functor. We also deduce that the set of isomorphism classes of dualizing complexes over a ring (or a scheme) form a group with respect to derived Hochschild cohomology, and that the twisted inverse image functor is a group homomorphism.

Key Words:

• Dualizing complex
• Hochschild cohomology

2010 Mathematics Subject Classification:

• 13D03
• 18E30
• 18D10
• 18D15

ACKNOWLEDGMENTS

The author would like to thank Professor Joseph Lipman and Professor Amnon Yekutieli for some useful suggestions. The author also wish to thank the Department of Mathematics at the Weizmann Institute of Science, where this work was carried out.